Discontinuous phase transition in an open-ended Naming Game

In this work we study on a 2-dimensional square lattice a recent version of the Naming Game, an agent-based model used for describing the emergence of linguistic structures. The system is open-ended and agents can invent new words all along the evolution of the game, picking them up from a pool characterised by a Gaussian distribution with standard deviation $\sigma$. The model displays a nonequilibrium phase transition at a critical point $\sigma_{c}\approx 25.6$, which separates an absorbing consensus state from an active fragmented state where agents continuously exchange different words. The finite-size scaling analysis of our simulations strongly suggests that the phase transition is discontinuous.

💡 Research Summary

In this paper the authors investigate a variant of the Naming Game (NG) in which agents are placed on a two‑dimensional square lattice and are allowed to invent an unlimited number of new words throughout the dynamics. New words are drawn from a Gaussian distribution centered at zero with standard deviation σ, which therefore controls the breadth of the lexical repertoire that can be generated. The interaction rules are the same as in the original NG: at each elementary step a randomly chosen pair of nearest‑neighbour agents interacts; the speaker either selects a word from its inventory or, if empty, creates a new one; the hearer checks whether it already possesses the transmitted word. If the hearer knows the word the interaction is successful and both agents prune their inventories to keep only that word; otherwise the interaction fails and the speaker invents another new word (again sampled from the Gaussian pool).

The authors first compare the dynamics on the lattice with the previously studied fully‑connected (mean‑field) version. While both systems display an early burst of word creation followed by a maximum in the total number of words, the lattice shows no prolonged plateau: after the peak the number of words drops sharply and coarsening proceeds rapidly, whereas the mean‑field case exhibits a long, slowly decreasing plateau because agents can interact with any other agent, leading to many failed communications and sustained word proliferation. Consequently, the success rate S(t) grows much faster on the lattice, reflecting the locality of interactions.

A central focus of the work is the identification and characterization of a nonequilibrium phase transition driven by σ. For small σ the variance of the word pool is limited, allowing the system to reach an absorbing consensus state in which all agents share a single word. For large σ the continual invention of highly diverse words overwhelms the ability of local interactions to align inventories, and the system settles into an active fragmented phase where clusters of agents hold different words indefinitely. By scanning σ the authors locate a critical value σ_c≈25.6 that separates these two regimes.

Finite‑size scaling analyses are performed on lattices of various linear sizes L (up to L=100). The order parameter (e.g., the normalized total number of words) exhibits a discontinuous jump at σ_c, and the Binder cumulant shows a pronounced minimum that deepens with increasing system size. Histograms of the order parameter near σ_c become bimodal, indicating coexistence of the two phases. Moreover, the width of the transition region shrinks as L grows, consistent with a first‑order (discontinuous) transition rather than a continuous one such as directed percolation.

The convergence time T_c, defined as the average time needed to reach consensus, scales linearly with the total number of agents P=L² (T_c∝P) on the lattice, in contrast with the mean‑field case where T_c also grows linearly but with a smaller prefactor. As σ approaches σ_c from below, T_c increases exponentially, and for σ≫σ_c it appears to diverge, reflecting the impossibility of reaching consensus in the fragmented phase.

The paper discusses the broader implications of these findings. The presence of an unlimited lexical space combined with a tunable diversity constraint produces a novel universality class for nonequilibrium ordering processes, distinct from the well‑studied Ising, directed‑percolation, or generalized voter classes. The results suggest that in real social or linguistic systems, even when agents can generate arbitrarily many new conventions, a moderate restriction on the diversity of inventions (analogous to a finite σ) is essential for collective agreement to emerge. Moreover, the strong influence of spatial locality highlights that short‑range interactions can dramatically accelerate consensus formation compared with fully mixed populations.

In summary, the study provides a comprehensive numerical investigation of an open‑ended Naming Game on a two‑dimensional lattice, demonstrates that the model undergoes a discontinuous phase transition at σ_c≈25.6, and elucidates how the interplay between word invention diversity and local interaction topology governs the emergence of consensus versus persistent fragmentation. This contributes valuable insights to the fields of language evolution, cultural dynamics, and nonequilibrium statistical physics.