A Bayesian Approach to the Naming Game Model

A basic question in complexity theory is how the interactions between the units of the system lead to the emergence of ordered states from initially disordered configurations Castellano-2009a ; Baronchelli-2018a . This general question concerns phenomena ranging from phase transitions in condensed matter systems and self-organization in living matter to the appearance of norm conventions and cultural paradigms in social systems. Various models were used in order to study social interactions and cooperation, e.g. models of condensed matter systems (such as spin systems), statistical mechanical models (e.g. based on the master equation), ecological competition models Castellano-2009a , many-agents game-theoretical models Xia-2017a ; Xia-2018a ; Zhang-2017a . Opinion dynamics and cultural spreading models represent suitable theoretical frameworks for a quantitative description of the emergence of social consensus Baronchelli-2018a .

In this respect, the emergence of human language remains a challenging, multi-fold question, related in turn to biological, ecological, social, logical, and cognitive aspects Mufwene-2001a ; Lass1997 ; Berruto-2004a ; Edelman-2007a ; Tenenbaum-1999 . Language dynamics Wichmann-2008b ; Wichmann-2008c has provided models describing phenomena of language competition and change that focus on the mutual interactions of linguistic traits (sounds, phonemes, grammatical rules, or languages understood as fixed entities) under the influence of ecological and social factors, modeling such interactions in analogy to biological competition and evolution.

However, the basic learning process of a word has a complex dynamics due to its cognitive dimension. In fact, learning a word means to learn a concept (understood as a pointer to a subset of objects, see Refs. Tenenbaum-1999 ; Tenenbaum-2000a ; Xu-2007a ) and a linguistic label —for example the name of the object— used for communicating the concept. The double concept$\leftrightarrow$name nature of words has been studied through semiotic dynamics models, such as the models of Hurford Hurford-1989a and Nowak Nowak-1999a (see also Nowak-2000a ; Trapa-2000a ) and the naming game (NG) model Baronchelli-2006c ; guanrong2019 .

In the basic version of the model of Nowak Nowak-1999a , the language spoken by each agent $i$ ($i=1,\dots,N$) is defined by two personal matrices, representing the links of a bipartite network joining $Q$ names and $R$ concepts: (1) an active matrix ${\mathcal{U}}^{(i)}$ representing the concept$\,\rightarrow\,$name links, where the element ${\mathcal{U}}^{(i)}_{q,r}$ ($q\in(1,Q),~r\in(1,R)$) gives the probability that agent $i$ will utter the $q$th name to communicate the $r$th concept; (2) the passive matrix $H^{(i)}$, representing the name$\,\rightarrow\,$concept links, in which the element ${\mathcal{H}}^{(i)}_{q,r}$ represents the probability that an agent interprets the $q$th name as referring to the $r$th concept. In the models of Hurford and in the model of Nowak, the languages of each individual evolve with time according to a game-theoretical dynamics, with agents gaining a reproductive advantage if their matrices have a higher communication efficiency. These studies have achieved interesting results, such as the emergence of non-ambiguous one-to-one links between objects and sounds, and explain why homonyms are more frequent than synonyms Hurford-1989a ; Nowak-1999a ; Nowak-2000a ; Trapa-2000a .

In the NG model Baronchelli-2006c ; guanrong2019 there is only one concept ($R=1$) that can be linked to a set of $Q>1$ different names. The model can be reformulated through the agents’ lists ${\mathcal{L}}_{i}$ of the name$\leftrightarrow$concept connections known to each agent $i$. In the case of two-conventions models, where the conventions are the names $A$ and $B$, the list of the $i$th agent can be ${\mathcal{L}}_{i}=\emptyset$ (no connection), ${\mathcal{L}}_{i}=(A)$ or $(B)$ (one name is known), or ${\mathcal{L}}_{i}=(A,B)$ (both name$\leftrightarrow$concept connections are known).

