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-Noname manuscript No.
-(will be inserted by the editor)
-The advent and fall of a vocabulary learning bias from
-communicative eciency
-David Carrera-Casado 
Ramon
-Ferrer-i-Cancho
-Received: date / Accepted: date
-Abstract Biosemiosis is a process of choice-making between simultaneously alter-
-native options. It is well-known that, when suciently young children encounter
-a new word, they tend to interpret it as pointing to a meaning that does not
-have a word yet in their lexicon rather than to a meaning that already has a word
-attached. In previous research, the strategy was shown to be optimal from an infor-
-mation theoretic standpoint. In that framework, interpretation is hypothesized to
-be driven by the minimization of a cost function: the option of least communication
-cost is chosen. However, the information theoretic model employed in that research
-neither explains the weakening of that vocabulary learning bias in older children or
-polylinguals nor reproduces Zipf's meaning-frequency law, namely the non-linear
-relationship between the number of meanings of a word and its frequency. Here
-we consider a generalization of the model that is channeled to reproduce that law.
-The analysis of the new model reveals regions of the phase space where the bias
-disappears consistently with the weakening or loss of the bias in older children or
-polylinguals. The model is abstract enough to support future research on other
-levels of life that are relevant to biosemiotics. In the deep learning era, the model is
-a transparent low-dimensional tool for future experimental research and illustrates
-the predictive power of a theoretical framework originally designed to shed light
-on the origins of Zipf's rank-frequency law.
-Keywords biosemiosis
vocabulary learning 
mutual exclusivity 
Zipan laws

-information theory 
quantitative linguistics
-David Carrera-Casado & Ramon Ferrer-i-Cancho
-Complexity and Quantitative Linguistics Lab
-LARCA Research Group
-Departament de Ci encies de la Computaci o
-Universitat Polit ecnica de Catalunya
-Campus Nord, Edici Omega
-Jordi Girona Salgado 1-3
-08034 Barcelona, Catalonia, Spain
-E-mail: david.carrera@estudiantat.upc.edu,rferrericancho@cs.upc.eduarXiv:2105.11519v3  [cs.CL]  20 Jul 20212 David Carrera-Casado, Ramon Ferrer-i-Cancho
-Contents
-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
-2 The mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
-3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
-4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
-A The mathematical model in detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
-B Form degrees and number of links . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
-C Complementary heatmaps for other values of . . . . . . . . . . . . . . . . . . . . . 48
-D Complementary gures with discrete degrees . . . . . . . . . . . . . . . . . . . . . . 61
-1 Introduction
-Biosemiotics can be dened as a science of signs in living systems (Kull, 1999, p.
-386). Here we join the eort of developing such a science. Focusing on the problem
-of \learning" new signs, we hope to contribute (i) to place choice at the core of
-semiotic theory of learning (Kull, 2018) and (ii) to make biosemiotics compatible
-with the information theoretic perspective that is regarded as currently dominant
-in physics, chemistry, and molecular biology (Deacon, 2015).
-Languages use words to convey information. From a semantic perspective,
-words stand for meanings (Fromkin et al., 2014). Correlates of word meaning
-have been investigated in other species (e.g. Hobaiter and Byrne, 2014; Genty and
-Zuberb uhler, 2014; Moore, 2014). From a neurobiological perspective, words can
-be seen as the counterparts of cell assemblies with distinct cortical topographies
-(Pulvermuller, 2001; Pulverm uller, 2013). From a formal standpoint, the essence
-of that research is some binding between a sign or a form, e.g., a word or an ape
-gesture, and a counterpart, e.g. a 'meaning' or an assembly of cortical cells. Math-
-ematically, that binding can be formalized as a bipartite graph where vertices are
-forms and their counterparts (Fig. 1). Such abstract setting allows for a powerful
-exploration of natural systems across levels of life, from the mapping of animal
-vocal or gestural behaviors (Fig. 2 (a)) into their \meanings" down to the map-
-ping from codons into amino acids (Figure 2 (b)) while allowing for a comparison
-against \articial" coding systems such as the Morse code (Fig. 2 (c)) or those
-emerging in articial naming games (Hurford, 1989; Steels, 1996). In that setting,
-almost connectedness has been hypothesized to be the mathematical condition re-
-quired for the emergence of a rudimentary form of syntax and symbolic reference
-(Ferrer-i-Cancho et al., 2005; Ferrer-i-Cancho, 2006). By symbolic reference, we
-mean here Deacon's revision of Pierce's view (Deacon, 1997). The almost connect-
-edness condition is met when it is possible to reach practically any other vertex of
-the network by starting a walk from any possible vertex (as in Fig. 1 (a)-(b) but
-not in Figs. 1 (c)-(d)).
-Since the pioneering research of G. K. Zipf (1949), statistical laws of language
-have been interpreted as manifestations of the minimization of cognitive costs
-(Zipf, 1949; Ellis and Hitchcock, 1986; Ferrer-i-Cancho and D az-Guilera, 2007;
-Gustison et al., 2016; Ferrer-i-Cancho et al., 2019). Zipf argued that the law of
-abbreviation, the tendency of more frequent words to be shorter, resulted from a
-minimization of a cost function involving, for every word, its frequency, its \mass"
-and its \distance", which in turn implies the minimization of the size of words
-(Zipf, 1949, p.59). Recently, it as been shown mathematically that the minimiza-
-tion of the average of the length of words (the mean code length in the languageThe advent and fall of a vocabulary learning bias from communicative eciency 3
-(a) (b)
-(c) (d)
-Fig. 1 A bipartite graph linking forms (white circles) with their counterparts (black circles).
-(a) a connected graph (b) an almost connected graph (c) a one-to-one mapping between forms
-and counterparts (d) a mapping where only one form is linked with counterparts.
-of information theory) predicts a correlation between frequency and duration that
-cannot be positive, extending and generalizing previous results from information
-theory (Ferrer-i-Cancho et al., 2019). The framework addresses the general prob-
-lem of assigning codes as short as possible to counterparts represented by distinct
-numbers while warranting certain constraints, e.g., that every number will receive
-a distinct code (e.g. non-singular coding in the language of information theory). If
-the counterparts are word types from a vocabulary, it predicts the law of abbre-
-viation as it occurs in the vast majority of languages (Bentz and Ferrer-i-Cancho,
-2016). If these counterparts are meanings, it predicts that more frequent mean-
-ings should tend to be assigned smaller codes (e.g., shorter words) as found in real
-experiments (Kanwal et al., 2017; Brochhagen, 2021). Table 1 summarizes these
-and other predictions of compression.4 David Carrera-Casado, Ramon Ferrer-i-Cancho
-(a) (b)
-(c)
-Fig. 2 Real bipartite graphs linking forms (white circles) with their counterparts (black
-circles). (a) Chimpanzee gestures and their meaning (Hobaiter and Byrne, 2014, Table S3).
-This table was chosen for its broad coverage of gesture types (see other tables satisfying other
-constraints, e.g. only gesture-meaning associations employed by a suciently large number of
-individuals). (b) Codon translation into amino acids, where forms are 64 codons and counter-
-parts are 20 amino acids (c) The international Morse code, where forms are strings of dots
-and dashed and the counterparts are letters of the English alphabet ( A;B;:::;Z ) and digits
-(0;1;:::;9).The advent and fall of a vocabulary learning bias from communicative eciency 5
-linguistic laws ! principles ! predictions
-(K ohler, 1987; Altmann, 1993)
-Zipf's law of abbreviation !compression !Menzerath's law
-(Gustison et al., 2016; Ferrer-i-Cancho et al., 2019)
-!Zipf's rank-frequency law
-(Ferrer-i-Cancho, 2016a)
-!\shorter words" for more frequent \meanings"
-(Ferrer-i-Cancho et al., 2019; Kanwal et al., 2017; Brochhagen, 2021)
-Zipf's rank-frequency law !mutual information maximization
-+
-surprisal minimization!a vocabulary learning bias
-(Ferrer-i-Cancho, 2017a)
-!the principle of contrast
-(Ferrer-i-Cancho, 2017a)
-!range or variation of 
-(Ferrer-i-Cancho, 2005a, 2006)
-Table 1 The application of the scientic method in quantitative linguistics (italics) with various concrete examples (roman). is the exponent of Zipf's
-rank-frequency law (Zipf, 1949). The prediction that is the target of the current article is shown in boldface.6 David Carrera-Casado, Ramon Ferrer-i-Cancho
-1.1 A family of probabilistic models
-The bipartite graph of form-counterpart associations is the skeleton (Figs. 1 and
-2) on which a family of models of communication has been built (Ferrer-i-Cancho
-and D az-Guilera, 2007; Ferrer-i-Cancho and Vitevitch, 2018). The target of the
-rst of these models (Ferrer-i-Cancho and Sole, 2003) was Zipf's rank-frequency
-law, that denes the relationship between the frequency of a word fand its rank
-i, approximately as
-fi:
-These early models were aimed at shedding light on mainly three questions:
-1. The origins of this law (Ferrer-i-Cancho and Sole, 2003; Ferrer-i-Cancho, 2005b).
-2. The range of variation of in human language (Ferrer-i-Cancho, 2005a, 2006).
-3. The relationship between and the syntactic and referential complexity of a
-communication system (Ferrer-i-Cancho et al., 2005; Ferrer-i-Cancho, 2006).
-The main assumption of these models is that word frequency is an epiphenomenon
-of the structure of the skeleton or the probability of the meanings. Following the
-metaphor of the skeleton, the models are bodies whose 
esh are probabilities that
-are calculated from the skeleton. The rst models dened p(sijrj), the probabil-
-ity that a speaker produces sigiven a counterpart rj, as the same for all words
-connected to rj. In the language of mathematics,
-p(sijrj) =aij
-!j; (1)
-whereaijis a boolean (0 or 1) that indicates if siandrjare connected and !jis
-the degree of rj, namely the number of connections of rjwith forms, i.e.
-!j=X
-iaij:
-These models are often portrayed as models of the assignment of meanings to forms
-(Futrell, 2020; Piantadosi, 2014) but this description falls short because:
-{They are indeed models of production as they dene the probability of pro-
-ducing a form given some counterparts (as in Eq. 1) or simply the marginal
-probability of a form. The claim that theories of language production or discourse
-do not explain the law (Piantadosi, 2014) has no basis and raises the questions
-of which theories of language production are deemed acceptable.
-{They are also models of understanding, as they dene symmetric conditional
-probabilities such as p(rjjsi), the probability that a listener interprets rjwhen
-receivingsi.
-{The models are 
exible. In addition to \meaning", other counterparts were
-deemed possible from their birth. See for instance the use of the term \stimuli"
-(e.g. Ferrer-i-Cancho and D az-Guilera, 2007), as a replacement for meaning
-that was borrowed from neurolinguistics (Pulvermuller, 2001).
-{The models t in the distributional semantics framework (Lund and Burgess,
-1996) for two reasons: their 
exibility, as counterparts can be dimensions in
-some hidden space, and also because of representing a form as a vector of their
-joint or conditional probabilities with \counterparts" that is inferred from the
-network structure, as we have already explained (Ferrer-i-Cancho and Vite-
-vitch, 2018).The advent and fall of a vocabulary learning bias from communicative eciency 7
-Contrary to the conclusions of (Piantadosi, 2014), there are derivations of Zipf's
-law that do account for psychological processes of word production, especially the
-intentionality of choosing words in order to convey a desired meaning.
-The family of models assume that the skeleton that determines all the prob-
-abilities, the bipartite graph, is shaped by a combination of minimization of the
-entropy (or surprisal) of words ( H) and the maximization of the mutual infor-
-mation between words and meanings ( I), two principles that are cognitively mo-
-tivated and that capture speaker and listener's requirements (Ferrer-i-Cancho,
-2018). When only the entropy of words is minimized, congurations where only
-one form is linked as in Fig. 1 (d) are predicted. When only the mutual informa-
-tion between forms and counterparts is maximized, one-to-one mappings between
-forms and counterparts are predicted (when the number of forms and counter-
-parts is the same) as in Figure 1 (c) or Fig. 2 (d). Real language is argued to be
-in-between these two extreme congurations (Ferrer-i-Cancho and D az-Guilera,
-2007). Such a trade-o between simplicity (Zipf's unication) and eective com-
-munication (Zipf's diversication) is also found in information theoretic models
-of communication based on the information bottleneck approach (see Zaslavsky
-et al. (2021) and references there in).
-In quantitative linguistics, scientic theory is not possible without taking into
-consideration language laws (K ohler, 1987; Debowski, 2020). Laws are seen as
-manifestations of principles (also referred as \requirements" by K ohler (1987)),
-which are key components of explanations of linguistic phenomena. As part of
-the scientic method cycle, novel predictions are key aim (Altmann, 1993) and
-key to validation and renement of theory (Bunge, 2001). Table 1 synthesizes this
-general view as chains of the form: laws,principles that are inferred from them,
-and predictions that are made from those principles, giving concrete examples from
-previous research.
-Although one of the initial goals of the family of models was to shed light on
-the origins of Zipf's law for word frequencies, a member of the family of mod-
-els turned out to generate a novel prediction on vocabulary learning in children
-and the tendency of words to contrast in meaning (Ferrer-i-Cancho, 2017a): when
-encountering a new word, children tend to infer that it refers to a concept that
-does not have a word attached to it (Markman and Wachtel, 1988; Merriman
-and Bowman, 1989; Clark, 1993). The nding is cross-linguistically robust: it has
-been found in children speaking English (Markman and Wachtel, 1988), Canadian
-French (Nicoladis and Laurent, 2020), Japanese (Haryu, 1991), Mandarin Chinese
-(Byers-Heinlein and Werker, 2013; Hung et al., 2015), Korean (Eun-Nam, 2017).
-These languages correspond to four distinct linguistic families (Indo-European,
-Japonic, Sino-Tibetan, Koreanic). Furthermore, the nding has also been repli-
-cated in adults (Hendrickson and Perfors, 2019; Yurovsky and Yu, 2008) and
-other species Kaminski et al. (2004). This phenomenon is a example of biosemio-
-sis, namely a process of choice-making between simultaneously alternative options
-(Kull, 2018, p. 454).
-As an explanation for vocabulary learning, the information theoretic model
-suers from some limitations that motivate the present article. The rst one is that
-the vocabulary learning bias weakens in older children (Kalashnikova et al., 2016;
-Yildiz, 2020) or in polylinguals (Houston-Price et al., 2010; Kalashnikova et al.,
-2015), while the current version of the model predicts the vocabulary learning bias8 David Carrera-Casado, Ramon Ferrer-i-Cancho
-Casea(b) Vertex degrees do not exceed one
-Casea(a) Counterpart degrees do not exceed one
-µk= 2 µk= 1ωj= 1 ωj= 1
-Caseb
-Caseb
-Fig. 3 Strategies for linking a new word to a meaning. Strategy aconsists of linking a word to
-a free meaning, namely an unlinked meaning. Strategy bconsists of linking a word to a meaning
-that is already linked. We assume that the meaning that is already linked is connected to a
-single word of degree k. Two simplifying assumptions are considered. (a) Counterpart degrees
-do not exceed one, implying k1. (b) Vertex degrees do not exceed one, implying k= 1.
-only provided that mutual information maximization is not neglected (Ferrer-i-
-Cancho, 2017a).
-The second limitation is inherited from the family of models, where the de-
-nition of the probabilities over the bipartite graph skeleton leads to a linear rela-
-tionship between the frequency of a form and its number of counterparts (Ferrer-i-
-Cancho and Vitevitch, 2018). However, this is inconsistent with Zipf's prediction,
-namely that the number of meanings a word of frequency fshould follow (Zipf,
-1945)
-f; (2)
-with= 0:5. Eq. 2 is known as Zipf's meaning-frequency law (Zipf, 1949). To over-
-come such a limitation, Ferrer-i-Cancho and Vitevitch (2018) proposed dierent
-ways of modifying the denition of the probabilities from the skeleton. Here we
-borrow a proposal of dening the joint probability of a form and its counterpart
-as
-p(si;rj)/aij(i!j); (3)
-whereis a parameter of the model and iand!jare, respectively, the degree
-(number of connections) of the form siand the counterpart rj. Previous research
-on vocabulary learning in children with these models (Ferrer-i-Cancho, 2017a)
-assumed= 0, which leads to = 1 (Ferrer-i-Cancho, 2016b). When = 1, the
-system is channeled to reproduce Zipf's meaning-frequency law, i.e. Eq. 2 with
-= 0:5 (Ferrer-i-Cancho and Vitevitch, 2018).
-1.2 Overview of the present article
-It has been argued that there cannot be meaning without interpretation (Eco,
-1986). As Kull (2020) puts it, \ Interpretation (which is the same as primitive decision-
-making) assumes that there exists a choice between two or more options. The options
-can be described as dierent codes applicable simultaneously in the same situation. "
-The main aim to of this article is to shed light on the choice between strategy a,The advent and fall of a vocabulary learning bias from communicative eciency 9
-i.e. attaching the new form to a counterpart that is unlinked, and strategy b, i.e.
-attaching the new form to a counterpart that is already linked (Fig. 3).
-The remainder of the article is organized as follows. Section 2 considers a model
-of a communication system that has three components:
-1. A skeleton that is dened by a binary matrix Athat indicates the form-
-counterpart connections.
-2. A 
esh that is dened over the skeleton with Eq. 3,
-3. A cost function , that denes the cost of communication as
-
-=I+ (1)H; (4)
-whereis a parameter that regulates the weight of mutual information ( I)
-maximization and word entropy ( H) minimization such that 0 1.Iand
-Hare inferred from matrix Aand Eq. 3 (further details are given in Section
-2).
-This section introduces 
, i.e. the dierence in the cost of communication between
-strategyaand strategy baccording to 
-(Fig. 3).
 < 0 indicates that the cost
-of communication of strategy ais lower than that of b. Our main hypothesis is
-that interpretation is driven by the 
-cost function and that a receiver will choose
-the option that minimizes the resulting 
-. By doing this, we are challenging the
-longstanding and limiting belief that information theory is dissociated from semi-
-otics and not concerned about meaning (e.g. Deacon, 2015). This article is a just
-one counterexample (see also Zaslavsky et al. (2018)). Information theory, as any
-abstract powerful mathematical tool, can serve applications that do not assume
-meaning (or meaning-making processes) as in the original setting of telecommu-
-nication where it was developed by Shannon, as well as others that do, although
-they were not his primary concern for historical and sociological reasons.
-In general, the formula of 
is complex and the analysis of the conditions where
-ais advantageous (namely 
<0) requires making some simplifying assumptions.
