A note on the consensus time of mean-field majority-rule dynamics

In this work, it is pointed out that in the mean-field version of majority-rule opinion dynamics, the dependence of the consensus time on the population size exhibits two regimes. This is determined by the size distribution of the groups that, at each evolution step, gather to reach agreement. When the group size distribution has a finite mean value, the previously known logarithmic dependence on the population size holds. On the other hand, when the mean group size diverges, the consensus time and the population size are related through a power law. Numerical simulations validate this semi-quantitative analytical prediction.

💡 Research Summary

The paper investigates how the time required for a population to reach full consensus in the mean‑field version of the majority‑rule opinion model depends on the size of the population, N, when the size of the interacting group, G, is drawn from a probability distribution rather than being fixed. In the classic setting where G is constant (e.g., G = 3), the average number of update steps needed to reach consensus, S_c, scales as N log N for large N, a result that follows from the equivalence of the dynamics to a biased random walk with absorbing boundaries.

The authors generalize the model by allowing G to be a random variable drawn at each step from a distribution p_G that decays as a power law for large G: p_G ∝ G^{‑γ} with γ > 1. When G ≥ N, a single update instantly forces the whole population into consensus; this is interpreted as a “large‑jump” event in the random‑walk picture. The average waiting time for such an event, S_w, can be calculated analytically and scales as S_w ∝ N^{γ‑1} (Eq. 5).

Comparing S_w with the conventional random‑walk time S_c yields a critical exponent γ_c = 2. For γ > 2 the mean group size ⟨G⟩ is finite, S_w ≫ S_c, and the consensus dynamics are dominated by the usual incremental random‑walk mechanism, preserving the N log N scaling. For γ ≤ 2 the mean group size diverges, S_w ≪ S_c, and consensus is typically achieved by a single large‑G event; consequently the consensus time follows a power‑law dependence on N, namely S ∝ N^{γ‑1} (with the special case γ = 2 giving linear scaling S ∝ N).

To validate these predictions, extensive Monte‑Carlo simulations were performed for populations ranging from 10² to 10⁵ agents. Group sizes were generated as odd numbers G = 2g + 1 with g drawn from a normalized power‑law distribution p_g ∝ g^{‑γ}. The results, displayed in Figures 1‑3, confirm the theoretical expectations: for γ > 2 the normalized consensus time S/N exhibits a logarithmic dependence on N, matching Eq. (2); for γ = 2 the data collapse onto a straight line indicating S ∝ N; and for γ < 2 the measured consensus times coincide with the analytically derived waiting times S_w, demonstrating that large‑G events dominate. Figure 3 further shows that the fraction of realizations in which consensus is achieved via a large‑G event approaches unity for γ < 2 and decreases sharply for γ > 2, while remaining roughly constant (≈0.57) at the critical point γ = 2.

The study concludes that the consensus time in majority‑rule dynamics exhibits two distinct scaling regimes determined solely by the tail of the group‑size distribution. A fast‑decaying distribution (finite ⟨G⟩) reproduces the classic logarithmic scaling, whereas a slowly decaying power‑law (divergent ⟨G⟩) leads to a power‑law scaling driven by rare, system‑spanning interaction events. This dichotomy mirrors the transition between normal and anomalous diffusion in random‑walk theory, highlighting how occasional large‑scale social interactions (e.g., mass meetings, viral online events) can dramatically accelerate opinion convergence.