Extending semiotic dynamics models is not trivial and already two-opinion variants of the NG model, taking into account committed groups, show a remarkable phase diagram Xie-2012a ; and trying to describe actual cognitive effects requires entirely new features Fan-2018a . This paper presents a minimal model to study the interplay of the cognitive and social dynamical dimensions, assuming for simplicity the two-conventions NG model as a semiotic framework Baronchelli-2016a ; guanrong2019 and making a cognitive generalization within the experimentally validated Bayesian framework of Tenenbaum-1999 (see also Refs. Tenenbaum2001 ; Tenenbaum-2000a ; Griffiths2006 ; Xu-2007a ; Perfors-2011a ; Lake2015 ). In that framework, an individual can learn a concept from a small number of examples, a most remarkable feature of human learning Tenenbaum-1999 ; Tenenbaum-1999b ; Tenenbaum-2011a , to be contrasted with machine learning algorithms, which require a large amount of examples for generalizing successfully barber2012 ; Murphy-2012a ; Evgeniou-2000a .

The paper is organized as follows. The new model is introduced in Sec. II. In Sec. III, we present and discuss the features of the semiotic dynamics emerging from the numerical simulations and quantitatively compare them with those of the two-conventions NG model. Future directions in the study of the interplay of the cognitive and the social dynamics are outlined in Sec. V.

In this section we study numerically the Bayesian NG model introduced above and discuss its main features. We limit ourselves to study the model dynamics on a fully-connected network.

In the new learning scheme, which replaces the one-shot learning of the two-conventions NG model, an individual generalizes concept $C$ on a suitable time scale $\Delta t>1$, rather than during a single interaction. However, a few examples are sufficient for an agent to generalize concept $C$, as in a realistic concept-learning process. This is visible from the Bayesian probabilities $p_{A}$ and $p_{B}$ computed by agents in the role of hearer, according to Eq. (4), once at least $n^{\ast}_{ex,A}=5$ and $n^{\ast}_{ex,B}=6$ examples “+”, respectively, have been stored in the inventory associated to the name $A$ and $B$: Figure 3 shows the histograms of the $p_{A}$’s and $p_{B}$’s computed from the initial time until consensus for a single run with $N=2000$ agents and starting with SIC. The low frequencies at small values of $p_{A}$ and $p_{B}$ and the highest frequencies at values close to unity are due to the fact that the Bayesian probabilities reach values $p_{A}\approx p_{B}\approx 1$ very fast, after a few learning attempts, consistently with the size principle, on which the Bayesian learning paradigm, and in turn Eq. (4), are based Tenenbaum-1999 .

In order to visualize how the system approaches consensus, it is useful to consider some global observables, such as the fractions $n_{A}(t)$, $n_{B}(t)$, and $n_{AB}(t)$ of agents that have generalized concept $C$ in association with name $A$ only, name $B$ only, or both names $A$ and $B$, respectively, or the success rate $S(t)$. The dynamics of a population of $N=1000$ agents (panels (A) and (B)) using different initial conditions, SIC, AIC, and AICr, and that of a population of $N=100$ agents starting with SIC (panels (C) and (D)) are shown in Fig. 4.

Panel (A) of Fig. 4 shows only the population fractions corresponding to the name found at consensus, for the sake of clarity (the remaining population fractions eventually go to zero). For asymmetrical initial condition (AIC or AICr), it is the initial majority that determines the convention found at consensus (that is $B$ for AIC and $A$ for AICr). If the system starts from SIC, the convention $A$, for which agents can generalize earlier ($n^{\ast}_{ex,A}=5<n^{\ast}_{ex,B}=6$), is always found at consensus — in this case it is the asymmetry in the thresholds $n^{\ast}_{ex,A}$ and $n^{\ast}_{ex,B}$, characterizing the Bayesian learning process, to determine consensus.

Panel (B) of Fig. 4 shows the success rate $S(t=t_{k})$, representing the average over different runs of the instantaneous success rate $S_{k}$ of the $k$th interaction at time $t_{k}$, defined as follows: $S_{k}=1$ in case of agreement between the two agents or when a successful learning of the hearer takes places, following a Bayes probability $p>1/2$; or $S_{k}=0$ in case of unsuccessful generalization, when $p<1/2$ and only reinforcement takes place. The success rate $S(t)$ varies between $S(0)\approx(n_{A}(0))^{2}+(n_{B}(0))^{2}$, due to the respective fractions of agents that initially know the two conventions $A$ and $B$, to $S\approx 1$ at consensus, following a typical S-shaped curve of learning processes Baronchelli2006 . In the case of SIC, the initial value is $S(0)\approx 0.5^{2}+0.5^{2}=0.5$, while for AIC or AICr the initial value is $S(0)\approx(0.3)^{2}+(0.7)^{2}\approx 0.58$.