-If= 0, then one obtains that Ferrer-i-Cancho (2017a)
-
=(!j+ 1) log(!j+ 1)!jlog(!j)
-M+ 1; (5)
-whereMis the number of edges in the skeleton and !jis the degree of the al-
-ready linked counterpart that is selected in strategy b(Fig. 3). Eq. 5 indicates that
-strategyawill be advantageous provided that mutual information maximization
-matters (i.e.  >0) and its advantage will increase as mutual information max-
-imization becomes more important (i.e. for larger ), the linked counterpart has
-more connections (i.e. larger !j) or when the skeleton has less connections (i.e.
-smallerM). To be able to analyze the case >0, we will examine two classes of
-skeleta that are presented next.
-Counterpart degrees do not exceed one. In this class, the degrees of counterparts
-are restricted to not exceed one, namely a counterpart can only be disconnected
-or connected to just one form. If meanings are taken as counterparts, this class
-matches the view that \no two words ever have exactly the same meaning" (Fromkin
-et al., 2014, p. 256), based on the notion of absolute synonymy (Dangli and Abazaj,
-2009). This class also mirrors the linguistic principle that any two words should10 David Carrera-Casado, Ramon Ferrer-i-Cancho
-contrast in meaning (Clark, 1987). Alternatively, if synonyms are deemed real
-to some extent, this class may capture early stages of language development in
-children or early stages in the evolution of languages where synonyms have not
-been learned or developed. From a theoretical standpoint, this class is required
-by the maximization of the mutual information between forms and counterparts
-when the number of forms does not exceed that of counterparts (Ferrer-i-Cancho
-and Vitevitch, 2018).
-We usekto refer to degree of the word that will be connected to meaning
-selected in strategy b(Fig. 3). We will show that, in this class, 
is determined by
-,,kand the degree distribution of forms, namely the vector of form degrees
-~ = (1;:::;i;:::n).
-Vertex degrees do not exceed one. In this class, the degrees of any vertex are re-
-stricted to not exceed one, namely a form (or a meaning) can only be discon-
-nected or connected to just one counterpart (just one form). This class is narrower
-than the previous one because it imposes that degrees do not exceed one both for
-forms and counterparts. Words lack homonymy (or polysemy). We believe that this
-class would correspond to even earlier stages of language development in children
-(where children have learned at most one meaning of a word) or earlier stages
-in the evolution of languages (where the communication system has not devel-
-oped any homonymy). From a theoretical stand point, that class is a requirement
-of maximizing mutual information between forms and counterparts when n=m
-(Ferrer-i-Cancho and Vitevitch, 2018). We will show that 
is determined just by
-,andM, the number of links of the bipartite skeleton.
-Notice that meanings with synonyms have been found in chimpanzee gestures
-(Hobaiter and Byrne, 2014), which suggests that the two classes above do not
-capture the current state of the development of form-counterpart mappings in
-adults of other species. Section 2 presents the formulae of 
for each classes. Section
-3 uses this formulae to explore the conditions that determine when strategy ais
-more advantageous, namely 
 < 0, for each of the two classes of skeleta above,
-that correspond to dierent stages of the development of language in children.
-While the condition = 0 implies that strategy ais always advantageous when
->0, we nd regions of the space of parameters where this is not the case when
->0 and>0. In the more restrictive class, where vertex degrees do not exceed
-one, we nd a region where ais not advantageous when is suciently small and
-Mis suciently large. The size of that region increases as increases. From a
-complementary perspective, we nd a region where ais not advantageous ( 
0)
-whenis suciency small and is suciently large; the size of the region increases
-asMincreases. As Mis expected to be larger in older children or in polylinguals
-(if the forms of each language are mixed in the same skeleton), the model predicts
-the weakening of the bias in older children and polylinguals (Liittschwager and
-Markman, 1994; Kalashnikova et al., 2016; Yildiz, 2020; Houston-Price et al., 2010;
-Kalashnikova et al., 2015, 2019). To ease the exploration of the phase space for
-the class where the degrees of counterparts do not exceed one, we will assume
-that word frequencies follow Zipf's rank-frequency law. Again, regions where a
-is not advantageous ( 
0) also appear but the conditions for the emergence
-of this regions are more complex. Our preliminary analyses suggest that the bias
-should weaken in older children even for this class. Section 4 discusses the ndings,The advent and fall of a vocabulary learning bias from communicative eciency 11
-suggests future research directions and reviews the research program in light of
-the scientic method.
-2 The mathematical model
-Below we give more details about the model that we use to investigate the learning
-of new words and outlines the arguments that take from Eq. 3 to concrete formulae
-of
. Section 2.1 just presents the concrete formulae 
for each of the two classes
-of skeleta. Full details are given in Appendix A. The model has four components
-that we review next.
-Skeleton (A=aij).A bipartite graph that denes the associations between nforms
-andmcounterparts that are dened by an adjacency matrix A=faijg.
-Flesh (p(si;rj)).The 
esh consist of a denition of p(si;rj), the joint probability
-of a form (or word) and a counterpart (or meaning) and a series of probability
-denitions stemming from it. Probabilities depart from previous work (Ferrer-i-
-Cancho and Sole, 2003; Ferrer-i-Cancho, 2005b) by the addition of the parameter
-. Eq. 3 denes p(si;rj) as proportional to the product of the degrees of the form
-and the counterpart to the power of , which is a parameter of the model. By
-normalization, namely
-nX
-i=1mX
-j=1p(si;rj) = 1;
-Eq. 3 leads to
-p(si;rj) =1
-Maij(i!j); (6)
-where
-M=nX
-i=1mX
-j=1aij(i!j): (7)
-From these expressions, the marginal probabilities of a form p(si) and a counter-
-partp(rj) are obtained easily thanks to
-p(si) =mX
-j=1p(si;rj)
-p(rj) =nX
-i=1p(si;rj):
-The cost of communication ( 
-).The cost function is initially dened in Eq. 4 as in
-previous research (e.g. Ferrer-i-Cancho and D az-Guilera, 2007). In more detail,
-
-=I(S;R) + (1)H(S); (8)
-whereI(S;R) is the mutual information between forms from a repertoire Sand
-counterparts from a repertoire R, andH(S) is the entropy (or surprisal) of forms12 David Carrera-Casado, Ramon Ferrer-i-Cancho
-from a repertoire S. Knowing that I(S;R) =H(S) +H(R)H(S;R) Cover and
-Thomas (2006), the nal expression for the cost function in this article is
-
-() = (12)H(S)H(R) +H(S;R): (9)
-The entropies H(S),H(R) andH(S;R) are easy to calculate applying the deni-
-tions ofp(si),p(rj) andp(si;rj), respectively.
-The dierence in the cost of learning a new word ( 
).There are two possible strate-
-gies to determine the counterpart with which a new form (a previously unlinked
-form) should connect (Fig. 3):
-a. Connect the new form to a counterpart that is not already connected to any
-other forms.
-b. Connect the new form to a counterpart that is connected to at least one other
-form.
-The question we intend to answer is \when does strategy aresult in a smaller cost
-than strategy b?" Or, in the terminology of child language research, \for which
-strategy is the assumption of mutual exclusivity more advantageous?" To answer
-these questions, we dene 
, as a the dierence between the cost of each strategy.
-More precisely,
-
() =
-0
-a()
-0
-b(); (10)
-where
-0a() and
-0
-b() are the new value of 
-when a new link is created using
-strategyaorbrespectively. Then, our research question becomes \When is 
 <
-0?".
-Formulae for 
-0a() and
-0
-b() are derived in two steps. First, analyzing a
-general problem, i.e. 
-0, the new value of 
-after producing a single mutation in
-A(Appendix A.2). Second, deriving expressions for the case where that mutation
-results from linking a new form (an unlinked form) to a counterpart, that can be
-linked or unlinked (Appendix A.3).
-2.1
in two classes of skeleta
-In previous work, the value of 
was already calculated for = 0, obtaining
-expressions equivalent to Eq. 5 (see Appendix A.3.1 for a derivation). The next
-sections just summarize the more complex formulae that are obtained for each
-class of skeleta for 0 (see Appendix A for details on the derivation).
-2.1.1 Vertex degrees do not exceed one
-Here forms and counterparts both either have a single connection or are discon-
-nected. Mathematically, this can be expressed as
-i2f0;1gfor eachisuch that 1in
-!j2f0;1gfor eachjsuch that 1jm:
-Fig. 3 (b) oers a visual representation of a bipartite graph of this class. In case b,
-the counterpart we connect the new form to is connected to only one form ( !j= 1)The advent and fall of a vocabulary learning bias from communicative eciency 13
-and that form is connected to only one counterpart ( k= 1). Under this class, 

-becomes
-
() = (12)
-log
-1 +2(21)
-M+ 1
-+2+1log(2)
-M+ 2+11
-2+1log(2)
-M+ 2+11;(11)
-which can be rewritten as linear function of , i.e.
-
() =a+b;
-with
-a= 2 log
-1 +2(21)
-M+ 1
-(2+ 1)2+1log(2)
-M+ 2+11
-b=log
-1 +2(21)
-M+ 1
-+2+1log(2)
-M+ 2+11:
-Importantly, notice that this expression of 
is determined only by ,andM(the
-total number of links in the model). See Appendix A.3.3 for thorough derivations.
-2.1.2 Counterpart degrees do not exceed one
-This class of skeleta is a relaxation of the previous class. Counterparts are either
-connected to a single form or disconnected. Mathematically,
-!j2f0;1gfor eachjsuch that 1jm:
-Fig. 3 (a) oers a visual representation of a bipartite graph of this class. The
-number of forms the counterpart in case bis connected to is still 1 ( !j= 1) but
-this form may be connected to any number of counterparts; khas to satisfy
-1km.
-Under this class, 
becomes
-
() = (12)(
-log 
-M+ 1
-M+(21)
-k+ 2!
-+1
-M+(21)
-k+ 2"
-(+ 1)X(S;R)(21)(
-k+ 1)
-M+ 1
-2log(2) +
-kh
-log(k)(k+)
-(k1 + 2) log(k1 + 2)i#)
-1
-M+(21)
-k+ 2"
-
-
-k+ 1
2log
-
-k+ 1

-(1)2
-klog(k)#
-;(12)14 David Carrera-Casado, Ramon Ferrer-i-Cancho
-where
-X(S;R) =nX
-i=1+1
-ilogi (13)
-M=nX
-i=1+1
-i: (14)
-Eq. 12 can also be expressed as a linear function of as
-
() =a+b;
-with
-a= 2 log 
-M+ (21)
-k+ 2
-M+ 1!
-1
-M+ (21)
-k+ 2(
-2h
-(
-k+ 1) log(
-k+ 1) +
-klog(k)i
-+2h
-(+ 1)X(S;R)(21)
-k+ 1
-M+ 1
-+2log(2)
-klog(k)(k+)(k1 + 2) log(k1 + 2)i)
-b=log 
-M+ (21)
-k+ 2
-M+ 1!
-+1
-M+ (21)
-k+ 2(
-2
-klog(k)(+ 1)X(S;R)(21)
-k+ 1
-M+ 1
-+2log(2)
-kh
-log(k)(k+)(k1 + 2) log(k1 + 2)i)
-:
-Being a relaxation of the previous class, the resulting expressions of 
are more
-complex than those of the previous class, which are an in turn more complex than
-those of the case = 0 (Eq. 5). See Appendix A.3.2 for further details on the
-derivation of 
.
-Notice that X(S;R) (Eq. 13) and M(Eq. 14) are determined by the degrees
-of the forms ( i's). To explore the phase space with a realistic distribution of i's,
-we assume, without any loss of generality, that the i's are sorted decreasingly,
-i.e.12:::ii+1:::n. In addition, we assume
-1.n= 0, because we are investigating the problem of linking and unlinked form
-with counterparts.
-2.n1= 1.
-3. Form degrees are continuous.
-4. The relationship between iand its frequency rank is a right-truncated power-
-law, i.e.
-i=ci(15)
-for 1in1.The advent and fall of a vocabulary learning bias from communicative eciency 15
-Appendix B shows that forms then follow Zipf's rank-frequency law, i.e.
-p(si) =c0i
-with
-=(+ 1)
-c0=(n1)
-M:
-The value of 
is determined by ,,kand the sequence of degrees of the
-forms, which we have parameterized with andn. When=
-+1= 0, namely
-when= 0 or when !1 , we recover the class where vertex degrees do not
-exceed one but with just one form that is unlinked.
-A continuous approximation to the number of edges gives (Appendix B)
-M= (n1)
-+1n1X
-i=1i
-+1: (16)
-We aim to shed some light on the possible trajectory that children will describe
-on Fig. 4 as they become older. One expects that Mtends to increase as children
-become older, due to word learning. It is easy to see that Eq. 16 predicts that, if 
-andremain constant, Mis expected to increase as nincreases (Fig. 4). Besides,
-whennremains constant, a reduction of implies a reduction of Mwhen= 0
-but that eect vanishes for >0 (Fig. 4). Obviously, ntends to increase as a child
-becomes older (Saxton, 2010) and thus children's trajectory will be from left to
-right in Fig. 4. As for the temporal evolution of , there are two possibilities. Zipf's
-pioneering investigations suggest that remains close to 1 over time in English
-children (Zipf, 1949, Chapter IV). In contrast, a wider study reported a tendency of
-to decrease over time in suciently old children of dierent languages (Baixeries
-et al., 2013) but the study did not determine the actual number of children where
-that trend was statistically signicant after controlling for multiple comparisons.
-Then children, as they become older, are likely to move either from left to right,
-keepingconstant, or from the left-upper corner (high , lown) to the bottom-
-right corner (low , highn) within each panel of Fig. 4. When is suciently
-large, the actual evolution of some children (decrease of jointly with an increase
-ofn) is dominated by the increase of Mthat the growth of nimplies in the long
-run (Fig. 4).
-When exploring the space of parameters, we must warrant that kdoes not
-exceed the maximum degree that n,andyield, namely k1, where1is
-dened according to Eq. 15 with i= 1, i.e.
-k1
-=c
-= (n1)
-= (n1)
-+1: (17)16 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0.00.51.01.52.0
-0 250 500 750 1000
-nα
-0246log10 Mφ = 0(a)
-0.00.51.01.52.0
-0 250 500 750 1000
-nα
-01234log10 Mφ = 0.5(b)
-0.00.51.01.52.0
-0 250 500 750 1000
-nα
-0123log10 Mφ = 1(c)
-0.00.51.01.52.0
-0 250 500 750 1000
-nα
-0123log10 Mφ = 1.5(d)
-0.00.51.01.52.0
-0 250 500 750 1000
-nα
-0123log10 Mφ = 2(e)
-0.00.51.01.52.0
-0 250 500 750 1000
-nα
-0123log10 Mφ = 2.5(f)
-Fig. 4 log10M, the logarithm of the number of links M, as a function of n(x-axis) and
-(y-axis) according to Eq. 16. log10Mis used instead of Mto capture changes in order of
-magnitude of M. (a)= 0, (b)= 0:5, (c)= 1, (d)= 1:5, (e)= 2 and (f) = 2:5.
-3 Results
-Here we will analyze 
, that takes a negative value when strategy a(linking a
-new form to a new counterpart) is more advantageous than strategy blinking a
-new form to an already connected counterpart), and a positive value otherwise.
-j
jindicates the strength of the bias towards strategy aif
<0; towards strategyThe advent and fall of a vocabulary learning bias from communicative eciency 17
-bif
>0. Therefore, when 
<0, the smaller the value of 
, the higher the bias
-for strategy awhereas when 
>0, the greater the value of 
, the higher the bias
-for strategy b. Each class of skeleta is analyzed separately, beginning by the most
-restrictive class.
-3.1 Vertex degrees do not exceed one
-In this class of skeleta, corresponding to younger children, 
depends only on ,M
-and. We will explore the phase space with the help of two-dimensional heatmaps
-of
where thex-axis is always and they-axis isMor.
-Figs. 5 and 6 reveal regions where strategy ais more advantageous (red) and
-regions where bis more advantageous (blue) according to 
. The extreme situation
-is found when = 0 where a single red region covers practically all space except for
-= 0 (Fig. 5, top-left) as expected from previous work (Ferrer-i-Cancho, 2017a)
-and Eq. 5. Figs. 7 and 8 summarize these nding of regions, displaying the curve
-that denes the boundary between strategies aandb(
= 0).
-Figs. 7 and 8 show that strategy bis the optimal only if is suciently low,
-namely when the weight of entropy minimization is suciently high compared to
-that of mutual information maximization. Fig. 7 shows that the larger the value of
-the larger the number of links ( M) that is required for strategy bto be optimal.
-Fig. 7 also indicates that the larger the value of , the broader the blue region
-wherebis optimal. From a symmetric perspective, Fig. 8 shows that the larger the
-value ofthe larger the value of that is required for strategy bto be optimal and
-also that the larger the number of links ( M), the broader the blue region where b
-is optimal.
-3.2 Counterpart degrees do not exceed one
-For this class of skeleta, corresponding to older children, we have assumed that
-word frequencies follow Zipf's rank-frequency law, namely the relationship between
-the probability of a form (the number of counterparts connected to each form) and
-its frequency rank follows a right-truncated power-law with exponent (Section
-2). Then
depends only on (the exponent of the right-truncated power law),
-n(the number of forms), k(the degree of the form linked to the counterpart
-in strategy bas shown in Fig. 3), and. We will explore the phase space with
-the help of two-dimensional heatmaps of 
where the x-axis is always and the
-y-axis isk,orn. While in the class where vertex degrees do not exceed one
-we have found only one blue region (a region where 
 > 0 meaning that bis
-more advantageous), this class yields up to two distinct blue regions located in
-opposite corners of the heatmap while keeping always a red region as show in
-Figs. 10, 12 and 14 for = 1 from dierent perspectives. For the sake of brevity,
-this section only presents heatmaps of 
for= 0 or= 1 (see Appendix C for
-the remainder). A summary of exploration of the parameter space follows.
-Heatmaps of 
as a function of andk.The heatmaps of 
for dierent com-
-binations of parameters in Figs. 9, 10, 16, 17, 18 and 19 are summarized in Fig.
