Naming Game on Adaptive Weighted Networks

We examine a naming game on an adaptive weighted network. A weight of connection for a given pair of agents depends on their communication success rate and determines the probability with which the agents communicate. In some cases, depending on the parameters of the model, the preference toward successfully communicating agents is basically negligible and the model behaves similarly to the naming game on a complete graph. In particular, it quickly reaches a single-language state, albeit some details of the dynamics are different from the complete-graph version. In some other cases, the preference toward successfully communicating agents becomes much more relevant and the model gets trapped in a multi-language regime. In this case gradual coarsening and extinction of languages lead to the emergence of a dominant language, albeit with some other languages still being present. A comparison of distribution of languages in our model and in the human population is discussed.

💡 Research Summary

The paper extends the classic Naming Game—a model of cultural transmission in which agents negotiate a shared vocabulary—to an adaptive weighted network where the strength of each link evolves according to the past success of communication between the two agents. For a pair of agents (i, j) the success rate sᵢⱼ is defined as the ratio of successful interactions to total attempts. The link weight is then wᵢⱼ = sᵢⱼ + ε, where ε > 0 is a small constant that guarantees a non‑zero probability of interaction even when the pair has never succeeded. When an agent i is chosen as speaker (uniformly at random), the hearer j is selected with probability pᵢⱼ = wᵢⱼ / ∑ₖwᵢₖ (roulette wheel selection). The interaction follows the minimal Naming Game rules: the speaker utters a randomly chosen word from its lexicon (or invents a new one if empty); if the hearer already knows the word the game succeeds and both agents keep only that word, otherwise the hearer adds the word to its lexicon. After each interaction sᵢⱼ and consequently wᵢⱼ are updated.

The authors explore the dynamics for populations N ranging from 10² to 10⁴ and for ε = 10⁻⁴ and 10⁻⁵. They monitor three main observables: (1) the instantaneous success rate s, computed over the most recent N interactions; (2) the total number of distinct words L (interpreted as the number of languages when most agents hold a single word); and (3) N_d, the number of agents possessing the most common word. Simulations are averaged over many independent runs.

Two qualitatively different regimes emerge, governed primarily by the composite parameter N ε²:

1. Single‑language (complete‑graph‑like) regime (large N ε²).
   When ε is relatively large or the population is sufficiently big, the adaptive weights remain close to uniform. Consequently the interaction pattern approximates that of a complete graph. Initially many words coexist, but a rapid coarsening process reduces L dramatically. The success rate s climbs to near‑unity after a characteristic transient of about t ≈ 750 interaction units, a value that is essentially independent of N. After this transient the system converges to a consensus state where every agent shares the same word, reproducing the well‑known behavior of the Naming Game on static complete graphs. The authors note that the time scale of the transient is set by the rate of “outside‑cluster” interactions, which scales as ε N, and the subsequent incorporation of an outsider into the dominant cluster occurs in two steps whose combined probability scales as N ε².

2. Multi‑language (glassy) regime (small N ε²).
   For smaller ε or smaller populations, the adaptive weights quickly become highly heterogeneous. Agents preferentially interact with a limited set of partners with whom they have high past success, leading to the formation of tightly‑connected clusters. Within each cluster a single word dominates, but inter‑cluster communication is rare because the baseline ε‑term is too weak to overcome the weight disparity. As a result L stabilizes at a value far above one and persists for very long times, reminiscent of a glassy phase in statistical physics where the system is trapped in a metastable configuration. The number of languages remains roughly constant after a brief initial drop, and the overall success rate stays below unity, indicating sustained multilingual coexistence.

The authors argue that N ε² acts as the effective control parameter because the key processes that allow an outsider to join the dominant cluster—(i) a failure that introduces the dominant word into the outsider’s lexicon (rate ∝ ε) and (ii) a subsequent successful interaction that eliminates all competing words (rate ∝ N ε)—combine to give a transition rate proportional to N ε². Their numerical data, plotted for constant N ε², show good collapse across different system sizes, supporting this scaling hypothesis.

Finally, the paper compares the model’s language‑size distribution with empirical data on human languages. While the model is highly abstract, it reproduces two salient features of real linguistic landscapes: (i) the existence of a dominant language spoken by a large fraction of the population, and (ii) a long‑tail of many smaller languages that persist over extended periods. By tuning ε (which can be interpreted as the willingness to interact with unfamiliar speakers) the model can shift between rapid homogenization and stable multilingualism, offering a simple yet powerful framework for investigating how social network adaptation influences language evolution.

In summary, the study demonstrates that embedding the Naming Game in an adaptive weighted network yields rich dynamical behavior controlled by the product N ε². Large values drive the system toward fast consensus, reproducing the classic complete‑graph results, whereas small values generate heterogeneous, long‑lived multilingual states. This work bridges the gap between stylized models of language emergence and the complex, heterogeneous interaction patterns observed in real societies, and it provides a quantitative tool for exploring the interplay between network adaptation, cultural transmission, and linguistic diversity.