We now investigate how the modified Bayesian dynamics affects the convergence times to consensus. The study of the size-dependence of the convergence to consensus shows that there is a critical value $N^{\ast}\approx 500$ in the case of SIC, such that for $N\leq N^{\ast}$ there is a non-negligible probability that the final absorbing state is $B$. Panels (C) and (D) of Fig. 4, representing the results for a system starting with SIC and a smaller size $N=100$, show the existence of two possible final absorbing states and that there are different times scales associated to the convergence to consensus: name $A$ is found at consensus in about $90\%$ of cases and name $B$ in the remaining cases. The branching probability into $A$ or $B$ consensus is further investigated in panel (A) of Fig. 5, where we plot the branching probabilities $p_{e,A},p_{e,B}$ versus the system sizes $N$. The nonlinear behavior (symmetrical sigmoid) signals the presence of finite-size effects, particularly clear for relatively small $N$-values. In fact, when the fluctuations in the system are larger, the system size can play an important role in the dynamics of social systems, as an actual thermodynamic limit is only allowed for simulations of macroscopic physical systems Toral2007a .

The convergence time $t_{conv}$ follows a simple scaling rule with the system size $N$, related to the average number of examples $\bar{n}_{ex,A},\bar{n}_{ex,B}$ relative to $A,B$ respectively, stored in the agents’ inventories at consensus. These values depend on the number of learning and reinforcement processes, and hence are related to the system size $N$. The average number of interactions undergone by the agents until the system reaches the consensus is given by the sum $\bar{n}_{int}=\bar{n}_{ex,A}+\bar{n}_{ex,B}$ ¹¹1The $n_{ex,A}=n_{ex,B}=4$ examples given initially to each agent are not accounted for by $\bar{n}_{ex,A}$ and $\bar{n}_{ex,B}$.. One expects that

$$t_{conv}\approx\bar{n}_{int}N\,,$$

(5)

which suggests a linear scaling law ($t_{conv}\sim N$) for convergence time with the system size $N$ for all the possible initial conditions. A linear behavior is indeed confirmed by the numerical simulations with population sizes $N=50,100,500,1000,1500,2000$ starting from SIC, AIC, AICr. The relative numerical results are reported in Table 2. Moreover, in Eq. (5) the size-dependence of $\bar{n}_{int}$ is ignored as it shows a weak dependence upon $N$, see panel (B) in Fig. 5.

From the above mentioned scaling law, it is clear that the average number of examples stored by the agents at consensus plays an important role in the semiotic dynamics. In particular, it is found that if the final absorbing state is $A$ (or B), then $\bar{n}_{ex,A}>\bar{n}_{ex,B}$ ( $\bar{n}_{ex,B}>\bar{n}_{ex,A}$). Moreover, the average number of examples, relative to the absorbing state, always increases monotonically with the system size while a size-independent behavior is observed in the opposite case, see the right panel (B) of Fig. 5.

Finally, we compare the convergence time of the Bayesian word-learning model, $t_{conv}$, with that of two-conventions NG model, $\bar{t}_{conv}$ Castello2009 , by studying the corresponding ratio $R=t_{conv}/\bar{t}_{conv}$ for common initial conditions and population sizes. When starting with SIC, the values of the convergence times obtained from the two models become of the same order by increasing $N$: $R$ decreases with $N$, reaching unity for $N=10000$, see Fig. 6. In other words, the time scales of the two models become equivalent for relatively large system sizes, i.e., the learning processes of the two models perform equivalently and the Bayesian approach roughly gives rise to the one-shot learning that characterizes the two-conventions NG model. In the next section we discuss how the Bayesian model becomes asymptotically equivalent to the minimal NG model. The inset of Fig. 6 represents $R$ versus $N$, for $N<2000$, given different starting configurations, with SIC, AIC and AICr, and different population sizes. In the following, we focus on the case of SIC.

In this section we investigate the stability properties of the mean-field dynamics of the Bayesian NG model, in which statistical fluctuations and correlations are neglected. In the Bayesian NG model, as in the basic NG, agents can use two non-excluding options $A$ and $B$ to refer to the same concept $C$. The main difference between the Bayesian model and the basic NG model is in the learning process: a one-shot learning process in the basic NG and a Bayesian process in the Bayesian NG model. In the latter case the presence of a name in the word list indicates that the agent has generalized the corresponding concept from a set of positive recorded examples.