-11, showing the frontiers between regions where 
= 0. Notice how, for = 0,18 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λM
--0.6-0.4-0.2Δ < 0
-0Δ ≥ 0φ = 0(a)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λM
--0.6-0.4-0.2Δ < 0
-0.0050.010Δ ≥ 0φ = 0.5(b)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λM
--0.6-0.4-0.2Δ < 0
-0.000.010.020.030.040.05Δ ≥ 0φ = 1(c)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λM
--0.6-0.4-0.2Δ < 0
-0.0250.0500.0750.1000.125Δ ≥ 0φ = 1.5(d)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λM
--0.6-0.4-0.2Δ < 0
-0.000.050.100.150.20Δ ≥ 0φ = 2(e)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λM
--0.8-0.6-0.4-0.2Δ < 0
-0.10.20.3Δ ≥ 0φ = 2.5(f)
-Fig. 5
, the dierence between the cost of strategy aand strategy b, as a function of M, the
-number of links and , the parameter that controls the balance between mutual information
-maximization and entropy minimization, when vertex degrees do not exceed one (Eq. 11). Red
-indicates that strategy ais more advantageous while blue indicates that bis more advantageous.
-The lighter the red, the stronger the bias for strategy a. The lighter the blue, the stronger the
-bias for strategy b. (a)= 0, (b)= 0:5, (c)= 1, (d)= 1:5, (e)= 2 and (f) = 2:5.The advent and fall of a vocabulary learning bias from communicative eciency 19
-0.02.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λφ
--1.00-0.75-0.50-0.25Δ < 0
-0.00.10.20.30.4Δ ≥ 0M = 2(a)
-0.02.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λφ
--1.0-0.5Δ < 0
-0.00.20.40.6Δ ≥ 0M = 3(b)
-0.02.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λφ
--1.5-1.0-0.5Δ < 0
-0.000.250.500.751.00Δ ≥ 0M = 5(c)
-0.02.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λφ
--2.0-1.5-1.0-0.5Δ < 0
-0.00.40.81.21.6Δ ≥ 0M = 10(d)
-0.02.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λφ
--3-2-1Δ < 0
-0123Δ ≥ 0M = 50(e)
-0.02.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λφ
--4-3-2-1Δ < 0
-0123Δ ≥ 0M = 150(f)
-Fig. 6
, the dierence between the cost of strategy aand strategy b, as a function of ,
-the parameter that denes how the 
esh of the model from the skeleton, and , the parameter
-that controls the balance between mutual information maximization and entropy minimization
-(Eq. 11). Red indicates that strategy ais more advantageous while blue indicates that bis
-more advantageous. The lighter the red, the stronger the bias for strategy a. The lighter the
-blue, the stronger the bias for strategy b. (a)M= 2, (b)M= 3, (c)M= 5, (d)M= 10, (e)
-M= 50 and (f) M= 150.20 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λMφ
-0
-0.5
-1
-1.5
-2
-2.5
-Fig. 7 Summary of the boundaries between positive and negative values of 
when vertex
-degrees do not exceed one (Fig. 5). Each curve shows the points where 
= 0 (Eq. 12) as a
-function of andMfor distinct values of .
-strategyais optimal for all values of >0, as one would expect from Eq. 5. The
-remainder of the gures show how the shape of the two areas changes with each
-of the parameters. For small nand, a single blue region indicates that strategy
-bis more advantageous than awhenis closer to 0 and kis higher. For higher
-noran additional blue region appears indicating that strategy bis also optimal
-for high values of and low values of k.
-Heatmaps of 
as a function of and.The heatmaps of 
for dierent combi-
-nations of parameters in Figs. 12, 20, 21, 22 and 23 are summarized in Fig. 13,
-showing the frontiers between regions. There is a single region where strategy bis
-optimal for small values of kand, but for larger values a second blue region
-appears.
-Heatmaps of 
as a function of andn.The heatmaps of 
for dierent combina-
-tions of parameters in Figs. 14, 24, 25, 26 and 27 are summarized in Fig. 15. Again,
-one or two blue regions appear depending on the combination of parameters.
-See Appendix D for the impact of using discrete form degrees on the results
-presented in this section.The advent and fall of a vocabulary learning bias from communicative eciency 21
-0.02.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λφM
-2
-3
-5
-10
-50
-150
-Fig. 8 Summary of the boundaries between positive and negative values of 
when vertex
-degrees do not exceed one (Fig. 6). Each curve shows the points where 
= 0 (Eq. 12) as a
-function of andfor distinct values of M.22 David Carrera-Casado, Ramon Ferrer-i-Cancho
-1.01.52.02.53.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.075-0.050-0.025Δ < 0
-0Δ ≥ 0φ = 0 α = 0.5 n = 10(a)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.05-0.04-0.03-0.02-0.01Δ < 0
-0Δ ≥ 0φ = 0 α = 1 n = 10(b)
-01020
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.025-0.020-0.015-0.010-0.005Δ < 0
-0Δ ≥ 0φ = 0 α = 1.5 n = 10(c)
-2.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.006-0.004-0.002Δ < 0
-0Δ ≥ 0φ = 0 α = 0.5 n = 100(d)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.0025-0.0020-0.0015-0.0010-0.0005Δ < 0
-0Δ ≥ 0φ = 0 α = 1 n = 100(e)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λμk
--5e-04-4e-04-3e-04-2e-04-1e-04Δ < 0
-0Δ ≥ 0φ = 0 α = 1.5 n = 100(f)
-0102030
-0.00 0.25 0.50 0.75 1.00
-λμk
--6e-04-4e-04-2e-04Δ < 0
-0Δ ≥ 0φ = 0 α = 0.5 n = 1000(g)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.00015-0.00010-0.00005Δ < 0
-0Δ ≥ 0φ = 0 α = 1 n = 1000(h)
-0100002000030000
-0.00 0.25 0.50 0.75 1.00
-λμk
--1.6e-05-1.2e-05-8.0e-06-4.0e-06Δ < 0
-0Δ ≥ 0φ = 0 α = 1.5 n = 1000(i)
-Fig. 9
, the dierence between the cost of strategy aand strategy b, as a function of k,
-the degree of the form linked to the counterpart in strategy bas shown in Fig. 3, the number of
-links and, the parameter that controls the balance between mutual information maximization
-and entropy minimization, when the degrees of counterparts do not exceed one (Eq. 11) and
-= 0. Red indicates that strategy ais more advantageous while blue indicates that bis more
-advantageous. The lighter the red, the stronger the bias for strategy a. The lighter the blue,
-the stronger the bias for strategy b. Each heatmap corresponds to a distinct combination of
-nand. The heatmaps are arranged, from left to right, with = 0:5;1;1:5 and, from top to
-bottom, with n= 10;100;1000. (a)= 0:5 andn= 10, (b)= 1 andn= 10, (c)= 1:5
-andn= 10, (d)= 0:5 andn= 100, (e)= 1 andn= 100, (f)= 1:5 andn= 100, (g)
-= 0:5 andn= 1000, (h) = 1 andn= 1000, (i) = 1:5 andn= 1000.The advent and fall of a vocabulary learning bias from communicative eciency 23
-1.01.21.41.6
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.25-0.20-0.15-0.10-0.05Δ < 0
-0.020.040.06Δ ≥ 0φ = 1 α = 0.5 n = 10(a)
-1.01.52.02.53.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.20-0.15-0.10-0.05Δ < 0
-0.020.040.06Δ ≥ 0φ = 1 α = 1 n = 10(b)
-12345
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.10-0.05Δ < 0
-0.010.020.030.040.05Δ ≥ 0φ = 1 α = 1.5 n = 10(c)
-1.01.52.02.53.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.04-0.03-0.02-0.01Δ < 0
-0.0050.0100.0150.0200.025Δ ≥ 0φ = 1 α = 0.5 n = 100(d)
-2.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.04-0.03-0.02-0.01Δ < 0
-0.010.020.03Δ ≥ 0φ = 1 α = 1 n = 100(e)
-0102030
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.020-0.015-0.010-0.005Δ < 0
-0.0050.0100.015Δ ≥ 0φ = 1 α = 1.5 n = 100(f)
-12345
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.0100-0.0075-0.0050-0.0025Δ < 0
-0.0020.0040.006Δ ≥ 0φ = 1 α = 0.5 n = 1000(g)
-0102030
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.010-0.005Δ < 0
-0.00250.00500.00750.01000.0125Δ ≥ 0φ = 1 α = 1 n = 1000(h)
-050100150
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.004-0.003-0.002-0.001Δ < 0
-0.0010.0020.0030.004Δ ≥ 0φ = 1 α = 1.5 n = 1000(i)
-Fig. 10 Same as in Fig. 9 but with = 1.24 David Carrera-Casado, Ramon Ferrer-i-Cancho
-1.21.51.82.1
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 0.5 n = 10(a)
-234
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1 n = 10(b)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1.5 n = 10(c)
-1234
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 0.5 n = 100(d)
-05101520
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1 n = 100(e)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1.5 n = 100(f)
-2.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 0.5 n = 1000(g)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1 n = 1000(h)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1.5 n = 1000(i)
-Fig. 11 Summary of the boundaries between positive and negative values of 
when the
-degrees of counterparts do not exceed one (gures 9, 10, 16, 17, 18 and 19). Each curve shows
-the points where 
= 0 (Eq. 12) as a function of andkfor distinct values of . (a)= 0:5
-andn= 10, (b)= 1 andn= 10, (c)= 1:5 andn= 10, (d)= 0:5 andn= 100, (e)= 1
-andn= 100, (f)= 1:5 andn= 100, (g)= 0:5 andn= 1000, (h) = 1 andn= 1000, (i)
-= 1:5 andn= 1000.The advent and fall of a vocabulary learning bias from communicative eciency 25
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.10-0.05Δ < 0
-0.0050.0100.0150.020Δ ≥ 0φ = 1 μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.0100-0.0075-0.0050-0.0025Δ < 0
-0.0010.0020.0030.004Δ ≥ 0φ = 1 μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.00075-0.00050-0.00025Δ < 0
-1e-042e-043e-044e-045e-04Δ ≥ 0φ = 1 μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.3-0.2-0.1Δ < 0
-0.0250.0500.0750.100Δ ≥ 0φ = 1 μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.025-0.020-0.015-0.010-0.005Δ < 0
-0.0020.0040.006Δ ≥ 0φ = 1 μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.0020-0.0015-0.0010-0.0005Δ < 0
-1e-042e-043e-044e-045e-04Δ ≥ 0φ = 1 μk = 2 n = 1000(f)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.6-0.4-0.2Δ < 0
-0.10.20.30.4Δ ≥ 0φ = 1 μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.06-0.04-0.02Δ < 0
-0.010.020.030.04Δ ≥ 0φ = 1 μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.006-0.004-0.002Δ < 0
-0.0010.0020.003Δ ≥ 0φ = 1 μk = 4 n = 1000(i)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--1.0-0.5Δ < 0
-0.30.60.9Δ ≥ 0φ = 1 μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.16-0.12-0.08-0.04Δ < 0
-0.050.10Δ ≥ 0φ = 1 μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.016-0.012-0.008-0.004Δ < 0
-0.00250.00500.00750.01000.0125Δ ≥ 0φ = 1 μk = 8 n = 1000(l)
-Fig. 12
, the dierence between the cost of strategy aand strategy b, as a function of , the
-exponent of the rank-frequency law, and , the parameter that controls the balance between
-mutual information maximization and entropy minimization, when the degrees of counterparts
-do not exceed one (Eq. 11) and = 1. Red indicates that strategy ais more advantageous
-while blue indicates that bis more advantageous. The lighter the red, the stronger the bias for
-strategya. The lighter the blue, the stronger the bias for strategy b. Each heatmap corresponds
-to a distinct combination of nandk. The heatmaps are arranged, from left to right, with
-n= 10;100;1000 and, from top to bottom, with k= 1;2;4;8. Gray indicates regions where
-kexceeds the maximum degree according to other parameters (Eq. 17). (a) k= 1 and
-n= 10, (b)k= 1 andn= 100, (c)k= 1 andn= 1000, (d) k= 2 andn= 10, (e)k= 2
-andn= 100, (f) k= 2 andn= 1000, (g) k= 4 andn= 10, (h)k= 4 andn= 100,
-(i)k= 4 andn= 1000, (j) k= 8 andn= 10, (k)k= 8 andn= 100, (l)k= 8 and
-n= 1000.26 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-1
-1.5
-2
-2.5μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 n = 1000(f)
-0.60.81.01.21.4
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 4 n = 1000(i)
-1.01.11.21.31.41.5
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 8 n = 1000(l)
-Fig. 13 Summary of the boundaries between positive and negative values of 
when the
-degrees of counterparts do not exceed one (Figs. 12, 20, 21, 22 and 23). Each curve shows
-the points where 
= 0 (Eq. 12) as a function of andfor distinct values of . Points are
-restricted to combinations of parameters where kdoes not exceed the maximum (Eq. 17).
-Each distinct heatmap corresponds to a distinct combination of kandn. (a)k= 1 and
-n= 10, (b)k= 1 andn= 100, (c)k= 1 andn= 1000, (d) k= 2 andn= 10, (e)k= 2
-andn= 100, (f) k= 2 andn= 1000, (g) k= 4 andn= 10, (h)k= 4 andn= 100,
-(i)k= 4 andn= 1000, (j) k= 8 andn= 10, (k)k= 8 andn= 100, (l)k= 8 and
-n= 1000.The advent and fall of a vocabulary learning bias from communicative eciency 27
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.10-0.05Δ < 0φ = 1 μk = 1 α = 0.5(a)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.06-0.04-0.02Δ < 0
-0.0010.0020.003Δ ≥ 0φ = 1 μk = 1 α = 1(b)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.06-0.04-0.02Δ < 0
-0.0050.0100.0150.020Δ ≥ 0φ = 1 μk = 1 α = 1.5(c)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.3-0.2-0.1Δ < 0
-0.0250.0500.0750.100Δ ≥ 0φ = 1 μk = 2 α = 0.5(d)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.09-0.06-0.03Δ < 0
-3e-046e-049e-04Δ ≥ 0φ = 1 μk = 2 α = 1(e)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.06-0.04-0.02Δ < 0
-0.00250.00500.00750.01000.0125Δ ≥ 0φ = 1 μk = 2 α = 1.5(f)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.6-0.4-0.2Δ < 0
-0.10.20.30.4Δ ≥ 0φ = 1 μk = 4 α = 0.5(g)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.3-0.2-0.1Δ < 0
-0.040.080.120.16Δ ≥ 0φ = 1 μk = 4 α = 1(h)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.08-0.06-0.04-0.02Δ < 0
-0.0020.0040.006Δ ≥ 0φ = 1 μk = 4 α = 1.5(i)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.0-0.5Δ < 0
-0.30.60.9Δ ≥ 0φ = 1 μk = 8 α = 0.5(j)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.6-0.4-0.2Δ < 0
-0.10.20.30.40.5Δ ≥ 0φ = 1 μk = 8 α = 1(k)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.2-0.1Δ < 0
-0.050.100.15Δ ≥ 0φ = 1 μk = 8 α = 1.5(l)
-Fig. 14
, the dierence between the cost of strategy aand strategy b, as function of n, the
-number of forms, and , the parameter that controls the balance between mutual information
-maximization and entropy minimization, when the degrees of counterparts do not exceed one
-(Eq. 11) and = 1. We are taking values of nfrom 10 onwards (instead of one onwards) to see
-more clearly the light regions that are re
ected on the color scales. Red indicates that strategy
-ais more advantageous while blue indicates that bis more advantageous. The lighter the red,
-the stronger the bias for strategy a. The lighter the blue, the stronger the bias for strategy b.
-Each heatmap corresponds to a distinct combination of kand. The heatmaps are arranged,
-from left to right, with = 0:5;1;1:5 and, from top to bottom, with k= 1;2;4;8. Gray
-indicates regions where kexceeds the maximum degree according to other parameters (Eq.
-17). (a)k= 1 and= 0:5, (b)k= 1 and= 1, (c)k= 1 and= 1:5, (d)k= 2 and
-= 0:5, (e)k= 2 and= 1, (f)k= 2 and= 1:5, (g)k= 4 and= 0:5, (h)k= 4
-and= 1, (i)k= 4 and= 1:5, (j)k= 8 and= 0:5, (k)k= 8 and= 1, (l)k= 8
-and= 1:5.28 David Carrera-Casado, Ramon Ferrer-i-Cancho
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-1.5
-2
-2.5μk = 1 α = 0.5(a)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 1 α = 1(b)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 1 α = 1.5(c)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 α = 0.5(d)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 α = 1(e)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 α = 1.5(f)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1μk = 4 α = 0.5(g)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 4 α = 1(h)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 4 α = 1.5(i)
-2505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5μk = 8 α = 0.5(j)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2μk = 8 α = 1(k)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 8 α = 1.5(l)
-Fig. 15 Summary of the boundaries between positive and negative values of 
when the
-degrees of counterparts do not exceed one (Figs. 14, 24, 25, 26 and 27). Each curve shows
-the points where 
= 0 (Eq. 12) as a function of andnfor distinct values of . Points are
-restricted to combinations of parameters where kdoes not exceed the maximum (Eq. 17).
-Each distinct heatmap corresponds to a distinct combination of kand. (a)k= 1 and
-= 0:5, (b)k= 1 and= 1, (c)k= 1 and= 1:5, (d)k= 2 and= 0:5, (e)k= 2
-and= 1, (f)k= 2 and= 1:5, (g)k= 4 and= 0:5, (h)k= 4 and= 1, (i)k= 4
-and= 1:5, (j)k= 8 and= 0:5, (k)k= 8 and= 1, (l)k= 8 and= 1:5.The advent and fall of a vocabulary learning bias from communicative eciency 29
-4 Discussion
-4.1 Vocabulary learning
-In previous research with = 0, we predicted that the vocabulary learning bias
-(strategya) would be present provided that mutual information minimization is
-not disabled (  > 0) (Ferrer-i-Cancho, 2017a) as show in Eq. 5. However, the
-\decision" on whether assigning a new label to a linked or to an unlinked object
-is in
uenced by the age of a child and his/her degree of polylingualism. As for the
-eect of the latter, polylingual children tend to pick familiar objects more often
-than monolingual children, violating mutual exclusivity. This has been found for
-younger children below two years of age (17-22 months old in one study, 17-18
-in another) (Houston-Price et al., 2010; Byers-Heinlein and Werker, 2013). From
-three years onward, the dierence between polylinguals and monolinguals either
-vanishes, namely both violate mutual exclusivity similarly (Nicoladis and Laurent,
-2020; Frank and Poulin-Dubois, 2002), or polylingual children are still more willing
-to accept lexical overlap (Kalashnikova et al., 2015). One possible explanation for
-this phenomenon is the lexicon structure hypothesis (Byers-Heinlein and Werker,
-2013), which suggests that children that already have many multiple-word-to-
-single-object mappings may be more willing to suspend mutual exclusivity.
