Nonequilibrium dynamics of language games on complex networks

Diversity, competition, extinction: the ecophysics of language change

As indicated early by Charles Darwin, languages behave and change very much like living species. They display high diversity, differentiate in space and time, emerge and disappear. A large body of literature has explored the role of information exchanges and communicative constraints in groups of agents under selective scenarios. These models have been very helpful in providing a rationale on how complex forms of communication emerge under evolutionary pressures. However, other patterns of large-scale organization can be described using mathematical methods ignoring communicative traits. These approaches consider shorter time scales and have been developed by exploiting both theoretical ecology and statistical physics methods. The models are reviewed here and include extinction, invasion, origination, spatial organization, coexistence and diversity as key concepts and are very simple in their defining rules. Such simplicity is used in order to catch the most fundamental laws of organization and those universal ingredients responsible for qualitative traits. The similarities between observed and predicted patterns indicate that an ecological theory of language is emerging, supporting (on a quantitative basis) its ecological nature, although key differences are also present. Here, we critically review some recent advances and outline their implications and limitations as well as highlight problems for future research.

Language structure in the n -object naming game

We examine a naming game with two agents trying to establish a common vocabulary for n objects. Such efforts lead to the emergence of language that allows for an efficient communication and exhibits some degree of homonymy and synonymy. Although homonymy reduces the communication efficiency, it seems to be a dynamical trap that persists for a long, and perhaps indefinite, time. On the other hand, synonymy does not reduce the efficiency of communication but appears to be only a transient feature of the language. Thus, in our model the role of synonymy decreases and in the long-time limit it becomes negligible. A similar rareness of synonymy is observed in present natural languages. The role of noise, that distorts the communicated words, is also examined. Although, in general, the noise reduces the communication efficiency, it also regroups the words so that they are more evenly distributed within the available "verbal" space.

Effects of social diversity on the emergence of global consensus in opinion dynamics

We propose a variant of the voter model by introducing the social diversity in the evolution process. Each individual is assigned a weight that is proportional to the power of its degree, where the power exponent alpha is an adjustable parameter that controls the level of diversity among individuals in the network. At each time step, a pair of connected individuals, say i and j , are randomly selected to update their opinions. The probability p(i) of choosing is opinion as their common opinion is proportional to i s weight. We consider the scale-free topology and concentrate on the efficiency of reaching the final consensus, which is significant in characterizing the self-organized systems. Interestingly, it is found that there exists an optimal value of alpha, leading to the shortest consensus time. This phenomenon indicates that, although a strong influence of high-degree individuals is helpful for quick consensus achievement, over strong influence inhibits the convergence process. Other quantities, such as the probability of an individual's initial opinion becomes the final opinion as a function of degree, the evolution of the number of opinion clusters, as well as the relationship between average consensus time and the network size, are also studied. Our results are helpful for better understanding the role of degree heterogeneity of the individuals in the opinion dynamics.

Consequence of reputation in an open-ended naming game

We study a modified version of the naming game, a recently introduced model which describes how shared vocabulary can emerge spontaneously in a population without any central control. In particular, we introduce a mechanism that allows a continuous interchange with the external inventory of words. A playing strategy, influenced by the hierarchical structure that individuals' reputation defines in the community, is implemented. We analyze how these features influence the convergence times, the cognitive efforts of the agents, and the scaling behavior in memory and time.

Coevolutionary dynamics of opinions and networks: from diversity to uniformity

We investigate the coevolutionary dynamics of opinions and networks based upon majority-preference (MP) and minority-avoidance (MA) rules. Under MP, individuals adopt the majority opinion among their neighbors; while in MA individuals can break the link to one holding a minority and different opinion, and rewire either to neighbors of their neighbors with the same opinion or to a random one from the whole population except their nearest neighbors. We study opinion formation as a result of combination of these two competing rules, with a parameter tuning the balance between them. We find that the underlying network can be self-organized into connected communities with like-minded individuals belonging to the same group; thus a broad variety of opinions coexist. Diverse opinions disappear in a population in which all individuals share a uniform opinion, when the model parameter exceeds a critical value. Furthermore, we show that an increasing tendency to redirect to neighbors of neighbors is more likely to result in a consensus of opinion.

Random walks on complex trees

We study the properties of random walks on complex trees. We observe that the absence of loops is reflected in physical observables showing large differences with respect to their looped counterparts. First, both the vertex discovery rate and the mean topological displacement from the origin present a considerable slowing down in the tree case. Second, the mean first passage time (MFPT) displays a logarithmic degree dependence, in contrast to the inverse degree shape exhibited in looped networks. This deviation can be ascribed to the dominance of source-target topological distance in trees. To show this, we study the distance dependence of a symmetrized MFPT and derive its logarithmic profile, obtaining good agreement with simulation results. These unique properties shed light on the recently reported anomalies observed in diffusive dynamical systems on trees.