The NG model belongs to the wide class of models with two non-excluding options $A$ and $B$, such as many models of bilingualism Patriarca2012a , in which transitions between state $(A)$ and state $(B)$ are allowed only through an intermediate (“bilingual”) state $(A,B)$, as schematized in Fig. 7. The mean-field equations for the fractions $n_{A}(t)$ and $n_{B}(t)$ can be obtained considering the gain and loss contributions of the transitions depicted in Fig. 7,

	$\displaystyle\dot{n}_{A}=p_{AB\rightarrow A^{\,}}\,n_{AB}-p_{A\rightarrow AB}\,n_{A}\,,$
	$\displaystyle\dot{n}_{B}=p_{AB\rightarrow B}\,n_{AB}-p_{B\rightarrow AB}\,n_{B}\,.$		(6)

Here $\dot{n_{a}}(t)=dn_{a}(t)/dt$ and the quantities $p_{a\rightarrow b}$ represent the respective transition rates per individual, corresponding to the arrows in Fig. 7 ($a,b=A,B,AB$). The equation for $n_{AB}(t)$ was omitted, since it is determined by the condition that the total number of agents is constant, $n_{A}(t)+n_{B}(t)+n_{AB}(t)=1$.

The details of the possible pair-wise interactions in the Bayesian naming game are listed in Table 1. From the various contributions, one obtains the master equation

	$\displaystyle\dot{n}_{A}=-p_{B}n_{A}n_{B}+n_{AB}^{2}+\frac{3-p_{B}}{2}n_{A}n_{AB}\,,$
	$\displaystyle\dot{n}_{B}=-p_{A}n_{A}n_{B}+n_{AB}^{2}+\frac{3-p_{A}}{2}n_{B}n_{AB}\,,$		(7)

which can be rewritten in the form (IV) with transition rates per individual given by

		$\displaystyle p_{A\rightarrow AB}=p_{B}n_{B}+\frac{1}{2}p_{B}n_{AB}\,,\qquad$	$\displaystyle p_{B\rightarrow AB}=p_{A}n_{A}+\frac{1}{2}p_{A}n_{AB}\,,$			(8)
		$\displaystyle p_{AB\rightarrow A}=\frac{3}{2}n_{A}+n_{AB}\,,\qquad$	$\displaystyle p_{AB\rightarrow B}=\frac{3}{2}n_{B}+n_{AB}\,.$			(9)

Equations (8) provide the transition rates of learning processes, while Eqs. (9) give the transition rates of agreement processes. Setting $x\equiv n_{A}$, $y\equiv n_{B}$, and $z=n_{AB}\equiv 1-x-y$, the autonomous system (IV) becomes

	$\displaystyle\dot{x}=f_{x}\left(x,y\right)\equiv-p_{B}xy+(1-x-y)^{2}+\frac{1}{2}(3-p_{B})x(1-x-y)\,,$		(10)
	$\displaystyle\dot{y}=f_{y}\left(x,y\right)\equiv-p_{A}xy+(1-x-y)^{2}+\frac{1}{2}(3-p_{A})y(1-x-y)\,,$		(11)

where $\mathbf{v}=\left(f_{x}(x,y),f_{y}(x,y)\right)$ is the velocity field in the phase plane. For the following analysis, it is convenient to write the Bayesian probabilities $p_{A}$ and $p_{B}$ appearing in these equations as time-dependent parameters of the model, but they are actually highly non-linear functions of the variables. In fact, they can be thought as averages of the microscopic Bayesian probability in Eq. (4) over the possible dynamical realizations. For this reason, they have also a complex non-local time-dependence on the previous history of the interactions between agents. For the moment, we assume $p_{A}(t)=p_{B}(t)=p(t)$, returning later to the general case.

From the conditions defining the critical points, $f_{x}\left(x,y\right)=f_{y}\left(x,y\right)=0$, one obtains $\left(x-y\right)z=0$. Setting $z=0$, one obtains two solutions that correspond to consensus in $A$ or $B$, given by $(x_{1},y_{1},z_{1})=(1,0,0)$ and $(x_{2},y_{2},z_{2})=(0,1,0)$. Instead, setting $\left(x-y\right)=0$ leads to the equation

$$2x^{2}-(p+5)x+2=0\,,$$

(12)

that has the solutions

$$x_{\pm}=\frac{p+5\pm\sqrt{(p+5)^{2}-16}}{4}\,.$$

(13)

For $p\in\left(0,1\right]$, the corresponding solutions $(x_{\pm},x_{\pm},1-2x_{\pm})$ are not suitable solutions, because $z_{\pm}=1-2x_{\pm}<0$.