-As for the eect of age on monolingual children, the so-called mutual exclusivity
-bias has been shown to appear at an early age and, as time goes on, it is more easily
-suspended. Starting at 17 months old, children tend to look at a novel object rather
-than a familiar one when presented with a new word while 16-month-olds do not
-show a preference (Halberda, 2003). Interestingly, in the same study, 14-month-olds
-systematically look at a familiar object instead of a newer one. Reliance on mutual
-exclusivity is shown to improve between 18 and 30 months (Bion et al., 2013).
-Starting at least at 24 months of age, children may suspend mutual exclusivity to
-learn a second label for an object (Liittschwager and Markman, 1994). In a more
-recent study, it has been shown that three year old children will suspend mutual
-exclusivity if there are enough social cues present (Yildiz, 2020). Four to ve year
-old children continue to apply mutual exclusivity to learn new words but are able
-to apply it 
exibly, suspending it when given appropriate contextual information
-(Kalashnikova et al., 2016) in order to associate multiple labels to the same familiar
-object. As seen before, at 3 years of age both monolingual and polylingual children
-have similar willingness to suspend mutual exclusivity (Nicoladis and Laurent,
-2020; Frank and Poulin-Dubois, 2002), although polylinguals may still have a
-greater tendency to accept multiple labels for the same object (Kalashnikova et al.,
-2015).
-Here we have made an important contribution with respect to the precursor
-of the current model (Ferrer-i-Cancho, 2017a): we have shown that the bias is not
-theoretically inevitable (even when  > 0) according a more realistic model. In
-a more complex setting, research on deep neural networks has shed light on the
-architectures, learning biases and pragmatic strategies that are required for the
-vocabulary learning bias to emerge (e.g. Gandhi and Lake, 2020; Gulordava et al.,
-2020). In section 3, we have discovered regions of the space of parameters where
-strategyais not advantageous for two classes of skeleta. In the restrictive class,
-where one where vertex degrees do no exceed one, as expected in the earliest stages
-of vocabulary learning in children, we have unveiled the existence of a region of the30 David Carrera-Casado, Ramon Ferrer-i-Cancho
-phase space where strategy ais not advantageous (Figs. 7 and 6). In the broader
-class of skeleta where the degree of counterparts does not exceed one we have
-found up to two distinct regions where ais not advantageous (Figs. 11 and 13).
-Crucially, our model predicts that the bias should be lost in older children.
-The argument is as follows. Suppose a child that has not learned a word yet.
-Then his skeleton belongs to the class where vertex degrees do not exceed one.
-Then suppose that the child learns a new word. It could be that he/she learns
-it following strategy aorb. If he applies bthen the bias is gone at least for this
-word. Let us suppose that the child learns words adhering to strategy afor as long
-as possible. By doing this, he/she will increasing the number of links ( M) of the
-skeleton keeping as invariant a one-to-one mapping between words and meanings
-(Figs. 1 (c) and 2 (d)), which satises that vertex degrees do not exceed one. Then
-Figs. 7 and 8 predict that the longer the time strategy ais kept (when >0) the
-larger the region of the phase space where ais not advantageous. Namely, as times
-goes on, it will become increasingly more dicult to keep aas the best option.
-Then it is not surprising that the bias weakens either in older children (e.g., Yildiz,
-2020; Kalashnikova et al., 2016), as they are expected to have more links (larger M)
-because of their continued accretion of new words (Saxton, 2010), or in polylinguals
-(e.g., Nicoladis and Secco, 2000; Greene et al., 2013), where the mapping of words
-into meanings combining all their languages, is expected to yield more links than
-in monolinguals. Polylinguals make use of code-mixing to compensate for lexical
-gaps, as reported for from one-year-olds onward (Nicoladis and Secco, 2000) as
-well as in older children (ve year olds) (Greene et al., 2013). As a result, the
-bipartite skeleton of a polylingual integrates the words and association in all the
-languages spoken and thus polylinguals are expected to have a larger value of M.
-Children who know more translation equivalents (words from dierent languages
-but with same meaning), adhere to mutual exclusivity less than other children
-(Byers-Heinlein and Werker, 2013). Therefore, our theoretical framework provides
-an explanation for the lexicon structure hypothesis (Byers-Heinlein and Werker,
-2013), but shedding light on the possible origin of the mechanism, that is not the
-fact that there are already synonyms but rather the large number of links (Fig.
-8) as well as the capacity of words of higher degree to attract more meanings, a
-consequence of Eq. 3 with >0 in the vocabulary learning process (Fig. 3). Recall
-the stark contrast between Fig. 10 for = 1 and the Fig. 9 with = 0, where
-such attraction eect is missing. Our models oer a transparent theoretical tool
-to understand the failure of deep neural networks to reproduce the vocabulary
-learning bias (Gandhi and Lake, 2020): in its simpler form (vertex degrees do not
-exceed one), whether it is due to an excessive (Fig. 7) or an excessive M(Fig.
-8).
-We have focused on the loss of the bias in older children. However, there is
-evidence that the bias is missing initially in children, by the age of 14 months
-(Halberda, 2003). We speculate that this could be related to very young children
-having lower values of or larger values of as suggested by Figs. 7 and 6. This
-issue should be the subject of future research. Methods to estimate andin real
-speakers should be investigated.
-Now we turn our attention to skeleta where only the degree of the counterparts
-does not exceed one, that we believe to be more appropriate for older children.
-Whereas,andMsuced for the exploration of the phase space when vertex
-degrees do not exceed one, the exploration of that kind of skeleta involved manyThe advent and fall of a vocabulary learning bias from communicative eciency 31
-parameters: ,,n,kand. The more general class exhibits behaviors that we
-have already seen in the more restrictive class. While an increase in Mimplies a
-widening of the region where ais not advantageous in the more restrictive class,
-the more general class experiences an increase of Mwhennis increased but and
-remain constant (Section 2.1.2). Consistently with the more restrictive class,
-such increase of Mleads to a growth of the regions where ais not advantageous as
-it can be seen in Figs. 16, 10, 17, 18 and 19 when selecting a column (thus xing
-and) and moving from the top to the bottom increasing n. The challenge is
-thatmay not remain constant in real children as they become older and how to
-involve the remainder of the parameters in the argument. In fact, some of these
-parameters are known to be correlated with child's age:
-{ntends to increase over time in children, as children are learning new words
-over time (Saxton, 2010). We assume that the loss of words can be neglected
-in children.
-{Mtends to increase over time in children. In this class of skeleta, the growth
-ofMhas two sources: the learning of new words as well as the learning of new
-meanings for existing words. We assume that the loss of connections can be
-neglected in children.
-{The ambiguity of the words that children learn over time tends to increase over
-time (Casas et al., 2018). This does not imply that children are learning all
-the meanings of the word according to some online dictionary but rather than
-as times go on, children are able to handle words that have more meanings
-according to adult standards.
-{remains stable over time or tends to decrease over time in children depending
-on the individual (Baixeries et al., 2013; Zipf, 1949, Chapter IV).
-For other parameters, we can just speculate on their evolution with child's age.
-The growth of Mand the increase in the learning of ambiguous words over time
-leads to expect that the maximum value of kwill be larger in older children. It
-is hard to tell if older children will have a chance to encounter larger values of
-k. We do not know the value of in real language but the higher diversity of
-vocabulary in older children and adults (Baixeries et al., 2013) suggests that 
-may tend to increase over time, because the lower the value of , the higher the
-pressure to minimize the entropy of words (Eq. 4), namely the higher the force
-towards unication in Zipf's view (Zipf, 1949). We do not know the real value of 
-but a reasonable choice for adult language is = 1 (Ferrer-i-Cancho and Vitevitch,
-2018).
-Given the complexity of the space of parameters in the more general class
-of skeleta where only the degrees of counterparts cannot exceed one, we cannot
-make predictions that are as strong as those stemming from the class where vertex
-degrees cannot exceed one. However, we wish to make some remarks suggesting
-that a weakening of the vocabulary learning bias is also expected in older children
-for this class (provided that >0). The combination of increasing nand a value
-ofthat is stable over time suggests a weakening of the strategy aover time from
-dierent perspectives
-{Children evolve on a column of panels (constant ) of the matrix of panels
-in Figs. 16, 10, 17, 18 and 19, moving from top (low n) to the bottom (large
-n). That trajectory implies an increase of the size of the blue region, where
-strategyais not advantageous.32 David Carrera-Casado, Ramon Ferrer-i-Cancho
-{We do not know the temporal evolution of kbut oncekis xed, namely a
-row of panels is selected in Figs. 20, 12, 21, 22 and 23, children evolve from
-left (lower n) to right (higher n), which implies an increase of the size of the
-blue region where strategy ais not advantageous as children become older.
-{Within each panel in Figs. 24, 14, 25, 26 and 27, an increase of n, as a results
-of vocabulary learning over time, implies a widening of the blue region.
-In the preceding analysis we have assumed that remains stable over time. We
-wish to speculate on the combination of increasing nand decreasing as time goes
-on in certain children. In that case, children would evolve close to the diagonal of
-the matrix of panels, starting from the right-upper corner (low n, high, panel
-(c)) towards the lower-left corner (high n, low, panel (g)) in Figs. 16, 10, 17,
-18 and 19, which implies an increase of the size of the blue region where strategy
-ais not advantageous. Recall that we have argued that a combined increase of n
-and decrease of is likely to lead in the long run to an increase of M(Fig. 4). We
-suggest that the behavior "along the diagonal" of the matrix is an extension of
-the weakening of the bias when Mis increased in the more restrictive class (Fig.
-8).
-In our exploration of the phase space for the class of the skeleta where the
-degrees of counterparts do not exceed one, we assumed a right-truncated power-law
-with two parameters, andnas a model for Zipf's rank-frequency law. However,
-distributions giving a better t have been considered (Li et al., 2010) and function
-(distribution) capturing the shape of the law of what Piotrowski called saturated
-samples (Piotrowski and Spivak, 2007) should be considered in future research. Our
-exploration of the phase space was limited by a brute force approach neglecting
-the negative correlation between nandthat is expected in children where 
-and time are negatively correlated: as children become older, nincreases as a
-result of word learning (Saxton, 2010) but decreases (Baixeries et al., 2013). A
-more powerful exploration of the phase space could be performed with a realistic
-mathematical relationship of the expected correlation between nand, which
-invites to empirical research. Finally, there might be deeper and better ways of
-parameterizing the class of skeleta.
-4.2 Biosemiotics
-Biosemiotics is concerned about building bridges between biology, philosophy, lin-
-guistics, and the communication sciences as announced in the front page of this
-journal https://www.springer.com/journal/12304 . As far as we know, there is lit-
-tle research on the vocabulary learning bias in other species. Its conrmation in
-a domestic dog suggests that \ the perceptual and cognitive mechanisms that may
-mediate the comprehension of speech were already in place before early humans began
-to talk " (Kaminski et al., 2004). We hypothesize that the cost function 
-cap-
-tures the essence of these mechanisms. A promising target for future research are
-ape gestures, where there has been signicant progress recently on their meaning
-(Hobaiter and Byrne, 2014). As far as we know, there is no research on that bias
-in other domains that also fall into the scope of biosemiotics, e.g., in unicellu-
-lar organisms such as bacteria. Our research has established some mathematical
-foundations for research on the accretion and interpretation of signs across theThe advent and fall of a vocabulary learning bias from communicative eciency 33
-living world, not only among great apes, a key problem in research program of
-biosemiotics (Kull, 2018).
-The remainder of the discussion section is devoted to examine general chal-
-lenges that are shared by biosemiotics and quantitative linguistics, a eld that, as
-biosemiotics, aspires to contribute to develop a science beyond human communi-
-cation.
-4.3 Science and its method
-It has been argued that a problem of research on the rank-frequency is law is the
-The absence of novel predictions... which has led to a very peculiar situation in the
-cognitive sciences, where we have a profusion of theories to explain an empirical phe-
-nomenon, yet very little attempt to distinguish those theories using scientic methods.
-(Piantadosi, 2014). As we have already shown the predictive power of a model
-whose original target was the rank-frequency laws here and in previous research
-(Ferrer-i-Cancho, 2017a), we take this criticism as an invitation to re
ect on sci-
-ence and its method (Altmann, 1993; Bunge, 2001).
-4.3.1 The generality of the patterns for theory construction
-While in psycholinguisics and the cognitive sciences a major source of evidence are
-often experiments involving restricted tasks or sophisticated statistical analyses
-covering a handful of languages (typically English and a few other Indo-European
-languages), quantitative linguistics aims to build theory departing from statistical
-laws holding in a typologically wide range of languages (K ohler, 1987; Debowski,
-2020) as re
ected in Fig. 1. In addition, here we have investigated a specic vocab-
-ulary learning phenomenon that is, however, supported cross-linguistically (recall
-Section 1). A recent review on the eciency of languages, only pays attention to
-the law of abbreviation (Gibson et al., 2019) in contrast with the body of work
-that has been developed in the last decades linking laws with optimization princi-
-ples (Fig. 1), suggesting that this law is the only general pattern of languages that
-is shaped by eciency or that linguistic laws are secondary for deep theorizing
-on eciency. In other domains of the cognitive sciences, the importance of scaling
-laws has been recognized (Chater and Brown, 1999; Kello et al., 2010; Baronchelli
-et al., 2013).
-4.3.2 Novel predictions
-In section 4.1, we have checked predictions of our information theoretic framework
-that matches knowledge on the vocabulary learning bias from past research. Our
-theoretical framework allows the researcher to play the game of science in another
-direction: use the relevant parameters to guide the design of new experiments with
-children or adults where more detailed predictions of the theoretical framework
-can be tested. For children who have about the same nand, and= 1, our
-model predicts that strategy awill be discarded if (Fig. 10)
-(1)is low and k(Fig.3) is large enough.
-(2)is high and kis suciently low.34 David Carrera-Casado, Ramon Ferrer-i-Cancho
-Interestingly, there is a red horizontal band in Fig. 10, and even for other values of
-such that6= 1 but keeping >0 (Figs. 16, 17, 18, 19), indicating the existence
-of some value of kor a range of kwhere strategy ais always advantageous (notice
-however, that when >1, the band may become too narrow for an integer kto
-t as suggested by Figs. 31, 32, 33 in Appendix D). Therefore the 1st concrete
-prediction is that, for a given child, there is likely to be some range or value of k
-where the bias (strategy a) will be observed. The 2nd concrete prediction that can
-be made is on the conditions where the bias will not be observed. Although the
-true value of is not known yet, previous theoretical research with = 0 suggests
-that1=2 in real language (Ferrer-i-Cancho and Sole, 2003; Ferrer-i-Cancho,
-2005b, 2006, 2005a), which would imply that real speakers should satisfy only (1).
-Child or adult language researchers may design experiments where kis varied.
-If successful, that would conrm the lexicon structure hypothesis (Byers-Heinlein
-and Werker, 2013) but providing a deeper understanding. These are just examples
-of experiments that could be carried out.
-4.3.3 Towards a mathematical theory of language eciency
-Our past and current research on the eciency are supported by a cost function
-and a (analytical or numerical) mathematical procedure that links the minimiza-
-tion of the cost function with the target phenomena, e.g., vocabulary learning,
-as in research on how pressure for eciency gives rise to Zipf's rank-frequency
-law, the law of abbreviation or Menzerath's law (Ferrer-i-Cancho, 2005b; Gusti-
-son et al., 2016; Ferrer-i-Cancho et al., 2019). In the cognitive sciences, such a
-cost function and the mathematical linking argument are sometimes missing (e.g.,
-Piantadosi et al., 2011) and neglected when reviewing how languages are shaped
-by eciency (Gibson et al., 2019). A truly quantitative approach in the context
-of language eciency is two-fold: it has to comprise either a quantitative descrip-
-tion of the data and a quantitative theorizing, i.e. it has to employ both statistical
-methods of analysis and mathematical methods to dene the cost and the how cost
-minimization leads to the expected phenomena. Our framework relies on standard
-information theory (Cover and Thomas, 2006) and its extensions (Ferrer-i-Cancho
-et al., 2019; Debowski, 2020). The psychological foundations of the information
-theoretic principles postulated in that framework and the relationships between
-them have already been reviewed (Ferrer-i-Cancho, 2018). How the so-called noisy-
-channel \theory" or noisy-channel hypothesis explains the results in (Piantadosi
-et al., 2011), others reviewed recently (Gibson et al., 2019) or language laws in a
-broad sense has not yet shown, to our knowledge, with detailed enough information
-theory arguments. Furthermore, the major conclusions of the statistical analysis
-of (Piantadosi et al., 2011) have recently been shown to change substantially after
-improving the methods: eects attributable to plain compression are stronger than
-previously reported (Meylan and Griths, 2021). Theory is crucial to reduce false
-positives and replication failures (Stewart and Plotkin, 2021). In addition, higher
-order compression can explain more parsimoniously phenomena that are central
-in noisy-channel \theorizing" (Ferrer-i-Cancho, 2017b).The advent and fall of a vocabulary learning bias from communicative eciency 35
-4.3.4 The trade-o between parsimony and perfect t.
-Our emphasis is on generality and parsimony over perfect t. Piantadosi (2014)
-makes emphasis on what models of Zipf's rank-frequency law apparently do not
-explain while our emphasis is on what the models do explain and the many predic-
-tions they make (Table 1), in spite of their simple design. It is worth reminding a
-big lesson from machine learning, i.e. a perfect t can be obtained simply by over-
-tting the data and another big lesson from the philosophy of science to machine
-learning and AI: sophisticated models (specially deep learning ones) are in most
-cases black boxes that imitate complex behavior but neither explain nor yield un-
-derstanding. In our theoretical framework, the principle of contrast (Clark, 1987)
-or the mutual exclusivity bias (Markman and Wachtel, 1988; Merriman and Bow-
-man, 1989) are not principles per se (or core principles) but predictions of the prin-
-ciple of mutual information maximization involved in explaining the emergence of
-Zipf's rank-frequency law (Ferrer-i-Cancho and Sole, 2003; Ferrer-i-Cancho, 2005b)
-and word order patterns (Ferrer-i-Cancho, 2017b). Although there are computa-
-tional models that are able to account for that vocabulary learning bias and other
-phenomena (Frank et al., 2009; Gulordava et al., 2020), ours is much simpler,
-transparent (in opposition to black box modeling) and to the best our knowledge,
-the rst to predict that the bias will weaken over time providing a preliminary
-understanding of why this could happen.
-Acknowledgements We are grateful to two anonymous reviewers for their valuable feeback
-and recommendations to improve the article. We are also grateful to A. Hern andez-Fern andez
-and G. Boleda for their revision of the article and many recommendations to improve it. The
-article has beneted from discussions with T. Brochhagen, S. Semple and M. Gustison. Finally,
-we thank C. Hobaiter for her advice and inspiration for future research. DCC and RFC are
-supported by the grant TIN2017-89244-R from MINECO (Ministerio de Econom a, Industria
-y Competitividad). RFC is also supported by the recognition 2017SGR-856 (MACDA) from
-AGAUR (Generalitat de Catalunya).
