Divide-and-rule policy in the Naming Game

TL;DR

The paper investigates multi-opinion dynamics in the Naming Game with committed agents, revealing a tipping-point where the largest committed group can rapidly dominate a population on a complete graph. It develops a mean-field framework for the original model, then introduces a recursive density-evolution scheme that compresses the exponential state space to a tractable set of variables, enabling analysis for arbitrary configurations. Agent-based simulations on ER, small-world, and scale-free networks validate the divide-and-conquer policy: distributing minority committed agents across many small groups lowers the critical fraction required for dominance, with network topology modulating the effect. These findings illuminate how divided opposition and network structure influence opinion dominance and offer a quantitative toolkit for studying real-world polarization and persuasion dynamics.

Abstract

The Naming Game is a classic model for studying the emergence and evolution of language within a population. In this paper, we extend the traditional Naming Game model to encompass multiple committed opinions and investigate the system dynamics on the complete graph with an arbitrarily large population and random networks of finite size. For the fully connected complete graph, the homogeneous mixing condition enables us to use mean-field theory to analyze the opinion evolution of the system. However, when the number of opinions increases, the number of variables describing the system grows exponentially. To mitigate this, we focus on a special scenario where the largest group of committed agents competes with a motley of committed groups, each of which is smaller than the largest one, while initially, most of uncommitted agents hold one unique opinion. This scenario is chosen for its recurrence in diverse societies and its potential for complexity reduction by unifying agents from smaller committed groups into one category. Our investigation reveals that when the size of the largest committed group reaches the critical threshold, most of uncommitted agents change their beliefs to this opinion, triggering a phase transition. Further, we derive the general formula for the multi-opinion evolution using a recursive approach, enabling investigation into any scenario. Finally, we employ agent-based simulations to reveal the opinion evolution and dominance transition in random graphs. Our results provide insights into the conditions under which the dominant opinion emerges in a population and the factors that influence these conditions.

Figures (13)

• Figure 1: Illustration of model dynamics (Original version). Agents hold one or multiple opinions on the question "the most popular sport in the world", and may update their opinions after an interaction (indicated by the yellow border of the opinion box). (a) An uncommitted speaker sends one of its three opinions randomly ("soccer" in the example) to an uncommitted neighbor (listener). If the listener already holds this opinion, both agents retain only this sent opinion ("soccer") as their new state, which is considered a success towards consensus. Otherwise, the listener adds the sent opinion to its state, resulting in a failure. (b) A committed speaker sends the only opinion to an uncommitted listener. Only the listener may change its status depending on whether the consensus is reached. (c) An uncommitted speaker communicates with a committed listener. Similar to (b), only the speaker may change its status. (d) Both speaker and listener are committed to a single opinion. Their statuses are not updated regardless of whether it is a success or failure.
• Figure 2: Phase transition from $B$ dominance to $A$ dominance and the corresponding tipping points for two-opinion scenario ($m=2$). All uncommitted agents support $B$ initially. (a) The stable density of agents with opinion $A$, $n_A$, changes as a function of the committed fraction $P_A$ for different values of $P_B$. As $P_A$ increases, the system is dominated by $A$. (b) The critical point $P_A^{(c)}$ changes with $P_B$. The increase in $P_B$ raises the value of the critical points for $A$ to dominate. The blue dots represent the discontinuous transition of $n_A$ versus $P_A$, while the red ones represent the continuous change.
• Figure 3: Phase transition from $B$ or $C$ dominance to $A$ dominance and the corresponding tipping points for three-opinion scenario ($m=3$). All uncommitted agents support $B$ initially. (a) The stable fraction of agents with opinion $A$ changes as a function of the committed fractions $P_A$ and $P_C$. Similarly, (b) and (c) show the change of $n_B$ and $n_c$, respectively. As $P_A$ increases, the system is dominated by $A$, including continuous and discontinuous transitions. When $P_A$ is below the tipping point $P_A^{(c)}$, the system is dominated by $B$ for small $P_{C}$ ($<0.1$) and dominated by $C$ for large $P_{C}$. (d) The critical point $P_A^{(c)}$ changes with $P_C$. As $P_C$ increases, the transition $n_A$ versus $P_A$ changes from the discontinuous transition (blue dots) to the continuous transition (red dots).
• Figure 4: The density of agents supporting single opinions at the steady state in scenario $S_1$. The number of opinions is set as $m=6$. The fractions of agents holding opinions $A$, $B$, and $C$ (representing any single opinion $C_1$, $C_2$, $C_3$, or $C_4$ in group $\tilde{A}$) are shown in subfigures (a) -- (c), respectively. As $P_A$ increases, the system is dominated by $A$, including continuous and discontinuous transitions. When $P_A$ is below the tipping point $P_A^{(c)}$, the system is dominated by $B$ for small $P_{\tilde{A}}$ ($<0.2$) and dominated by the unified group $\tilde{A}$ for large $P_{\tilde{A}}$.
• Figure 5: The phase transition and the critical points in scenario $S_1$. For different values of $m$ ($m=4, 5, 6, 7, 8, 9$), the critical point $p_A^{(c)}$ changes with (a) $p_0$ and (b) $P_{\tilde{A}}$ ($P_{\tilde{A}} = (m-2) p_0$). The initial decrease of $P_A^{(c)}$ with $p_0$ (or $P_{\tilde{A}}$) indicates that the minority group facilitates the dominance of $A$, corresponding to the discontinuous transition. The linear increasing regime suggests the competition between $A$ and the unified group $\tilde{A}$, corresponding to the continuous transition.