This analysis is valid for $p>0$. In fact, $p=p(t)$ is a function of time and for a finite interval of time after the initial time one has that $p=0$, which defines a different dynamical system. In the initial conditions used, $z(0)=0$, which implies $z(t)=0$, $x(t)=x(0)$, and $y(t)=y(0)$ at any later time $t$ as long as $p(t)=0$, since $\dot{x}(t)=\dot{y}(t)=\dot{z}(t)=0$ (see Eq. (IV); in fact, the whole line $x+y=1$ (for $0<x,y<1$) represents a continuous set of equilibrium points. The reason why in this model $p(0)=0$ at $t=0$ and also during a subsequent finite interval of time is twofold. First, agents do not have any examples associated to the name not known and they have to receive at least $n_{ex,A}^{\ast}$ or $n_{ex,B}^{\ast}$ examples, before being able to compute the corresponding Bayesian probability $p_{A}(t)$ or $p_{B}(t)$ — thus it is to be expected that $p(t)=0$ meanwhile. Furthermore, even when agents can compute the Bayesian probabilities, the effective probability to generalize is actually zero, due to the threshold $p^{\ast}=0.5$ for a generalization to take place. The existence of the (temporary) equilibrium points on the line $x+y=1$ ends as soon as the parameter $p(t)>p^{\ast}$ and, according to Eqs. (IV), the two $A$- and $B$-consensus states become the only stable equilibrium points. The representative point in the $x$-$y$-plane is deemed to leave the initial conditions on the $z=1-x-y=0$ line, due to the stochastic nature of the dynamics, which is not invariant under time reversal hinrichsen2006 .

To determine the nature of the critical points $(x_{1},y_{1})=(1,0)$ and $(x_{2},y_{2})=(0,1)$, one needs to evaluate at the equilibrium points the $2\times 2$ Jacobian matrix $A(x,y)=\{\partial_{i}f_{j}\}$, where $i,j=x,y$. It is easy to show that both the critical points $(0,1)$ and $(1,0)$ are asymptotically stable strogatz2000 .

As long as the general case $p_{A}\neq p_{B}$, it can be shown that the trajectory of the system can point toward and eventually reach the consensus state with $A$ or $B$, depending on whether $p_{A}\left(t^{\ast}\right)>p_{B}\left(t^{\ast}\right)$ or $p_{A}\left(t^{\ast}\right)<p_{B}\left(t^{\ast}\right)$, where $t^{\ast}>0$ is the critical time at which the representative point leaves the initial position.

The convention $A$ or $B$ is selected randomly, depending on various factors related to the specific realization of the system evolution, such as the numbers of examples $\bar{n}_{ex,A}(t)$ and $\bar{n}_{ex,B}(t)$ recorded by the agents until time $t$, their quality from the point of view of the generalization, and the initial asymmetry of the thresholds for generalizing, $n^{\ast}_{ex,A}\neq n^{\ast}_{ex,B}$. The asymmetrical thresholds $n^{\ast}_{ex,A}=4<n^{\ast}_{ex,B}=5$ produce a bias toward consensus in $A$ and play a crucial role in the subsequent Bayesian semiotic dynamics; in fact, swapping the threshold values (setting $n^{\ast}_{ex,A}=5>n^{\ast}_{ex,B}=4$), the approach to consensus occurs with the outcomes $A$, $B$ swapped.

We observed that for $N\gtrsim N^{\ast}\approx 500$, the chances that the system converges to $(B)$ become negligible. This can be seen in panels (C) and (D) of Fig. 8, showing $\bar{n}_{ex,A}(t)$ and $\bar{n}_{ex,B}(t)$ versus time (averaged over the agents of the system) for single runs, a population of $N=100$ agents, and SIC, for different runs that relax toward consensus $A$ and $B$, respectively. After an initial transient, in which $\bar{n}_{ex,A}(t)\approx\bar{n}_{ex,B}(t)$, they differ more and more from each other at times $t>t^{\ast}$. In turn, also $p_{A}$ and $p_{B}$ begin to differ significantly from each other, thus affecting the rate of depletion of the populations during the subsequent dynamics. For instance, if $p_{A}>p_{B}$, then $p_{B\rightarrow AB}>p_{A\rightarrow AB}$, see Eqs. (8), which means that the depletion of $n_{B}$ occurs faster then that of $n_{A}$. In turn, this favours the decay of the mixed states $(A,B)$ into the state $(A)$, see Eqs. (9), being $n_{A}>n_{B}$.