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-A The mathematical model in detail
-This appendix is organized as follows. Section A.1 details the expressions for probabilities and
-entropies introduced in Section 2. Section A.2 addresses the general problem of the dynamic
-calculation of 
-(Eq. 8) when a cell of the adjacency matrix is mutated, deriving the formulae
-to update these entropies once a single mutation has taken place. Finally, Section A.3 applies
-these formulae to derive the expressions for 
presented in Section 2.1.40 David Carrera-Casado, Ramon Ferrer-i-Cancho
-A.1 Probabilities and entropies
-In section 2, we obtained an expression for the joint probability of a form and a counterpart (Eq.
-6) and the corresponding normalization factor, M(Eq. 7). Notice that M0is the number of
-edges of the bipartite graph. i.e. M=M0. To ease the derivation of the marginal probabilities,
-we dene
-;i=mX
-j=1aij!
-j(18)
-!;i=nX
-i=1aij
-i: (19)
-Notice that ;iand!;jshould not be confused with iand!i(the degree of the form iand
-of the counterpart jrespectively). Indeed, i=0;iand!j=!0;j. From the joint probability
-(Eq. 6), we obtain the marginal probabilities
-p(si) =mX
-j=1p(si;rj)
-=
-i;i
-M(20)
-p(rj) =nX
-i=1p(si;rj)
-=!
-j!;j
-M: (21)
-To obtain expressions for the entropies, we use the rule
-X
-ixi
-Tlogxi
-T= logT1
-TX
-ixilogxi; (22)
-which holds whenP
-ixi=T.
-We can now derive the entropies H(S;R),H(S) andH(R). Applying Eq. 6 to
-H(S;R) =nX
-i=1mX
-j=1p(si;rj) logp(si;rj);
-we obtain
-H(S;R) = logM
-MnX
-i=1mX
-j=1aij(i!j)log(i!j):
-Applying Eq. 20 and the rule in Eq. 22,
-H(S) =nX
-i=1p(si) logp(si)
-becomes
-H(S) = logM1
-MnX
-i=1
-
-i;i
-log
-
-i;i
-:
-By symmetry, equivalent formulae for H(R) can be derived easily using Eq. 21, obtaining
-H(R) = logM1
-MmX
-j=1
-!
-j!;j
-log
-!
-j!;j
-:The advent and fall of a vocabulary learning bias from communicative eciency 41
-Interestingly, when = 0, the entropies simplify as
-H(S;R) = logM0
-H(S) = logM01
-M0nX
-i=1ilogi
-H(R) = logM01
-M0mX
-j=1!ilog!i
-as expected from previous work (Ferrer-i-Cancho, 2005b). Given the formulae for H(S;R),
-H(S) andH(R) above, the calculation of 
-() (Eq. 9) is straightforward.
-A.2 Change in entropies after a single mutation in the adjacency matrix
-Here we investigate a general problem: the change in the entropies needed to calculate 
-when
-there is a single mutation in the cell ( i;j) of the adjacency matrix, i.e. when a link between
-a formiand a counterpart jis added (aijbecomes 1) or deleted ( aijbecomes 0). The goal
-of this analysis is to provide the mathematical foundations for research on the evolution of
-communication and in particular, the problem of learning of a new word, i.e. linking a form
-that was previously unlinked (Appendix A.3), which is a particular case of mutation where
-aij= 0 andi= 0 before the mutation ( aij= 1 andi= 1 after the mutation).
-Firstly, we express the entropies compactly as
-H(S;R) = logM
-MX(S;R) (23)
-H(S) = logM1
-MX(S) (24)
-H(R) = logM1
-MX(R) (25)
-with
-X(S;R) =X
-(i;j)2Ex(si;rj)
-X(S) =nX
-i=1x(si)
-X(R) =mX
-j=1x(rj) (26)
-x(si;rj) = (i!j)log(i!j) (27)
-x(si) =
-
-i;i
-log
-
-i;i
-(28)
-x(rj) =
-!
-j!;j
-log
-!
-j!;j
-: (29)
-We will use a prime mark to indicate the new value of a certain measure once a mutation has
-been produced in the adjacency matrix. Suppose that aijmutates. Then
-a0
-ij= 1aij
-0
-i=i+ (1)aij (30)
-!0
-j=!j+ (1)aij: (31)
-We deneS(i) as the set of neighbors of siin the graph and, similarly, R(j) as the set of
-neighbors of rjin the graph. Then 0
-;kcan only change if k=iork2R(j) (recall Eq. 18)42 David Carrera-Casado, Ramon Ferrer-i-Cancho
-and!0
-;lcan only change if l=jorl2R(i) (Eq. 19). Then, for any ksuch that 1kn,
-we have that
-0
-;k=8
-><
->:;kaij!
-j+ (1aij)!0
-jifk=i
-;k!
-j+!0
-jifk2R(j)
-;k otherwise:(32)
-Likewise, for any lsuch that 1lm, we have that
-!0
-;l=8
-<
-:!;laij
-i+ (1aij)0
-iifl=j
-!;l
-i+0
-iifl2S(i)
-!;l otherwise:(33)
-We then aim to calculate M0
-andX0(S;R) fromMandX(S;R) (Eq. 7 and Eq. 23) respec-
-tively. Accordingly, we focus on the pairs ( sk;rl), shortly (k;l), such that 0
-k!0
-l=k!lmay
-not hold. These pairs belong to E(i;j)[(i;j), whereE(i;j) is the set of edges having sior
-rjat one of the ends. That is, E(i;j) is the set of edges of the form ( i;l) wherel2S(i) or
-(k;j) wherek2R(j). Then the new value of Mwill be
-M0
-=M2
-4X
-(k;l)2E(i;j)(k!l)3
-5aij(i!j)
-+2
-4X
-(k;l)2E(i;j)(0
-k!0
-l)3
-5+ (1aij)(0
-i!0
-j):(34)
-Similarly, the new value of X(S;R) will be
-X0(S;R) =X(S;R)2
-4X
-(k;l)2E(i;j)x(sk;rl)3
-5aijx(si;rj)
-+2
-4X
-(k;l)2E(i;j)x0(sk;rl)3
-5+ (1aij)x0(si;rj):(35)
-x0(si;rj) can be obtained by applying 0
-iand!0
-j(Eqs. 30 and 31) to x(si;rj) (Eq. 27). The
-value ofH0(S;R) is then obtained applying M0
-(Eq. 34) and X(S;R)0(Eq. 35) to H(S;R)
-(Eq. 23).
-As forH0(S), notice that x0(sk) can only dier from x(sk) if0
-kand0
-;kchange, namely
-whenk=iork2R(j). Therefore
-X0(S) =X(S)2
-4X
-k2R(j)x(sk)3
-5aijx(si) +2
-4X
-k2R(j)x0(sk)3
-5+ (1aij)x0(si):(36)
-Similarly,x0(si) can be obtained by applying i(Eq. 30) and ;i(Eq. 32) to x(si) (Eq 28).
-ThenH0(S) is obtained by applying MandX0(S) (Eqs. 34 and 36) to H(S) (Eq. 24). By
-symmetry,
-X0(R) =X(R)2
-4X
-l2S(i)x(rl)3
-5aijx(rj) +2
-4X
-l2S(i)x0(rl)3
-5+ (1aij)x0(rj); (37)
-wherex0(rj) andH0(R) are obtained similarly, applying !j(Eq. 33)x(rj) (Eq. 29) and nally
-H(R) (Eq. 25).The advent and fall of a vocabulary learning bias from communicative eciency 43
-A.3 Derivation of 

-Following from the previous sections, we set o to obtain expressions for 
for each of the
-skeleton classes we set out to study. As before, we denote the value of a variable after applying
-either strategy with a prime mark, meaning that it is a modied value after a mutation in the
-adjacency matrix. We also use a subindex aorbto indicate the vocabulary learning strategy
-corresponding to the mutation. A value without prime mark then denotes the state of that
-variable before applying either strategy.
-Firstly, we aim to obtain an expression for 
that depends on the new values of the
-entropies after either strategy aorbhas been chosen. Combining 
() (Eq. 10) with 
-()
-(Eq. 9), one obtains
-
() = (12)(H0
-a(S)H0
-b(S))(H0
-a(R)H0
-b(R)) +(H0
-a(S;R)H0
-b(S;R)):
-The application of H(S;R) (Eq. 23), H(S) (Eq. 24) and H(R) (Eq. 25), yields
-
() = (12) logM0
-a
-M0
-b1
-M0
-aM0
-b
-(12)
X(S)
X(R)+
X(S;R)
-(38)
-with
-
X(S)=M0
-bX0
-a(S)M0
-aX0
-b(S)
-
X(R)=M0
-bX0
-a(R)M0
-aX0
-b(R)
-
X(S;R)=M0
-bX0
-a(S;R)M0
-aX0
-b(S;R):
-Now we nd expressions for M0
-a,X0
-a(S;R),X0
-a(S),X0
-a(R),M0
-b,X0
-b(S;R),X0
-b(S),X0
-b(R).
-To obtain generic expressions for M0
-,X0(S;R),X0(S) andX0(R) via Eqs. 34, 35, 36 and 37,
-we dene mathematically the state of the bipartite matrix before and after applying either
-strategyaorbwith the following restrictions
-{aija=aijb= 0. Formiand counterpart jare initially unconnected.
-{ia=ib= 0. Formihas initially no connections.
-{0
-ia=0
-ib= 1. Formiwill have one connection afterwards.
-{!ja= 0. In case a, counterpart jis initially disconnected.
-{!jb=!j>0. In caseb, counterpart jhas initially at least one connection.
-{!0
-ja= 1. In case a, counterpart jwill have one connection afterwards.
-{!0
-jb=!j+ 1. In case b, counterpart jwill have one more connection afterwards.
-{Sa(i) =Sb(i) =?. Formihas initially no neighbors.
-{Ra(j) =?. In casea, counterpart jhas initially no neighbors.
-{Rb(j)6=?. In caseb, counterpart jhas initially some neighbors.
-{Ea(i;j) =?. In casea, there are no links with iorjat one of their ends.
-{Eb(i;j) =f(k;j)jk2R(j)g. In caseb, there are no links with iat one of their ends, only
-withj.
-We can apply these restrictions to x(si;rj),x(si) andx(rj) (Eqs. 27, 28 and 29) to obtain
-expressions of x0
-a(si),x0
-b(si),x0
-b(rj) andx0
-b(si;rj) that depend only on the initial values of !j
-and!;j
-x0(si) =!0
-jlog!0
-j
-x0
-a(si) = 0 (39)
-x0
-a(rj) = 0 (40)
-x0
-b(si) =(!j+ 1)log(!j+ 1) (41)
-x0
-b(rj) = (!j+ 1)(!;j+ 1) log((!j+ 1)(!;j+ 1)) (42)
-x0
-a(si;rJ) = 0 (43)
-x0
-b(si;rj) = (!j+ 1)log(!j+ 1): (44)44 David Carrera-Casado, Ramon Ferrer-i-Cancho
-Additionally, for any forms sksuch thatk2Rb(j) (that is, for every form that counterpart
-jis connected to), we can also obtain expressions that depend only on the initial values of !j,
-!;j,kand;kusing the same restrictions and equations
-xb(sk;rj) =!
-j(
-klogk) + (!
-jlog!j)
-k(45)
-x0
-b(sk;rj) = (!j+ 1)(
-klogk) +h
-(!j+ 1)log(!j+ 1)i
-
-k(46)
-x0
-b(sk) =n
-
-k;k+
-kh
-!
-j+ (!j+ 1)io
-
log"
-(
-k;k) 
-;k!
-j+ (!j+ 1)
-;k!#
-=sb(sk) +h
-(!j+ 1)!
-ji
-
-klogn
-
-kh
-;k!
-j+ (!j+ 1)io
-+
-k;klog 
-;k!
-j+ (!j+ 1)
-;k!
-:(47)
-Applying the restrictions to M0
-(Eq. 34), we can also obtain an expression that depends only
-on some initial values
-M0
-a=M+ 1 (48)
-M0
-b=M+h
-(!j+ 1)!
-ji
-!;j+ (!j+ 1): (49)
-Applying now the expressions for x0
-a(si;rj) (Eq. 43), x0
-b(si;rj) (Eq. 44), xb(sk;rj) (Eq. 45)
-andx0
-b(sk;rj) (Eq. 46) to X0(S;R) (Eq. 35), along with the restrictions, we obtain
-X0
-a(S;R) =X(S;R) (50)
-X0
-b(S;R) =X(S;R) +h
-(!j+ 1)!
-jiX
-k2R(j)
-klogk
-+!;jh
-(!j+ 1)log(!j+ 1)!
-jlog(!j)i
-+ (!j+ 1)log(!j+ 1):(51)
-Similarly, we apply x0
-a(si) (Eq. 39), x0
-b(si) (Eq. 41) and x0
-b(sk) (Eq. 47) to X0(S) (Eq. 36) as
-well as the restrictions and obtain
-X0
-a(S) =X(S) (52)
-X0
-b(S) =X(S) +(!j+ 1)log(!j+ 1)
-+h
-(!j+ 1)!
-jiX
-k2R(j)
-klogn
-
-kh
-;k!
-j+ (!j+ 1)io
-+X
-k2R(j)
-k;klog 
-;k!
-j+ (!j+ 1)
-;k!
-:(53)
-We applyx0
-a(rj) (Eq. 40) and x0
-b(rj) (Eq. 42) to X0(R) (Eq. 37) along with the restrictions
-and obtain
-X0
-a(R) =X(R) (54)
-X0
-b(R) =X(R)!
-j!;jlog(!
-j!;j)
-+ (!j+ 1)(!;j+ 1) logh
-(!j+ 1)(!;j+ 1)i
-:(55)
-At this point we could attempt to build an expression for 
for the most general case. However,
-this expression would be extremely complex. Instead, we study the expression of 
in three
-simplifying conditions: the case = 0 and the two classes of skeleta.The advent and fall of a vocabulary learning bias from communicative eciency 45
-A.3.1 The case = 0
-The condition = 0 corresponds to a model that is a precursor of the current model Ferrer-
-i-Cancho (2017a), and that we use to ensure our that our general expressions are correct. We
-apply= 0 to the expressions in Section A.3. M0
-aandM0
-b(Eqs. 48 and 49) both simplify as
-M0
-a=M0
-b=M+ 1: (56)
-X0
-a(S;R) andX0
-b(S;R) (Eqs. 50 and 51) simplify as
-X0
-a(S;R) =X(S;R) (57)
-X0
-b(S;R) =X(S;R) + (!j+ 1) log(!j+ 1)!jlog(!j): (58)
-X0
-a(S) andX0
-b(S) (Eqs. 52 and 53) both simplify as
-X0
-a(S) =X0
-b(S) =X(S): (59)
-X0
-a(R) andX0
-b(R) (Eqs. 54 and 55) simplify as
-X0
-a(R) =X(R) (60)
-X0
-b(R) =X(R)!jlog(!j) + (!j+ 1) log(!j+ 1): (61)
-The application of Eqs. 56, 57, 58, 59, 60 and 61 into the expression of 
(Eq. 38) results in
-the expression for 
(Eq. 5) presented in Section 1.
-A.3.2 Counterpart degrees do not exceed one
-In this case we assume that !j2f0;1gfor everyrjand further simplify the expressions from
-A.3 under this assumption. This is the most relaxed of the conditions and so these expressions
-remain fairly complex.
-M0
-aandM0
-b(Eqs. 48 and 49) simplify as
-M0
-a=M+ 1 (62)
-M0
-b=M+ (21)
-k+ 2(63)
-with
-M=nX
-i=1+1
-i;
-X0
-a(S;R) andX0
-b(S;R) (Eqs. 50 and 51) simplify as
-X0
-a(S;R) =X(S;R) (64)
-X0
-b(S;R) =X(S;R) + (21)
-klogk+ (
-k+ 1)2log 2 (65)
-with
-X(S;R) =nX
-i=1+1
-ilogi: (66)
-X0
-a(S) andX0
-b(S) (Eqs. 52 and 53) simplify as
-X0
-a(S) =X(S) (67)
-X0
-b(S) =X(S) + (21)
-klogh
-
-k(k1 + 2)i
-++1
-klogk1 + 2
-k
-+2log(2)(68)46 David Carrera-Casado, Ramon Ferrer-i-Cancho
-with
-X(S) =nX
-i=1
-i;ilog(
-i;i)
-=nX
-i=1
-iilog(
-ii)
-= (+ 1)nX
-i=1+1
-ilogi
-= (+ 1)X(S;R):
-X0
-a(R) andX0
-b(R) (Eqs. 54 and 55) simplify as
-X0
-a(R) =X(R) (69)
-X0
-b(R) =X(R)
-klog(k) + 2(
-k+ 1) logh
-2(
-k+ 1)i
-(70)
-with
-X(R) =X(S;R): (71)
-The previous result on X(R) deserves a brief explanation as it is not straightforward. Firstly,
-we apply the denition of x(rj) (Eq. 29) to that of X(R) (Eq. 26)
-X(R) =mX
-j=1!
-j!;jlog(!
-j!;j):
-As counterpart degrees are one, !j= 1 and!;j=
-i j, wherei jis used to indicate that
-we refer to the form ithat the counterpart jis connected to (see Eq. 19). That leads to
-X(R) =mX
-j=1
-i jlog(i j):
-In order to change the summation over each j(every counterpart) to a summation over each
-i(every form) we must take into account that when summing over j, we accounted for each
-formia total ofitimes. Therefore we need to multiply by iin order for the summations to
-be equivalent, as otherwise we would be accounting for each form ionly once. This leads to
-X(R) =nX
-i=1+1
-ilogi
-and eventually Eq. 71 thanks to Eq. 66.
-The application of Eqs. 62, 63, 64, 65, 67, 68, 69 and 70 into the expression of 
(Eq. 38)
-results in the expression for 
(Eq. 12) presented in Section 1. If we apply the two extreme
-values of, i.e.= 0 and= 1, to that equation, we obtain the following expressions
-
(0) = log 
-M+ 1
-M+ (21)
-k+ 2!
-+1
-M+ (21)
-k+ 2(
-2
-klog(k)
-h
-(+ 1)X(S;R)(21)(
-k+ 1)
-M+ 12log(2)
-+
-kh
-log(k)(k+)(k1 + 2) log(k1 + 2)ii)The advent and fall of a vocabulary learning bias from communicative eciency 47
-
(1) =log 
-M+ 1
-M+ (21)
-k+ 2!
