Effect of higher-order interactions on noisy majority-rule dynamics with random group sizes

We study noisy majority-rule dynamics on annealed hypergraphs to clarify how variability in group interaction sizes reshapes collective ordering. At each update, a group is sampled from a prescribed size distribution and either follows the strict within-group majority or, with probability $q$, updates independently under an external bias $p$. At the symmetric point $p=1/2$, we obtain an explicit analytical expression for the critical independence threshold $q_c$, which separates macroscopic ordering from a fluctuating mixed state and can be interpreted as the largest fraction of independent behavior that can be sustained without destroying order. Because $q_c$ is governed by group-size statistics through an effective majority leverage, broad and heavy-tailed size distributions enhance robustness by enabling rare large-group events to realign a substantial fraction of the population. We further derive analytical predictions, benchmarked against Monte Carlo simulations, for the leading finite-size behavior of relaxation: for narrow distributions the characteristic relaxation time typically grows logarithmically with system size, whereas sufficiently heavy-tailed power laws produce strong crossovers and make the large-system dynamics sensitive to how $q$ approaches the transition. In the pure majority-rule limit, we find a crossover from conventional logarithmic consensus times to rapid ordering driven by occasional macroscopic groups, and the exit probability near coexistence collapses onto a universal error-function form controlled by a single structural parameter.

💡 Research Summary

The paper investigates a noisy majority‑rule (MR) opinion dynamics model defined on annealed hypergraphs where the size of the interacting group is drawn from a prescribed distribution P(n). At each elementary update a group of size n is sampled, then with probability 1‑q all n agents adopt the local majority opinion within the group, while with probability q each agent updates independently, choosing state +1 with probability p and ‑1 with probability 1‑p. The parameters q and p represent, respectively, the level of independence (noise) and an external bias (e.g., media influence).

The authors first formulate the stochastic process as a multi‑spin‑jump Markov chain and derive drift v(c) and diffusion D(c) coefficients for the fraction c of +1 opinions using a standard diffusion approximation. The drift consists of two contributions: (i) a restoring term proportional to (1‑q) that amplifies any small deviation from the symmetric state c=½ through a combinatorial factor ΔM_n(c) that quantifies the “majority leverage” of a group of size n, and (ii) a noise term proportional to q that pulls the system toward the unbiased fixed point c=p.

Focusing on the up‑down symmetric case p=½, the authors linearize the drift around c=½ and obtain a simple stability condition. The symmetric fixed point loses stability when

 (1‑q) A₁(P) > q ⟨n⟩,

where ⟨n⟩ is the mean group size and A₁(P) = ∑ₙ P(n)