The asymmetry discussed above also affects the convergence times $t^{A}_{conv}$ and $t^{B}_{conv}$ and we find $t^{B}_{conv}>t^{A}_{conv}$ in all the numerical simulations. Despite the noise, such a trend is already appreciable in a single run, as shown in panels (A) and (B) of Fig. 8. The mean fractions $n_{A}(t)$, $n_{B}(t)$, and $n_{AB}(t)$ versus time, obtained by averaging over many runs, result in less noisy outputs and provide a more clear picture of the difference, which is visible in Fig. 9, obtained using $600$ runs starting with SIC and for $N=100$ agents (panels (A) and (B)) and $N=200$ agents (panels (C) and (D)). In addition, the convergence times depend on the system size, increasing with the number of agents $N$: compare the left panels (A) and (B), where $N=100$ agents, with the right panels (C) and (D), where $N=200$ agents.

The possibility that the same system, starting with the same initial conditions and evolving with the same dynamical parameters, can reach either $A$ or $B$ is a consequence of the stochastic nature of dynamics. This does not happen for $N\gtrsim N^{\ast}$, when both $\bar{n}_{ex,A}$ and $\bar{n}_{ex,B}$ reach some threshold values close to those observed at $t_{conv}$, which is clearly a value sufficient for the agents to generalizing concept $C$. In fact, the scaling law of $t_{conv}$ with $N$ shows that the sum of $\bar{n}_{ex,A}$ with $\bar{n}_{ex,B}$ becomes nearly constant for $N\gtrsim N^{\ast}$, implying that the dynamics is uniquely determined, that is, the consensus always occurs at $A$ from SIC, once the agents have stored a threshold number of $\bar{n}_{ex,A}$, $\bar{n}_{ex,B}$. It is found that these threshold values correspond to $\bar{n}^{\ast}_{ex,A}=21$, $\bar{n}^{\ast}_{ex,B}=12$. Note that in $\bar{n}^{\ast}_{ex,A}$, $\bar{n}^{\ast}_{ex,B}$ we add values the four initial given examples stored in the agents’ inventories at the beginning. The reason is that the generalization function $p(t)$ outputs will effectively depend on them all. Therefore, at these threshold values, it would be very unlikely that $p_{B}>p_{A}$, and so it would be the same for the consensus at $B$.

Now we consider the Bayesian probabilities $p_{A}(t)$ and $p_{B}(t)$ computed by agents and the corresponding number of learning attempts $no_{A}(t)$ and $no_{B}(t)$ made by agents at time $t$ to learn concept $C$ in association with word $A$ or $B$, respectively, i.e. the number of times that the agents compute $p_{A}$ or $p_{B}$ (only the case of a system starting with SIC is considered). We consider a single run of a system with $N=5000$ agents and study the average values $\bar{p}_{A}(t),\bar{p}_{B}(t)$, obtained by averaging $p_{A}(t)$ and $p_{B}(t)$ over the agents of the system. We also assume a coarse-grained view, consisting in an additional average of $\bar{p}_{A}(t)$, $\bar{p}_{B}(t)$, and $no_{A}$, $no_{B}$, over a a temporal bin $\Delta t=16\times 10^{3}$, in order to reduce random fluctuations. Figure 10 shows the time evolution of the average probabilities $\bar{p}_{A}(t)$ and $\bar{p}_{B}(t)$ in the time-range where data allow a good statistics. The probabilities grow monotonically and eventually reach the value one. While this points at an equivalence between the mean-field regime of the Bayesian naming game and that of the two-conventions NG model, in which agents learn at the first attempt (one-shoot learning), such an equivalence is suggested but not fully reproduced by the coarse-grained analysis. The time evolution of the number of learning attempts $no_{A}(t)$ and $no_{B}(t)$ shows that they are negligible both at the beginning and at the end of the dynamics — see inset in Fig. 10. This is due to the fact that at the beginning it is most likely that either interactions between agents with the same conventions take place (starting with SIC, each agent has a probability of 50% to interact with an agent having the same convention) or interactions between agents with different conventions but with still too small inventories to be able to generalize concept $C$, leading to reinforcement processes only. When approaching consensus, agents with one of the conventions constitute the large majority of the population and thus they are again most likely to interact through reinforcements only. Thus, the largest numbers of attempt to learn concept $C$ in association with $A$ and $B$ are expected to occur at the intermediate stage of the dynamics. In fact, $no_{A}(t)$ and $no_{B}(t)$ are observed to reach a maximum at $t\approx t_{conv}/2$ for any given system size $N$, as visible in the inset of Fig.  10. Notice that also the fraction of agents $n_{AB}$, who know both conventions and can communicate using both name $A$ and name $B$, possibly allowing other agents to generalize in association with name $A$ or $B$, reaches its maximum roughly at the same time.