-1
-M+ (21)
-k+ 2(
-(
-k+ 1)2log(
-k+ 1)
-h
-(+ 1)X(S;R)(21)(
-k+ 1)
-M+ 12log(2)
-+
-kh
-log(k)(k+)(k1 + 2) log(k1 + 2)ii)
-:
-A.3.3 Vertex degrees do not exceed one
-As seen in Section 2.1, for this class we are working under the two conditions that !j2f0;1g
-for everyrjandi2f0;1gfor everysi. We can simplify the expressions from A.3. M0
-aand
-M0
-b(Eqs. 62 and 63) simplify as
-M0
-a=M+ 1 (72)
-M0
-b=M+ 2+11; (73)
-whereM=M0=M, the number of edges in the bipartite graph. X0
-a(S;R) andX0
-b(S;R)
-(Eqs. 64 and 65) simplify as
-X0
-a(S;R) = 0 (74)
-X0
-b(S;R) = 2+1log 2: (75)
-X0
-a(S) andX0
-b(S) (Eqs. 67 and 68) simplify as
-X0
-a(S) = 0 (76)
-X0
-b(S) =2+1log 2: (77)
-X0
-a(R) andX0
-b(R) (Eqs. 69 and 70) simplify as
-X0
-a(R) = 0 (78)
-X0
-b(R) = (+ 1)2+1log 2: (79)
-Combining Eqs. 72, 73, 74, 75, 76, 77, 78, 79 into the equation for 
(Eq. 38) results in the
-expression for 
(Eq.11) presented in Section 1. When the extreme values, i.e. = 0 and
-= 1, are applied to this equation, we obtain the following expressions
-
(0) =log
-1 +2(21)
-M+ 1
-+2+1log(2)
-M+ 2+11
-
(1) = log
-1 +2(21)
-M+ 1
-2+1(+ 1) log(2)
-M+ 2+11:
-B Form degrees and number of links
-Here we develop the implications of Eq. 15 with n1= 1 andn= 0. Imposing n1= 1,
-we get
-c= (n1):
-Inserting the previous results into the denition of p(si) when!j1, we have that
-p(si) =1
-M+1
-i
-=c0i;48 David Carrera-Casado, Ramon Ferrer-i-Cancho
-with
-=(+ 1)
-c0=(n1)
-M:
-A continuous approximation to vertex degrees and the number of edges gives
-M=nX
-i=1i
-=cn1X
-i=1i
-= (n1)n1X
-i=1i:
-Thanks to well-known integral bounds (Cormen et al., 1990, pp. 50-51), we have that
-Zn
-1idin1X
-i=1i1 +Zn1
-1idi:
-as0 by denition. When = 1, one obtains
-lognn1X
-i=1i11 + log(n1):
-When6= 1, one obtains
-1
-1
-1n1
-n1X
-i=1i1 +1
-1
-1(n1)1
-:
-Combining the results above, one obtains
-(n1) lognM(n1)[1 + log(n1)]
-for= 1 and
-(n1)1
-1
-1n1
-M(n1)
-1 +1
-1
-1(n1)1
-for6= 1.
-C Complementary heatmaps for other values of 
-In Section 3, heatmaps were used to analyze 
takes for distinct sets of parameters. For the
-class of skeleta where counterpart degrees do not exceed one, only heatmaps corresponding to
-= 0 (Fig. 9) and = 1 (Figs. 10, 12 and 14) were presented. The summary gures presented
-in that same section (Figs. 11, 13 and 15) already displayed the boundaries between positive
-and negative values of 
for the whole range of values of . Heatmaps for the remainder of
-values ofare presented next.
-Heatmaps of 
as a function of andkFigures 16, 17, 18 and 19 vary kon they-axis
-(while keeping on thex-axis, as with all others) and correspond to values of = 0:5,= 1:5,
-= 2 and= 2:5 respectively.
-Heatmaps of 
as a function of andFigures 20, 21, 22 and 23 vary on they-axis
-and correspond to values of = 0:5,= 1:5,= 2 and= 2:5 respectively.
-Heatmaps of 
as a function of andnFigures 24, 25, 26 and 27 vary non they-axis
-and correspond to values of = 0:5,= 1:5,= 2 and= 2:5 respectively.The advent and fall of a vocabulary learning bias from communicative eciency 49
-1.21.51.82.1
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.16-0.12-0.08-0.04Δ < 0
-0.0040.0080.0120.016Δ ≥ 0φ = 0.5 α = 0.5 n = 10(a)
-1234
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.09-0.06-0.03Δ < 0
-0.00250.00500.00750.01000.0125Δ ≥ 0φ = 0.5 α = 1 n = 10(b)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.06-0.04-0.02Δ < 0
-0.0010.0020.0030.004Δ ≥ 0φ = 0.5 α = 1.5 n = 10(c)
-1234
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.020-0.015-0.010-0.005Δ < 0
-0.0020.0040.006Δ ≥ 0φ = 0.5 α = 0.5 n = 100(d)
-05101520
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.0125-0.0100-0.0075-0.0050-0.0025Δ < 0
-0.0010.0020.0030.0040.005Δ ≥ 0φ = 0.5 α = 1 n = 100(e)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.003-0.002-0.001Δ < 0
-0.00050.00100.0015Δ ≥ 0φ = 0.5 α = 1.5 n = 100(f)
-2.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.003-0.002-0.001Δ < 0
-0.00040.00080.0012Δ ≥ 0φ = 0.5 α = 0.5 n = 1000(g)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.0020-0.0015-0.0010-0.0005Δ < 0
-0.00040.00080.0012Δ ≥ 0φ = 0.5 α = 1 n = 1000(h)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λμk
--3e-04-2e-04-1e-04Δ < 0
-1e-042e-04Δ ≥ 0φ = 0.5 α = 1.5 n = 1000(i)
-Fig. 16 Same as in Fig. 9 but with = 0:5.50 David Carrera-Casado, Ramon Ferrer-i-Cancho
-1.01.21.4
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.4-0.3-0.2-0.1Δ < 0
-0.050.100.15Δ ≥ 0φ = 1.5 α = 0.5 n = 10(a)
-1.01.52.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.3-0.2-0.1Δ < 0
-0.040.080.120.16Δ ≥ 0φ = 1.5 α = 1 n = 10(b)
-123
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.2-0.1Δ < 0
-0.050.10Δ ≥ 0φ = 1.5 α = 1.5 n = 10(c)
-1.01.52.02.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.100-0.075-0.050-0.025Δ < 0
-0.020.040.06Δ ≥ 0φ = 1.5 α = 0.5 n = 100(d)
-246
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.125-0.100-0.075-0.050-0.025Δ < 0
-0.0250.0500.0750.100Δ ≥ 0φ = 1.5 α = 1 n = 100(e)
-481216
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.06-0.04-0.02Δ < 0
-0.020.040.06Δ ≥ 0φ = 1.5 α = 1.5 n = 100(f)
-1234
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.020-0.015-0.010-0.005Δ < 0
-0.0050.0100.015Δ ≥ 0φ = 1.5 α = 0.5 n = 1000(g)
-481216
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.05-0.04-0.03-0.02-0.01Δ < 0
-0.010.020.030.04Δ ≥ 0φ = 1.5 α = 1 n = 1000(h)
-0204060
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.020-0.015-0.010-0.005Δ < 0
-0.0050.0100.0150.020Δ ≥ 0φ = 1.5 α = 1.5 n = 1000(i)
-Fig. 17 Same as in Fig. 9 but with = 1:5.The advent and fall of a vocabulary learning bias from communicative eciency 51
-1.01.11.21.31.4
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.5-0.4-0.3-0.2-0.1Δ < 0
-0.10.2Δ ≥ 0φ = 2 α = 0.5 n = 10(a)
-1.21.51.82.1
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.5-0.4-0.3-0.2-0.1Δ < 0
-0.10.20.3Δ ≥ 0φ = 2 α = 1 n = 10(b)
-1.01.52.02.53.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.4-0.3-0.2-0.1Δ < 0
-0.10.2Δ ≥ 0φ = 2 α = 1.5 n = 10(c)
-1.001.251.501.752.00
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.15-0.10-0.05Δ < 0
-0.050.10Δ ≥ 0φ = 2 α = 0.5 n = 100(d)
-1234
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.25-0.20-0.15-0.10-0.05Δ < 0
-0.050.100.150.20Δ ≥ 0φ = 2 α = 1 n = 100(e)
-2.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.15-0.10-0.05Δ < 0
-0.050.100.15Δ ≥ 0φ = 2 α = 1.5 n = 100(f)
-1.01.52.02.53.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.05-0.04-0.03-0.02-0.01Δ < 0
-0.010.020.030.04Δ ≥ 0φ = 2 α = 0.5 n = 1000(g)
-2.55.07.510.0
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.125-0.100-0.075-0.050-0.025Δ < 0
-0.0250.0500.0750.1000.125Δ ≥ 0φ = 2 α = 1 n = 1000(h)
-0102030
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.06-0.04-0.02Δ < 0
-0.020.040.06Δ ≥ 0φ = 2 α = 1.5 n = 1000(i)
-Fig. 18 Same as in Fig. 9 but with = 2.52 David Carrera-Casado, Ramon Ferrer-i-Cancho
-1.01.11.21.3
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.8-0.6-0.4-0.2Δ < 0
-0.10.20.30.4Δ ≥ 0φ = 2.5 α = 0.5 n = 10(a)
-1.001.251.501.75
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.6-0.4-0.2Δ < 0
-0.10.20.30.4Δ ≥ 0φ = 2.5 α = 1 n = 10(b)
-1.01.52.02.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.6-0.4-0.2Δ < 0
-0.10.20.30.4Δ ≥ 0φ = 2.5 α = 1.5 n = 10(c)
-1.001.251.501.75
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.3-0.2-0.1Δ < 0
-0.050.100.150.200.25Δ ≥ 0φ = 2.5 α = 0.5 n = 100(d)
-123
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.4-0.3-0.2-0.1Δ < 0
-0.10.20.30.4Δ ≥ 0φ = 2.5 α = 1 n = 100(e)
-246
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.3-0.2-0.1Δ < 0
-0.10.20.3Δ ≥ 0φ = 2.5 α = 1.5 n = 100(f)
-1.01.52.02.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.075-0.050-0.025Δ < 0
-0.020.040.060.08Δ ≥ 0φ = 2.5 α = 0.5 n = 1000(g)
-246
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.2-0.1Δ < 0
-0.050.100.150.200.25Δ ≥ 0φ = 2.5 α = 1 n = 1000(h)
-5101520
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.15-0.10-0.05Δ < 0
-0.050.100.15Δ ≥ 0φ = 2.5 α = 1.5 n = 1000(i)
-Fig. 19 Same as in Fig. 9 but with = 2:5.The advent and fall of a vocabulary learning bias from communicative eciency 53
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.100-0.075-0.050-0.025Δ < 0φ = 0.5 μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.006-0.004-0.002Δ < 0
-0.000250.000500.000750.00100Δ ≥ 0φ = 0.5 μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--6e-04-4e-04-2e-04Δ < 0
-0.000050.000100.00015Δ ≥ 0φ = 0.5 μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.12-0.08-0.04Δ < 0
-0.0050.010Δ ≥ 0φ = 0.5 μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.012-0.009-0.006-0.003Δ < 0
-0.000250.000500.00075Δ ≥ 0φ = 0.5 μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.00100-0.00075-0.00050-0.00025Δ < 0
-5e-051e-04Δ ≥ 0φ = 0.5 μk = 2 n = 1000(f)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.20-0.15-0.10-0.05Δ < 0
-0.020.040.06Δ ≥ 0φ = 0.5 μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.015-0.010-0.005Δ < 0
-0.0010.0020.0030.004Δ ≥ 0φ = 0.5 μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.0015-0.0010-0.0005Δ < 0
-1e-042e-043e-04Δ ≥ 0φ = 0.5 μk = 4 n = 1000(i)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.3-0.2-0.1Δ < 0
-0.050.100.15Δ ≥ 0φ = 0.5 μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.03-0.02-0.01Δ < 0
-0.0050.010Δ ≥ 0φ = 0.5 μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.002-0.001Δ < 0
-3e-046e-049e-04Δ ≥ 0φ = 0.5 μk = 8 n = 1000(l)
-Fig. 20 The same as in Fig. 12 but with = 0:5.54 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.20-0.15-0.10-0.05Δ < 0
-0.010.020.030.04Δ ≥ 0φ = 1.5 μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.016-0.012-0.008-0.004Δ < 0
-0.0020.0040.006Δ ≥ 0φ = 1.5 μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.0016-0.0012-0.0008-0.0004Δ < 0
-0.000250.000500.000750.00100Δ ≥ 0φ = 1.5 μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.6-0.5-0.4-0.3-0.2-0.1Δ < 0
-0.10.20.3Δ ≥ 0φ = 1.5 μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.06-0.04-0.02Δ < 0
-0.010.020.03Δ ≥ 0φ = 1.5 μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.006-0.005-0.004-0.003-0.002-0.001Δ < 0
-0.0010.002Δ ≥ 0φ = 1.5 μk = 2 n = 1000(f)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--1.5-1.0-0.5Δ < 0
-0.250.500.751.001.25Δ ≥ 0φ = 1.5 μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.20-0.15-0.10-0.05Δ < 0
-0.050.100.150.20Δ ≥ 0φ = 1.5 μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.020-0.015-0.010-0.005Δ < 0
-0.0050.0100.0150.020Δ ≥ 0φ = 1.5 μk = 4 n = 1000(i)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--3-2-1Δ < 0
-12Δ ≥ 0φ = 1.5 μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.8-0.6-0.4-0.2Δ < 0
-0.20.40.6Δ ≥ 0φ = 1.5 μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.075-0.050-0.025Δ < 0
-0.0250.0500.075Δ ≥ 0φ = 1.5 μk = 8 n = 1000(l)
-Fig. 21 The same as in Fig. 12 but with = 1:5.The advent and fall of a vocabulary learning bias from communicative eciency 55
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.3-0.2-0.1Δ < 0
-0.020.040.06Δ ≥ 0φ = 2 μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.03-0.02-0.01Δ < 0
-0.0030.0060.0090.012Δ ≥ 0φ = 2 μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.0025-0.0020-0.0015-0.0010-0.0005Δ < 0
-0.00040.00080.0012Δ ≥ 0φ = 2 μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--1.00-0.75-0.50-0.25Δ < 0
-0.20.40.6Δ ≥ 0φ = 2 μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.12-0.08-0.04Δ < 0
-0.0250.0500.0750.100Δ ≥ 0φ = 2 μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.015-0.010-0.005Δ < 0
-0.00250.00500.00750.0100Δ ≥ 0φ = 2 μk = 2 n = 1000(f)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--2.0-1.5-1.0-0.5Δ < 0
-0.51.01.52.0Δ ≥ 0φ = 2 μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.6-0.4-0.2Δ < 0
-0.20.40.6Δ ≥ 0φ = 2 μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.075-0.050-0.025Δ < 0
-0.020.040.060.08Δ ≥ 0φ = 2 μk = 4 n = 1000(i)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--3-2-1Δ < 0
-123Δ ≥ 0φ = 2 μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--2.5-2.0-1.5-1.0-0.5Δ < 0
-0.51.01.52.02.5Δ ≥ 0φ = 2 μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.4-0.3-0.2-0.1Δ < 0
-0.10.20.30.4Δ ≥ 0φ = 2 μk = 8 n = 1000(l)
-Fig. 22 The same as in Fig. 12 but with = 2.56 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.4-0.3-0.2-0.1Δ < 0
-0.050.10Δ ≥ 0φ = 2.5 μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.05-0.04-0.03-0.02-0.01Δ < 0
-0.0040.0080.0120.016Δ ≥ 0φ = 2.5 μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.004-0.003-0.002-0.001Δ < 0
-0.00050.00100.00150.0020Δ ≥ 0φ = 2.5 μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--1.0-0.5Δ < 0
-0.30.60.9Δ ≥ 0φ = 2.5 μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.3-0.2-0.1Δ < 0
-0.10.2Δ ≥ 0φ = 2.5 μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.03-0.02-0.01Δ < 0
-0.010.020.03Δ ≥ 0φ = 2.5 μk = 2 n = 1000(f)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--2-1Δ < 0
-12Δ ≥ 0φ = 2.5 μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--1.5-1.0-0.5Δ < 0
-0.51.01.5Δ ≥ 0φ = 2.5 μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.3-0.2-0.1Δ < 0
-0.10.20.3Δ ≥ 0φ = 2.5 μk = 4 n = 1000(i)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--3-2-1Δ < 0
-123Δ ≥ 0φ = 2.5 μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--4-3-2-1Δ < 0
-1234Δ ≥ 0φ = 2.5 μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--1.5-1.0-0.5Δ < 0
-0.51.01.5Δ ≥ 0φ = 2.5 μk = 8 n = 1000(l)
-Fig. 23 The same as in Fig. 12 but with = 2:5.The advent and fall of a vocabulary learning bias from communicative eciency 57
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.100-0.075-0.050-0.025Δ < 0φ = 0.5 μk = 1 α = 0.5(a)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.03-0.02-0.01Δ < 0
-0.000050.000100.000150.000200.00025Δ ≥ 0φ = 0.5 μk = 1 α = 1(b)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.03-0.02-0.01Δ < 0
-0.0010.0020.003Δ ≥ 0φ = 0.5 μk = 1 α = 1.5(c)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.12-0.08-0.04Δ < 0
-0.0050.010Δ ≥ 0φ = 0.5 μk = 2 α = 0.5(d)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.06-0.04-0.02Δ < 0
-2.5e-055.0e-057.5e-05Δ ≥ 0φ = 0.5 μk = 2 α = 1(e)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.02-0.01Δ < 0
-0.00050.00100.0015Δ ≥ 0φ = 0.5 μk = 2 α = 1.5(f)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.20-0.15-0.10-0.05Δ < 0
-0.020.040.06Δ ≥ 0φ = 0.5 μk = 4 α = 0.5(g)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.100-0.075-0.050-0.025Δ < 0
-0.0020.0040.0060.008Δ ≥ 0φ = 0.5 μk = 4 α = 1(h)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.02-0.01Δ < 0
-3e-046e-049e-04Δ ≥ 0φ = 0.5 μk = 4 α = 1.5(i)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.3-0.2-0.1Δ < 0
-0.050.100.15Δ ≥ 0φ = 0.5 μk = 8 α = 0.5(j)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.16-0.12-0.08-0.04Δ < 0
-0.010.020.030.040.050.06Δ ≥ 0φ = 0.5 μk = 8 α = 1(k)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.05-0.04-0.03-0.02-0.01Δ < 0
-2e-044e-046e-04Δ ≥ 0φ = 0.5 μk = 8 α = 1.5(l)
-Fig. 24 The same as in Fig. 14 but with = 0:5.58 David Carrera-Casado, Ramon Ferrer-i-Cancho
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.20-0.15-0.10-0.05Δ < 0
-0.00050.00100.00150.00200.0025Δ ≥ 0φ = 1.5 μk = 1 α = 0.5(a)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.08-0.06-0.04-0.02Δ < 0
-0.0020.0040.0060.008Δ ≥ 0φ = 1.5 μk = 1 α = 1(b)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.09-0.06-0.03Δ < 0
-0.010.020.030.04Δ ≥ 0φ = 1.5 μk = 1 α = 1.5(c)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.6-0.5-0.4-0.3-0.2-0.1Δ < 0
