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

TL;DR

The paper analyzes noisy majority-rule dynamics on annealed hypergraphs with a random group-size distribution $P(n)$. Using a drift-diffusion framework, it derives a general critical independence threshold $q_c(P)=\frac{\mathcal{A}_1(P)}{\mathcal{A}_1(P)+\langle n\rangle}$ at $p=1/2$, showing that heavier tails in $P(n)$ increase robustness by enabling rare large-group realignments. It then characterizes the finite-size scaling of consensus and disordering times, $T_{\mathrm{cons}}$ and $T_{\mathrm{dis}}$, with explicit prefactors for $n$-uniform, geometric, and power-law ensembles, including crossovers for heavy-tailed distributions. The exit probability near coexistence collapses to a universal error-function form $E(c_0)=\tfrac{1}{2}[1+\operatorname{erf}(\sqrt{\gamma(P)}(c_0-\tfrac{1}{2}))]$, with $\gamma(P)=\frac{\mathcal{A}_1(P)}{D_0(P)}$. Overall, the full $P(n)$ controls both ordering robustness and relaxation dynamics, highlighting how higher-order interaction heterogeneity shapes social consensus processes.

Abstract

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.

Figures (7)

• Figure 1: Schematic illustration of a single update in the model. A hyperedge of size $n$ (example: $n=3$) is sampled according to $P(n)$, and the within-hyperedge majority state is identified from the pre-update configuration. Each of the $n$ selected agents then updates as follows: with probability $1-q$ it adopts the within-hyperedge majority state, whereas with probability $q$ it acts independently (ignoring the local majority) and adopts state $+1$ with probability $p$ (red) or state $-1$ with probability $1-p$ (blue). Gray nodes indicate the selected agents prior to the update. Ties for even $n$ yield no change (not shown).
• Figure 2: Phase diagram of the noisy majority-rule dynamics on annealed hypergraphs in the up--down symmetric case $p=1/2$. (a) $q_c$ for the $n$-uniform ensemble [Eq. \ref{['eq:crit_q_uniform']}] and the shifted geometric ensemble [Eq. \ref{['eq:crit_q_geo']}], plotted versus the characteristic group size $n^*\equiv n=\langle n\rangle$. (b) $q_c(\alpha)$ for the truncated power-law ensemble as a function of the tail exponent $\alpha$ [Eq. \ref{['eq:qc_pl_asymp']}]. The ordered phase corresponds to $q<q_c$, while the disordered phase corresponds to $q>q_c$.
• Figure 3: Consensus time $T$ versus system size $N$ for three hyperedge-size distributions starting from the balanced initial condition $c(0)=1/2$: (a) annealed $n$-uniform for several $n$; (b) shifted geometric for several mean sizes $\langle n\rangle$; and (c) truncated power law for several tail exponents $\alpha$. Solid curves show the mean-field prediction from Eq. \ref{['eq:T_logN_generic']} with the corresponding prefactors, while symbols connected by dashed lines are MC data averages over $10^{5}$ runs. The data confirm the predicted crossover: logarithmic growth with a distribution-dependent prefactor for uniform/geometric ensembles and for power laws with $\alpha>2$, and saturation for heavy tails $\alpha\le 2$.
• Figure 4: Consensus time $T$ at fixed system size $N=10^{5}$ for three hyperedge-size distributions under two initial conditions. Panels (a)--(c): balanced start $c(0)=0.5$. Panels (d)--(f): unbalanced start $c(0)=0.7$. (a),(d) Annealed $n$-uniform (varying $n$); (b),(e) shifted geometric (varying $\langle n\rangle$); (c),(f) truncated power law (varying $\alpha$). Solid curves: mean-field predictions. Symbols with dashed lines: MC averages over $10^{5}$ realizations. Shaded bands: one standard deviation.
• Figure 5: Disordering time $T_{\mathrm{dis}}$ versus system size $N$ for three hyperedge-size distributions starting from the fully ordered state $c(0)=1$: (a) annealed $n$-uniform for various $n$; (b) shifted geometric for various $\langle n\rangle$; (c) truncated power law for various $\alpha$. Data are shown at a fixed relative distance from the threshold, $q=\kappa q_c$ with $\kappa=1.2$. Solid curves: mean-field prediction of Eq. \ref{['eq:Tdis_kappa']} with the corresponding prefactors. Symbols with dashed lines: MC averages over $10^{5}$ runs. The data confirm the logarithmic growth in the disordered phase ($q > q_c$) for all distributions, including power laws with $\alpha > 5/2$.