We introduced a novel agent-based model that describes the appearance of linguistic consensus through a word-learning process. The work presented is exploratory in nature, concerning the minimal problem of a single concept that can be associated to two different possible names $A$ or $B$, but is aimed at providing a prototype of general framework for describing the interaction between the social and the cognitive dimension. To this aim, the model is constructed on the basis of the semiotic dynamics of the NG model and is then extended by adding a Bayesian cognitive process, mimicking human learning processes.

The model describes in a natural way (1) the uncertainty accompanying the first phase of a learning process, (2) the gradual reduction of the uncertainty as more and more examples are provided, and (3) the ability to learn from a few examples. The semiotic dynamics of the synonyms is different from the basic NG, in that it depends on parameters that are of a strictly cognitive nature, such as the thresholds $n_{ex}^{\ast}$ of the number of examples necessary before an agent can try to generalize and the acceptance threshold $p^{\ast}$ for carrying out the generalization of a concept. The interplay between the asymmetry of the conventions $A$ and $B$, the system size, and the stochastic character of the time evolution, have dramatic consequences on the consensus dynamics: there is a critical time $t^{\ast}>0$, when the system begins to move in the phase-plane to eventually converge toward a consensus state; there is a critical system size $N^{*}$, such that for $N<N^{\ast}$ the system can end up in any of the two consensus states and the convergence times depends on $N$; there is an asymmetry in the branching probabilities that the system converges toward one of the two possible conventions and of the corresponding convergence times; the scaling laws of the convergence times versus $N$ differ from those observed in the basic NG model, because they depend on the learning experience of the agents.

The cognitive dimension offers additional possibilities for modelling in terms of specific cognitive parameters problems that are out of the reach of traditional social dynamics models. The model illustrated in this work represents a step toward a generalized Bayesian approach to social interactions, leading to cultural conventions.

Future work can address specific problems of current interest from the point of view of cognitive processes; or features relevant from the general standpoint of complexity theory. In the first case, it is possible to study in the cognitive dimension the semiotic dynamics of homonyms, synonyms, and innovation, e.g., the cognitive conditions leading to a name $A_{1}$, associated to a concept $C_{1}$, splitting into two names $A_{1}$ and $A_{2}$, associated to two related but distinct concepts $C_{1}$ and $C_{2}$, as more examples become available that make the two concepts eventually distinguishable from each other — a type of problems that cannot be tackled within models of cultural competition. In the second case, one can mention the classical problem of the interplay between a central information source (bias) and the local influences of individuals — this time in a cognitive framework.

Another question to be investigated within a cognitive framework would be the role of heterogeneity. In fact, heterogeneity is known to characterize most of the known complex systems at various levels — here the diversity could affect the dynamical parameters of e.g. the different competing names as well as those of the agents. Heterogeneity of individuals can lead to counter-intuitive effects, such as resonant behaviors Tessone2009a ; VazMartins2009a . Furthermore, the complex, heterogeneous nature of a local underlying social network can change drastically the co-evolution and the time-scales of the conventions in competition with each other Toivonen2009a .

The authors acknowledge support from the Estonian Ministry of Education and Research through Institutional Research Funding IUT (IUT39-1), the Estonian Research Council through Grant PUT (PUT1356), and the ERDF (European Development Research Fund) CoE (Center of Excellence) program through Grant TK133.

We also thank Andrea Baronchelli for providing useful remarks about the naming game model and the manuscript.