-0.10.20.3Δ ≥ 0φ = 1.5 μk = 2 α = 0.5(d)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.20-0.15-0.10-0.05Δ < 0
-0.020.040.06Δ ≥ 0φ = 1.5 μk = 2 α = 1(e)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.08-0.06-0.04-0.02Δ < 0
-0.0050.0100.0150.020Δ ≥ 0φ = 1.5 μk = 2 α = 1.5(f)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.5-1.0-0.5Δ < 0
-0.250.500.751.001.25Δ ≥ 0φ = 1.5 μk = 4 α = 0.5(g)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.8-0.6-0.4-0.2Δ < 0
-0.20.40.6Δ ≥ 0φ = 1.5 μk = 4 α = 1(h)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.3-0.2-0.1Δ < 0
-0.050.100.15Δ ≥ 0φ = 1.5 μk = 4 α = 1.5(i)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--3-2-1Δ < 0
-12Δ ≥ 0φ = 1.5 μk = 8 α = 0.5(j)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2.0-1.5-1.0-0.5Δ < 0
-0.51.01.52.0Δ ≥ 0φ = 1.5 μk = 8 α = 1(k)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.9-0.6-0.3Δ < 0
-0.250.500.751.00Δ ≥ 0φ = 1.5 μk = 8 α = 1.5(l)
-Fig. 25 The same as in Fig. 14 but with = 1:5.The advent and fall of a vocabulary learning bias from communicative eciency 59
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.3-0.2-0.1Δ < 0
-0.010.020.030.040.05Δ ≥ 0φ = 2 μk = 1 α = 0.5(a)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.075-0.050-0.025Δ < 0
-0.0030.0060.009Δ ≥ 0φ = 2 μk = 1 α = 1(b)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.15-0.10-0.05Δ < 0
-0.020.040.06Δ ≥ 0φ = 2 μk = 1 α = 1.5(c)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.00-0.75-0.50-0.25Δ < 0
-0.20.40.6Δ ≥ 0φ = 2 μk = 2 α = 0.5(d)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.5-0.4-0.3-0.2-0.1Δ < 0
-0.050.100.150.200.25Δ ≥ 0φ = 2 μk = 2 α = 1(e)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.075-0.050-0.025Δ < 0
-0.010.020.03Δ ≥ 0φ = 2 μk = 2 α = 1.5(f)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2.0-1.5-1.0-0.5Δ < 0
-0.51.01.52.0Δ ≥ 0φ = 2 μk = 4 α = 0.5(g)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.5-1.0-0.5Δ < 0
-0.51.01.5Δ ≥ 0φ = 2 μk = 4 α = 1(h)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.8-0.6-0.4-0.2Δ < 0
-0.20.40.6Δ ≥ 0φ = 2 μk = 4 α = 1.5(i)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--3-2-1Δ < 0
-123Δ ≥ 0φ = 2 μk = 8 α = 0.5(j)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--3-2-1Δ < 0
-123Δ ≥ 0φ = 2 μk = 8 α = 1(k)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2.0-1.5-1.0-0.5Δ < 0
-0.51.01.52.0Δ ≥ 0φ = 2 μk = 8 α = 1.5(l)
-Fig. 26 The same as in Fig. 14 but with = 2.60 David Carrera-Casado, Ramon Ferrer-i-Cancho
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.4-0.3-0.2-0.1Δ < 0
-0.050.10Δ ≥ 0φ = 2.5 μk = 1 α = 0.5(a)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.125-0.100-0.075-0.050-0.025Δ < 0
-0.0030.0060.0090.012Δ ≥ 0φ = 2.5 μk = 1 α = 1(b)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.15-0.10-0.05Δ < 0
-0.020.040.06Δ ≥ 0φ = 2.5 μk = 1 α = 1.5(c)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.0-0.5Δ < 0
-0.30.60.9Δ ≥ 0φ = 2.5 μk = 2 α = 0.5(d)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.8-0.6-0.4-0.2Δ < 0
-0.10.20.30.40.50.6Δ ≥ 0φ = 2.5 μk = 2 α = 1(e)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.2-0.1Δ < 0
-0.020.040.060.08Δ ≥ 0φ = 2.5 μk = 2 α = 1.5(f)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2-1Δ < 0
-12Δ ≥ 0φ = 2.5 μk = 4 α = 0.5(g)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2.0-1.5-1.0-0.5Δ < 0
-0.51.01.52.0Δ ≥ 0φ = 2.5 μk = 4 α = 1(h)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.2-0.8-0.4Δ < 0
-0.51.0Δ ≥ 0φ = 2.5 μk = 4 α = 1.5(i)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--4-3-2-1Δ < 0
-1234Δ ≥ 0φ = 2.5 μk = 8 α = 0.5(j)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--3-2-1Δ < 0
-123Δ ≥ 0φ = 2.5 μk = 8 α = 1(k)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2-1Δ < 0
-12Δ ≥ 0φ = 2.5 μk = 8 α = 1.5(l)
-Fig. 27 The same as in Fig. 14 but with = 2:5.The advent and fall of a vocabulary learning bias from communicative eciency 61
-D Complementary gures with discrete degrees
-To investigate the class of skeleta such that the degree of counterparts does not exceed one,
-we have assumed that the relationship between the degree of a vertex and its rank follows a
-power-law (Eq. 15). For the plots of the regions where strategy ais advantageous, we have
-assumed, for simplicity, that the degree of a form is a continuous variable. As form degrees are
-actually discrete in the model, here we show the impact of rounding form degrees dened by
-Eq. 15 to the nearest integer in previous gures.
-The correspondence between the gures in this appendix with rounded form degrees and
-the gures in other sections is as follows. Figs. 28, 29, 30, 31, 32 and 33 are equivalent to
-Figs. 9, 16, 10, 17, 18 and 19, respectively. These are the gures where is on thex-axis and
-kon they-axis of the heatmap. Fig. 34, that summarizes the boundaries of the heatmaps,
-corresponds to Fig. 11 after discretization. Figs. 35, 36, 37, 38 and 39 are equivalent to Figs.
-20, 12, 21, 22 and 23, respectively. In these gures, is placed on the y-axis instead. Fig.
-40 summarizes the boundaries and is the discretized version of Fig. 13. Finally, Fig. 41, 42,
-43, 44 and 45 are equivalent to Figs. 24, 14, 25, 26 and 27, respectively. This set places non
-they-axis. The boundaries in these last discretized gures are summarized by Fig. 46, that
-corresponds to Figure 15.
-We have presented two kinds of gures: heatmaps showing the value of 
and gures
-summarizing the boundaries between regions where 
 > 0 and
 < 0. Interestingly, the
-discretization does not change the presence of regions where 
< 0 and
> 0 and in general,
-it does not change the shape of the regions in a qualitative sense except in some cases where
-remarkable distortions appear (e.g., Figs. 32 or 33 have one or very few integer values on
-they-axis for certain combinations of parameters, forming one dimensional bands that don't
-change over that axis; see also the distorted shapes in Figs. 38 and specially 45). In contrast,
-123
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.075-0.050-0.025Δ < 0
-0Δ ≥ 0φ = 0 α = 0.5 n = 10(a)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.05-0.04-0.03-0.02-0.01Δ < 0
-0Δ ≥ 0φ = 0 α = 1 n = 10(b)
-01020
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.025-0.020-0.015-0.010-0.005Δ < 0
-0Δ ≥ 0φ = 0 α = 1.5 n = 10(c)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.006-0.004-0.002Δ < 0
-0Δ ≥ 0φ = 0 α = 0.5 n = 100(d)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.002-0.001Δ < 0
-0Δ ≥ 0φ = 0 α = 1 n = 100(e)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λμk
--5e-04-4e-04-3e-04-2e-04-1e-04Δ < 0
-0Δ ≥ 0φ = 0 α = 1.5 n = 100(f)
-0102030
-0.00 0.25 0.50 0.75 1.00
-λμk
--6e-04-4e-04-2e-04Δ < 0
-0Δ ≥ 0φ = 0 α = 0.5 n = 1000(g)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.00015-0.00010-0.00005Δ < 0
-0Δ ≥ 0φ = 0 α = 1 n = 1000(h)
-0100002000030000
-0.00 0.25 0.50 0.75 1.00
-λμk
--1.6e-05-1.2e-05-8.0e-06-4.0e-06Δ < 0
-0Δ ≥ 0φ = 0 α = 1.5 n = 1000(i)
-Fig. 28 Figure equivalent to Fig. 9 after discretization of the 0
-is.62 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0.51.01.52.02.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.15-0.10-0.05Δ < 0
-0.0050.0100.015Δ ≥ 0φ = 0.5 α = 0.5 n = 10(a)
-1234
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.09-0.06-0.03Δ < 0
-0.00250.00500.0075Δ ≥ 0φ = 0.5 α = 1 n = 10(b)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.06-0.04-0.02Δ < 0
-0.0010.0020.0030.0040.005Δ ≥ 0φ = 0.5 α = 1.5 n = 10(c)
-1234
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.015-0.010-0.005Δ < 0
-0.0010.0020.0030.004Δ ≥ 0φ = 0.5 α = 0.5 n = 100(d)
-05101520
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.0125-0.0100-0.0075-0.0050-0.0025Δ < 0
-0.0010.0020.0030.0040.005Δ ≥ 0φ = 0.5 α = 1 n = 100(e)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.003-0.002-0.001Δ < 0
-0.00050.00100.0015Δ ≥ 0φ = 0.5 α = 1.5 n = 100(f)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.003-0.002-0.001Δ < 0
-5e-041e-03Δ ≥ 0φ = 0.5 α = 0.5 n = 1000(g)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.0020-0.0015-0.0010-0.0005Δ < 0
-0.00050.00100.0015Δ ≥ 0φ = 0.5 α = 1 n = 1000(h)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λμk
--3e-04-2e-04-1e-04Δ < 0
-1e-042e-04Δ ≥ 0φ = 0.5 α = 1.5 n = 1000(i)
-Fig. 29 Figure equivalent to Fig. 16 after discretization of the 0
-is.
-the discretization has drastic impact on the summary plots of the boundary curves, where the
-curvy shapes of the continuous case are lost and altered substantially in many cases (Fig. 34,
-where some curves become one or a few points, or Fig. 40, re
ecting the loss of the curvy
-shapes).The advent and fall of a vocabulary learning bias from communicative eciency 63
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.16-0.12-0.08-0.04Δ < 0φ = 1 α = 0.5 n = 10(a)
-123
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.20-0.15-0.10-0.05Δ < 0
-0.010.020.030.040.05Δ ≥ 0φ = 1 α = 1 n = 10(b)
-12345
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.10-0.05Δ < 0
-0.010.020.030.040.05Δ ≥ 0φ = 1 α = 1.5 n = 10(c)
-123
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.05-0.04-0.03-0.02-0.01Δ < 0
-0.0050.0100.0150.020Δ ≥ 0φ = 1 α = 0.5 n = 100(d)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.03-0.02-0.01Δ < 0
-0.010.02Δ ≥ 0φ = 1 α = 1 n = 100(e)
-0102030
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.020-0.015-0.010-0.005Δ < 0
-0.0050.0100.015Δ ≥ 0φ = 1 α = 1.5 n = 100(f)
-12345
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.0075-0.0050-0.0025Δ < 0
-0.0020.0040.006Δ ≥ 0φ = 1 α = 0.5 n = 1000(g)
-0102030
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.010-0.005Δ < 0
-0.00250.00500.00750.0100Δ ≥ 0φ = 1 α = 1 n = 1000(h)
-050100150
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.004-0.003-0.002-0.001Δ < 0
-0.0010.0020.0030.004Δ ≥ 0φ = 1 α = 1.5 n = 1000(i)
-Fig. 30 Figure equivalent to Fig. 10 after discretization of the 0
-is.64 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.15-0.12-0.09-0.06Δ < 0φ = 1.5 α = 0.5 n = 10(a)
-0.51.01.52.02.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.2-0.1Δ < 0
-0.020.040.06Δ ≥ 0φ = 1.5 α = 1 n = 10(b)
-123
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.12-0.09-0.06-0.03Δ < 0
-0.010.020.030.040.05Δ ≥ 0φ = 1.5 α = 1.5 n = 10(c)
-0.51.01.52.02.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.06-0.04-0.02Δ < 0
-0.010.02Δ ≥ 0φ = 1.5 α = 0.5 n = 100(d)
-246
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.100-0.075-0.050-0.025Δ < 0
-0.0250.0500.075Δ ≥ 0φ = 1.5 α = 1 n = 100(e)
-051015
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.06-0.04-0.02Δ < 0
-0.010.020.030.040.050.06Δ ≥ 0φ = 1.5 α = 1.5 n = 100(f)
-123
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.015-0.010-0.005Δ < 0
-0.00250.00500.00750.0100Δ ≥ 0φ = 1.5 α = 0.5 n = 1000(g)
-051015
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.04-0.03-0.02-0.01Δ < 0
-0.010.020.030.04Δ ≥ 0φ = 1.5 α = 1 n = 1000(h)
-0204060
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.020-0.015-0.010-0.005Δ < 0
-0.0050.0100.0150.020Δ ≥ 0φ = 1.5 α = 1.5 n = 1000(i)
-Fig. 31 Figure equivalent to Fig. 17 after discretization of the 0
-is.The advent and fall of a vocabulary learning bias from communicative eciency 65
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.5-0.4-0.3-0.2-0.1Δ < 0
-0.050.100.150.20Δ ≥ 0φ = 2 α = 0.5 n = 10(a)
-0.51.01.52.02.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.4-0.3-0.2-0.1Δ < 0
-0.050.100.150.20Δ ≥ 0φ = 2 α = 1 n = 10(b)
-123
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.4-0.3-0.2-0.1Δ < 0
-0.050.100.150.200.25Δ ≥ 0φ = 2 α = 1.5 n = 10(c)
-0.51.01.52.02.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.15-0.10-0.05Δ < 0
-0.030.060.09Δ ≥ 0φ = 2 α = 0.5 n = 100(d)
-1234
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.10-0.05Δ < 0
-0.0250.0500.0750.100Δ ≥ 0φ = 2 α = 1 n = 100(e)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.125-0.100-0.075-0.050-0.025Δ < 0
-0.030.060.090.12Δ ≥ 0φ = 2 α = 1.5 n = 100(f)
-123
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.04-0.03-0.02-0.01Δ < 0
-0.010.020.030.04Δ ≥ 0φ = 2 α = 0.5 n = 1000(g)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.075-0.050-0.025Δ < 0
-0.020.040.060.08Δ ≥ 0φ = 2 α = 1 n = 1000(h)
-0102030
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.06-0.04-0.02Δ < 0
-0.020.040.06Δ ≥ 0φ = 2 α = 1.5 n = 1000(i)
-Fig. 32 Figure equivalent to Fig. 18 after discretization of the 0
-is.66 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.6-0.4-0.2Δ < 0
-0.10.20.3Δ ≥ 0φ = 2.5 α = 0.5 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.15-0.10-0.05Δ < 0φ = 2.5 α = 1 n = 10(b)
-0.51.01.52.02.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.20-0.15-0.10-0.05Δ < 0
-0.0250.0500.0750.1000.125Δ ≥ 0φ = 2.5 α = 1.5 n = 10(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.05-0.04-0.03-0.02-0.01Δ < 0
-0.00250.00500.0075Δ ≥ 0φ = 2.5 α = 0.5 n = 100(d)
-123
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.16-0.12-0.08-0.04Δ < 0
-0.050.10Δ ≥ 0φ = 2.5 α = 1 n = 100(e)
-246
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.3-0.2-0.1Δ < 0
-0.10.20.3Δ ≥ 0φ = 2.5 α = 1.5 n = 100(f)
-0.51.01.52.02.5
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.04-0.03-0.02-0.01Δ < 0
-0.010.020.03Δ ≥ 0φ = 2.5 α = 0.5 n = 1000(g)
-246
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.20-0.15-0.10-0.05Δ < 0
-0.050.100.150.20Δ ≥ 0φ = 2.5 α = 1 n = 1000(h)
-05101520
-0.00 0.25 0.50 0.75 1.00
-λμk
--0.15-0.10-0.05Δ < 0
-0.050.100.15Δ ≥ 0φ = 2.5 α = 1.5 n = 1000(i)
-Fig. 33 Figure equivalent to Fig. 19 after discretization of the 0
-is.The advent and fall of a vocabulary learning bias from communicative eciency 67
-1.001.251.501.752.00
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-2
-2.5α = 0.5 n = 10(a)
-2.02.53.03.54.0
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2α = 1 n = 10(b)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1.5 n = 10(c)
-1234
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 0.5 n = 100(d)
-05101520
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1 n = 100(e)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1.5 n = 100(f)
-2.55.07.5
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 0.5 n = 1000(g)
-0255075100
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1 n = 1000(h)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λμkφ
-0.5
-1
-1.5
-2
-2.5α = 1.5 n = 1000(i)
-Fig. 34 Figure equivalent to Fig. 11 after discretization of the 0
-is. It summarizes Figs. 29,
-30, 31, 32 and 33.68 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.09-0.06-0.03Δ < 0φ = 0.5 μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.008-0.006-0.004-0.002Δ < 0
-0.000250.000500.000750.00100Δ ≥ 0φ = 0.5 μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--6e-04-4e-04-2e-04Δ < 0
-0.000050.000100.00015Δ ≥ 0φ = 0.5 μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.15-0.10-0.05Δ < 0
-0.0050.0100.015Δ ≥ 0φ = 0.5 μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.0125-0.0100-0.0075-0.0050-0.0025Δ < 0
-0.000250.000500.00075Δ ≥ 0φ = 0.5 μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--9e-04-6e-04-3e-04Δ < 0
-5e-051e-04Δ ≥ 0φ = 0.5 μk = 2 n = 1000(f)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.25-0.20-0.15-0.10-0.05Δ < 0
-0.020.040.060.08Δ ≥ 0φ = 0.5 μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.015-0.010-0.005Δ < 0
-0.0010.0020.0030.004Δ ≥ 0φ = 0.5 μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.0015-0.0010-0.0005Δ < 0
-1e-042e-043e-04Δ ≥ 0φ = 0.5 μk = 4 n = 1000(i)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.4-0.3-0.2-0.1Δ < 0