Coherence thresholds in models of language change and evolution: the effects of noise, dynamics, and network of interactions

A simple model of language evolution proposed by Komarova, Niyogi, and Nowak is characterized by a payoff in communicative function and by an error in learning that measure the accuracy in language acquisition. The time scale for language change is generational, and the model's equations in the mean-field approximation are a particular case of the replicator-mutator equations of evolutionary dynamics. In well-mixed populations, this model exhibits a critical coherence threshold; i.e., a minimal accuracy in the learning process is required to maintain linguistic coherence. In this work, we analyze in detail the effects of different fitness-based dynamics driving linguistic coherence and of the network of interactions on the nature of the coherence threshold by performing numerical simulations and theoretical analyses of three different models of language change in finite populations with two types of structure: fully connected networks and regular random graphs. We find that although the threshold of the original replicator-mutator evolutionary model is robust with respect to the structure of the network of contacts, the coherence threshold of related fitness-driven models may be strongly affected by this feature.

Indications of a possible symmetry and its breaking in a many-agent model obeying quantum statistics

The results of computer simulations are presented which give evidence for the existence of an interesting symmetry in a many-agent model which demonstrates, in special cases, both Bose-Einstein and Fermi-Dirac statistics. This symmetry is expressed in the close vicinity of the mean values of the degree of ultrametricity and the fraction of isosceles of the sets of agent memories (histories) coded by two different information-loss coding schemes. It is shown that this (in some sense) approximate statistical supersymmetry is probably broken at low temperatures--below some condensation limit. This breaking leads to the appearance of specific coding schemes for boson and fermion histories. The meaning of this specificity is revealed by applying the interpretation of the many-agent model described earlier [A. A. Ezhov and A. Yu. Khrennikov, Phys. Rev. E 71, 016138 (2005)].

Asymmetric negotiation in structured language games

We propose an asymmetric negotiation strategy to investigate the influence of high-degree agents on the agreement dynamics in a structured language game, the naming game. We introduce a model parameter, which governs the frequency of high-degree agents acting as speakers in communication. It is found that there exists an optimal value of the parameter that induces the fastest convergence to a global consensus on naming an object for both scale-free and small-world naming games. This phenomenon indicates that, although a strong influence of high-degree agents favors consensus achievement, very strong influences inhibit the convergence process, making it even slower than in the absence of influence of high-degree agents. Investigation of the total memory used by agents implies that there is some trade-off between the convergence speed and the required total memory. Other quantities, including the evolution of the number of different names and the relationship between agents' memories and their degrees, are also studied. The results are helpful for better understanding of the dynamics of the naming game with asymmetric negotiation strategy.

Naming games in two-dimensional and small-world-connected random geometric networks

We investigate a prototypical agent-based model, the naming game, on two-dimensional random geometric networks. The naming game [Baronchelli, J. Stat. Mech.: Theory Exp. (2006) P06014] is a minimal model, employing local communications that captures the emergence of shared communication schemes (languages) in a population of autonomous semiotic agents. Implementing the naming games with local broadcasts on random geometric graphs, serves as a model for agreement dynamics in large-scale, autonomously operating wireless sensor networks. Further, it captures essential features of the scaling properties of the agreement process for spatially embedded autonomous agents. Among the relevant observables capturing the temporal properties of the agreement process, we investigate the cluster-size distribution and the distribution of the agreement times, both exhibiting dynamic scaling. We also present results for the case when a small density of long-range communication links are added on top of the random geometric graph, resulting in a "small-world"-like network and yielding a significantly reduced time to reach global agreement. We construct a finite-size scaling analysis for the agreement times in this case.

Consensus formation on adaptive networks

The structure of a network can significantly influence the properties of the dynamical processes that take place on them. While many studies have been paid to this influence, much less attention has been devoted to the interplay and feedback mechanisms between dynamical processes and network topology on adaptive networks. Adaptive rewiring of links can happen in real life systems such as acquaintance networks, where people are more likely to maintain a social connection if their views and values are similar. In our study, we consider different variants of a model for consensus formation. Our investigations reveal that the adaptation of the network topology fosters cluster formation by enhancing communication between agents of similar opinion, although it also promotes the division of these clusters. The temporal behavior is also strongly affected by adaptivity: while, on static networks, it is influenced by percolation properties, on adaptive networks, both the early and late time evolutions of the system are determined by the rewiring process. The investigation of a variant of the model reveals that the scenarios of transitions between consensus and polarized states are more robust on adaptive networks.

Role of connectivity-induced weighted words in language games

We present a modified naming game by introducing weights of words in the evolution process. We assign the weight of a word spoken by an agent according to its connectivity, which is a natural reflection of the agent's influence in population. A tunable parameter is introduced, governing the word weight based on the connectivity of agents. We consider the scale-free topology and concentrate on the efficiency of reaching the final consensus, which is of high importance in the self-organized system. Interestingly, it is found that there exists an optimal parameter value, leading to the fastest convergence. This indicates appropriate hub's effects favor the achievement of consensus. The evolution of distinct words helps to give a qualitative explanation of this phenomena. Similar nontrivial phenomena are observed in the total memory of agents with a peak in the middle range of parameter values. Other relevant characters are provided as well, including the time evolution of total memory and success rate for different parameter values as well as the average degree of the network, which are helpful for understanding the dynamics of the modified naming game in detail.