-0.050.100.15Δ ≥ 0φ = 0.5 μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.03-0.02-0.01Δ < 0
-0.0050.010Δ ≥ 0φ = 0.5 μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.002-0.001Δ < 0
-0.000250.000500.000750.00100Δ ≥ 0φ = 0.5 μk = 8 n = 1000(l)
-Fig. 35 Figure equivalent to Fig. 20 after discretization of the 0
-is.The advent and fall of a vocabulary learning bias from communicative eciency 69
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.16-0.12-0.08-0.04Δ < 0
-0.0050.0100.0150.020Δ ≥ 0φ = 1 μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.009-0.006-0.003Δ < 0
-0.0010.0020.0030.004Δ ≥ 0φ = 1 μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.00100-0.00075-0.00050-0.00025Δ < 0
-1e-042e-043e-044e-045e-04Δ ≥ 0φ = 1 μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.3-0.2-0.1Δ < 0
-0.0250.0500.0750.100Δ ≥ 0φ = 1 μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.02-0.01Δ < 0
-0.0020.0040.006Δ ≥ 0φ = 1 μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.0025-0.0020-0.0015-0.0010-0.0005Δ < 0
-1e-042e-043e-044e-045e-04Δ ≥ 0φ = 1 μk = 2 n = 1000(f)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.6-0.4-0.2Δ < 0
-0.10.20.30.4Δ ≥ 0φ = 1 μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.06-0.04-0.02Δ < 0
-0.010.020.030.04Δ ≥ 0φ = 1 μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.006-0.004-0.002Δ < 0
-0.0010.0020.003Δ ≥ 0φ = 1 μk = 4 n = 1000(i)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--1.5-1.0-0.5Δ < 0
-0.250.500.751.001.25Δ ≥ 0φ = 1 μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.15-0.10-0.05Δ < 0
-0.050.100.15Δ ≥ 0φ = 1 μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.015-0.010-0.005Δ < 0
-0.0050.010Δ ≥ 0φ = 1 μk = 8 n = 1000(l)
-Fig. 36 Figure equivalent to Fig. 12 after discretization of the 0
-is.70 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.15-0.10-0.05Δ < 0
-0.010.020.030.040.05Δ ≥ 0φ = 1.5 μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.015-0.010-0.005Δ < 0
-0.0020.0040.0060.008Δ ≥ 0φ = 1.5 μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.0015-0.0010-0.0005Δ < 0
-0.000250.000500.000750.00100Δ ≥ 0φ = 1.5 μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.5-0.4-0.3-0.2-0.1Δ < 0
-0.050.100.150.200.25Δ ≥ 0φ = 1.5 μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.06-0.04-0.02Δ < 0
-0.010.02Δ ≥ 0φ = 1.5 μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.006-0.004-0.002Δ < 0
-0.0010.002Δ ≥ 0φ = 1.5 μk = 2 n = 1000(f)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--1.0-0.5Δ < 0
-0.30.60.9Δ ≥ 0φ = 1.5 μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.25-0.20-0.15-0.10-0.05Δ < 0
-0.050.100.150.20Δ ≥ 0φ = 1.5 μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.02-0.01Δ < 0
-0.0050.0100.0150.020Δ ≥ 0φ = 1.5 μk = 4 n = 1000(i)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--2-1Δ < 0
-0.51.01.52.02.5Δ ≥ 0φ = 1.5 μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.8-0.6-0.4-0.2Δ < 0
-0.20.40.60.8Δ ≥ 0φ = 1.5 μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.100-0.075-0.050-0.025Δ < 0
-0.0250.0500.0750.100Δ ≥ 0φ = 1.5 μk = 8 n = 1000(l)
-Fig. 37 Figure equivalent to Fig. 21 after discretization of the 0
-is.The advent and fall of a vocabulary learning bias from communicative eciency 71
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.5-0.4-0.3-0.2-0.1Δ < 0
-0.050.100.150.20Δ ≥ 0φ = 2 μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.03-0.02-0.01Δ < 0
-0.00250.00500.00750.01000.0125Δ ≥ 0φ = 2 μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.0025-0.0020-0.0015-0.0010-0.0005Δ < 0
-0.00040.00080.00120.0016Δ ≥ 0φ = 2 μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--1.0-0.5Δ < 0
-0.250.500.751.00Δ ≥ 0φ = 2 μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.15-0.10-0.05Δ < 0
-0.030.060.09Δ ≥ 0φ = 2 μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.016-0.012-0.008-0.004Δ < 0
-0.00250.00500.00750.0100Δ ≥ 0φ = 2 μk = 2 n = 1000(f)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--2-1Δ < 0
-0.51.01.52.02.5Δ ≥ 0φ = 2 μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.8-0.6-0.4-0.2Δ < 0
-0.20.40.60.8Δ ≥ 0φ = 2 μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.100-0.075-0.050-0.025Δ < 0
-0.0250.0500.075Δ ≥ 0φ = 2 μk = 4 n = 1000(i)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--4-3-2-1Δ < 0
-123Δ ≥ 0φ = 2 μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--2-1Δ < 0
-12Δ ≥ 0φ = 2 μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.5-0.4-0.3-0.2-0.1Δ < 0
-0.10.20.30.40.5Δ ≥ 0φ = 2 μk = 8 n = 1000(l)
-Fig. 38 Figure equivalent to Fig. 22 after discretization of the 0
-is.72 David Carrera-Casado, Ramon Ferrer-i-Cancho
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.6-0.4-0.2Δ < 0
-0.10.20.3Δ ≥ 0φ = 2.5 μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.05-0.04-0.03-0.02-0.01Δ < 0
-0.0050.0100.015Δ ≥ 0φ = 2.5 μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.004-0.003-0.002-0.001Δ < 0
-0.00050.00100.00150.0020Δ ≥ 0φ = 2.5 μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--1.5-1.0-0.5Δ < 0
-0.51.0Δ ≥ 0φ = 2.5 μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.3-0.2-0.1Δ < 0
-0.10.20.3Δ ≥ 0φ = 2.5 μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.04-0.03-0.02-0.01Δ < 0
-0.010.020.03Δ ≥ 0φ = 2.5 μk = 2 n = 1000(f)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--3-2-1Δ < 0
-12Δ ≥ 0φ = 2.5 μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--2.0-1.5-1.0-0.5Δ < 0
-0.51.01.5Δ ≥ 0φ = 2.5 μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--0.3-0.2-0.1Δ < 0
-0.10.20.3Δ ≥ 0φ = 2.5 μk = 4 n = 1000(i)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--3-2-1Δ < 0
-123Δ ≥ 0φ = 2.5 μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--4-3-2-1Δ < 0
-1234Δ ≥ 0φ = 2.5 μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λα
--2.0-1.5-1.0-0.5Δ < 0
-0.51.01.52.0Δ ≥ 0φ = 2.5 μk = 8 n = 1000(l)
-Fig. 39 Figure equivalent to Fig. 23 after discretization of the 0
-is.The advent and fall of a vocabulary learning bias from communicative eciency 73
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-1
-1.5
-2
-2.5μk = 1 n = 10(a)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 1 n = 100(b)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 1 n = 1000(c)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 n = 10(d)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 n = 100(e)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 n = 1000(f)
-0.60.81.01.21.4
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1μk = 4 n = 10(g)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 4 n = 100(h)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 4 n = 1000(i)
-1.01.11.21.31.41.5
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5μk = 8 n = 10(j)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2μk = 8 n = 100(k)
-0.500.751.001.251.50
-0.00 0.25 0.50 0.75 1.00
-λαφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 8 n = 1000(l)
-Fig. 40 Figure equivalent to Fig. 13 after discretization of the 0
-is. It summarizes Figs. 35,
-36, 37, 38 and 39.74 David Carrera-Casado, Ramon Ferrer-i-Cancho
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.09-0.06-0.03Δ < 0φ = 0.5 μk = 1 α = 0.5(a)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.04-0.03-0.02-0.01Δ < 0
-1e-042e-04Δ ≥ 0φ = 0.5 μk = 1 α = 1(b)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.03-0.02-0.01Δ < 0
-0.0010.0020.003Δ ≥ 0φ = 0.5 μk = 1 α = 1.5(c)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.15-0.10-0.05Δ < 0
-0.0050.0100.015Δ ≥ 0φ = 0.5 μk = 2 α = 0.5(d)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.06-0.04-0.02Δ < 0
-2.5e-055.0e-057.5e-051.0e-04Δ ≥ 0φ = 0.5 μk = 2 α = 1(e)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.02-0.01Δ < 0
-0.00050.00100.0015Δ ≥ 0φ = 0.5 μk = 2 α = 1.5(f)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.25-0.20-0.15-0.10-0.05Δ < 0
-0.020.040.060.08Δ ≥ 0φ = 0.5 μk = 4 α = 0.5(g)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.09-0.06-0.03Δ < 0
-0.00250.00500.0075Δ ≥ 0φ = 0.5 μk = 4 α = 1(h)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.03-0.02-0.01Δ < 0
-3e-046e-049e-04Δ ≥ 0φ = 0.5 μk = 4 α = 1.5(i)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.4-0.3-0.2-0.1Δ < 0
-0.050.100.15Δ ≥ 0φ = 0.5 μk = 8 α = 0.5(j)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.15-0.10-0.05Δ < 0
-0.020.040.06Δ ≥ 0φ = 0.5 μk = 8 α = 1(k)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.05-0.04-0.03-0.02-0.01Δ < 0
-2e-044e-046e-04Δ ≥ 0φ = 0.5 μk = 8 α = 1.5(l)
-Fig. 41 Figure equivalent to Fig. 24 after discretization of the 0
-is.The advent and fall of a vocabulary learning bias from communicative eciency 75
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.16-0.12-0.08-0.04Δ < 0φ = 1 μk = 1 α = 0.5(a)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.06-0.04-0.02Δ < 0
-0.0010.0020.0030.004Δ ≥ 0φ = 1 μk = 1 α = 1(b)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.06-0.04-0.02Δ < 0
-0.0050.0100.0150.020Δ ≥ 0φ = 1 μk = 1 α = 1.5(c)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.3-0.2-0.1Δ < 0
-0.0250.0500.0750.100Δ ≥ 0φ = 1 μk = 2 α = 0.5(d)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.100-0.075-0.050-0.025Δ < 0
-5e-041e-03Δ ≥ 0φ = 1 μk = 2 α = 1(e)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.06-0.04-0.02Δ < 0
-0.00250.00500.00750.01000.0125Δ ≥ 0φ = 1 μk = 2 α = 1.5(f)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.6-0.4-0.2Δ < 0
-0.10.20.30.4Δ ≥ 0φ = 1 μk = 4 α = 0.5(g)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.3-0.2-0.1Δ < 0
-0.050.10Δ ≥ 0φ = 1 μk = 4 α = 1(h)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.075-0.050-0.025Δ < 0
-0.0020.0040.006Δ ≥ 0φ = 1 μk = 4 α = 1.5(i)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.5-1.0-0.5Δ < 0
-0.250.500.751.001.25Δ ≥ 0φ = 1 μk = 8 α = 0.5(j)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.6-0.4-0.2Δ < 0
-0.10.20.30.40.5Δ ≥ 0φ = 1 μk = 8 α = 1(k)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.3-0.2-0.1Δ < 0
-0.050.100.150.20Δ ≥ 0φ = 1 μk = 8 α = 1.5(l)
-Fig. 42 Figure equivalent to Fig. 14 after discretization of the 0
-is.76 David Carrera-Casado, Ramon Ferrer-i-Cancho
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.15-0.10-0.05Δ < 0φ = 1.5 μk = 1 α = 0.5(a)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.075-0.050-0.025Δ < 0
-0.0030.0060.0090.012Δ ≥ 0φ = 1.5 μk = 1 α = 1(b)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.12-0.09-0.06-0.03Δ < 0
-0.010.020.030.040.05Δ ≥ 0φ = 1.5 μk = 1 α = 1.5(c)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.5-0.4-0.3-0.2-0.1Δ < 0
-0.050.100.150.200.25Δ ≥ 0φ = 1.5 μk = 2 α = 0.5(d)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.2-0.1Δ < 0
-0.020.040.06Δ ≥ 0φ = 1.5 μk = 2 α = 1(e)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.100-0.075-0.050-0.025Δ < 0
-0.010.02Δ ≥ 0φ = 1.5 μk = 2 α = 1.5(f)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.0-0.5Δ < 0
-0.30.60.9Δ ≥ 0φ = 1.5 μk = 4 α = 0.5(g)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.75-0.50-0.25Δ < 0
-0.20.40.6Δ ≥ 0φ = 1.5 μk = 4 α = 1(h)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.2-0.1Δ < 0
-0.050.10Δ ≥ 0φ = 1.5 μk = 4 α = 1.5(i)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2-1Δ < 0
-0.51.01.52.02.5Δ ≥ 0φ = 1.5 μk = 8 α = 0.5(j)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2.0-1.5-1.0-0.5Δ < 0
-0.51.01.52.0Δ ≥ 0φ = 1.5 μk = 8 α = 1(k)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.00-0.75-0.50-0.25Δ < 0
-0.250.500.75Δ ≥ 0φ = 1.5 μk = 8 α = 1.5(l)
-Fig. 43 Figure equivalent to Fig. 25 after discretization of the 0
-is.The advent and fall of a vocabulary learning bias from communicative eciency 77
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.5-0.4-0.3-0.2-0.1Δ < 0
-0.050.100.150.20Δ ≥ 0φ = 2 μk = 1 α = 0.5(a)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.125-0.100-0.075-0.050-0.025Δ < 0
-0.010.02Δ ≥ 0φ = 2 μk = 1 α = 1(b)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.15-0.10-0.05Δ < 0
-0.020.040.06Δ ≥ 0φ = 2 μk = 1 α = 1.5(c)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.0-0.5Δ < 0
-0.250.500.751.00Δ ≥ 0φ = 2 μk = 2 α = 0.5(d)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.4-0.3-0.2-0.1Δ < 0
-0.050.100.150.20Δ ≥ 0φ = 2 μk = 2 α = 1(e)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.06-0.04-0.02Δ < 0
-0.010.020.03Δ ≥ 0φ = 2 μk = 2 α = 1.5(f)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2-1Δ < 0
-0.51.01.52.02.5Δ ≥ 0φ = 2 μk = 4 α = 0.5(g)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.6-1.2-0.8-0.4Δ < 0
-0.51.0Δ ≥ 0φ = 2 μk = 4 α = 1(h)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.8-0.6-0.4-0.2Δ < 0
-0.20.40.6Δ ≥ 0φ = 2 μk = 4 α = 1.5(i)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--4-3-2-1Δ < 0
-123Δ ≥ 0φ = 2 μk = 8 α = 0.5(j)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--3-2-1Δ < 0
-123Δ ≥ 0φ = 2 μk = 8 α = 1(k)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2.0-1.5-1.0-0.5Δ < 0
-0.51.01.52.0Δ ≥ 0φ = 2 μk = 8 α = 1.5(l)
-Fig. 44 Figure equivalent to Fig. 26 after discretization of the 0
-is.78 David Carrera-Casado, Ramon Ferrer-i-Cancho
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.6-0.4-0.2Δ < 0
-0.10.20.3Δ ≥ 0φ = 2.5 μk = 1 α = 0.5(a)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.15-0.10-0.05Δ < 0
-0.010.020.03Δ ≥ 0φ = 2.5 μk = 1 α = 1(b)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.20-0.15-0.10-0.05Δ < 0
-0.0250.0500.0750.1000.125Δ ≥ 0φ = 2.5 μk = 1 α = 1.5(c)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.5-1.0-0.5Δ < 0
-0.51.0Δ ≥ 0φ = 2.5 μk = 2 α = 0.5(d)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.5-0.4-0.3-0.2-0.1Δ < 0
-0.10.20.3Δ ≥ 0φ = 2.5 μk = 2 α = 1(e)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--0.09-0.06-0.03Δ < 0
-0.020.040.06Δ ≥ 0φ = 2.5 μk = 2 α = 1.5(f)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--3-2-1Δ < 0
-12Δ ≥ 0φ = 2.5 μk = 4 α = 0.5(g)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.5-1.0-0.5Δ < 0
-0.51.01.5Δ ≥ 0φ = 2.5 μk = 4 α = 1(h)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--1.00-0.75-0.50-0.25Δ < 0
-0.250.500.75Δ ≥ 0φ = 2.5 μk = 4 α = 1.5(i)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--4-3-2-1Δ < 0
-1234Δ ≥ 0φ = 2.5 μk = 8 α = 0.5(j)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--3-2-1Δ < 0
-123Δ ≥ 0φ = 2.5 μk = 8 α = 1(k)
-102505007501000
-0.00 0.25 0.50 0.75 1.00
-λn
--2.5-2.0-1.5-1.0-0.5Δ < 0
-0.51.01.52.0Δ ≥ 0φ = 2.5 μk = 8 α = 1.5(l)
-Fig. 45 Figure equivalent to Fig. 27 after discretization of the 0
-is.The advent and fall of a vocabulary learning bias from communicative eciency 79
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-1.5
-2
-2.5μk = 1 α = 0.5(a)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 1 α = 1(b)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 1 α = 1.5(c)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 α = 0.5(d)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 α = 1(e)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 2 α = 1.5(f)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1μk = 4 α = 0.5(g)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 4 α = 1(h)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 4 α = 1.5(i)
-2505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5μk = 8 α = 0.5(j)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2μk = 8 α = 1(k)
-02505007501000
-0.00 0.25 0.50 0.75 1.00
-λnφ
-0
-0.5
-1
-1.5
-2
-2.5μk = 8 α = 1.5(l)
-Fig. 46 Figure equivalent to Fig. 15 after discretization of the 0
-is. It summarizes Figs. 41,
-42, 43, 44 and 45.
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