Competition between group interactions and nonlinearity in voter dynamics on hypergraphs

TL;DR

This work investigates how higher-order group interactions compete with nonlinearity in voter-like dynamics on hypergraphs by introducing and analyzing the group-driven voter model (GVM). The authors derive transition probabilities, exit probabilities, and exit times using mean-field and backward Kolmogorov approaches on annealed hypergraphs, revealing a universal logarithmic scaling of the exit time with system size, modulated by the nonlinearity strength and group-size constraints. They present extensive Monte-Carlo simulations across simplicial and $s$-uniform hypergraphs, and develop analytic approximations that capture the observed optimal consensus times as a function of the group structure. The results highlight the nontrivial impact of polyadic interactions on dynamical processes, offering insights into how group size, nonlinearity, and network composition shape consensus formation in systems with polyadic interactions.

Abstract

Social dynamics are often driven by both pairwise (i.e., dyadic) relationships and higher-order (i.e., polyadic) group relationships, which one can describe using hypergraphs. To gain insight into the impact of polyadic relationships on dynamical processes on networks, we formulate and study a polyadic voter process, which we call the group-driven voter model (GVM), that incorporates the effect of group interactions by nonlinear interactions that are subject to a group (i.e., hyperedge) constraint. By examining the competition between nonlinearity and group sizes, we show that the GVM achieves consensus faster than standard voter-model dynamics, with an optimal minimizing exit time. We substantiate this finding by using mean-field theory on annealed uniform hypergraphs with $N$ nodes, for which the exit time scales as ${\cal A}\ln N$, where the prefactor ${\cal A}$ depends both on the nonlinearity and on group-constraint factors. Our results reveal how competition between group interactions and nonlinearity shapes GVM dynamics. We thereby highlight the importance of such competing effects in complex systems with polyadic interactions.

Figures (6)

• Figure S1: The drift function $v(\rho) = R(\rho) - L(\rho)$ for different values of $q$ when (a) $s = 7$ and (b) $s = N \rightarrow \infty$. The curves for $q = 2$ and $q = 3$ in (b) completely overlap.
• Figure S2: The exit time $T(\rho_0)$ of the GVM on annealed $7$-uniform hypergraphs for different initial densities $\rho_0$ when (a) $q = 2$ and (b) $q = 5$. The symbols give the means of $10^6$ (when $N \le 10^4$) or $10^3$ (when $N \ge 10^5$) independent MC simulations of the GVM on $N$-node hypergraphs. The solid lines are theoretical results from (a) Eq. (\ref{['eq:T_q2_3']}) and (b) Eq. (\ref{['eq:q_4_5']}). We obtain the dashed lines from (a) Eq. (\ref{['eq:leading_q2']}) and (b) Eq. (\ref{['eq:leading_q5']}).
• Figure S3: The dependence on the nonlinearity strength $q$ of the exit time $\tau$ for the GVM on hypergraphs with $N = 10^4$ nodes with a geometric $P(s)$ for several values of the mean group size $\langle s \rangle$. The markers are means of $10^4$ independent MC simulations of the GVM on annealed hypergraphs, and the analytical curves are from numerical solutions of the recursion relation (\ref{['eq:recur']}).
• Figure S4: The dependence on the hypergraph size $N$ of the exit time $\tau$ for the GVM with power law $P(k)$ for several values of the power-law exponent $\gamma$. The markers are means of $10^3$ independent MC simulations of the GVM with nonlinearity strength $q = 2$ on annealed $5$-uniform hypergraphs. We draw curves between these points as visual guides. The "regular" case corresponds to $\gamma \to \infty$ and reduces to the GVM that we studied in the main manuscript. We use semilogarithmic coordinates, and we observe that the exit time $\tau$ scales logarithmically in $N$.
• Figure S5: Dependence of the exit time $\tau$ on the system size $N$ for a GVM without duplicate selections for nonlinearity strength $q=2$ and hyperedge sizes $s = 3$, $s = 7$, and $s = N$. The markers indicate the means of $10^6$ (when $N \le 10^4$) or $10^3$ (when $N \ge 10^5$) independent MC simulations of this GVM on annealed $s$-uniform hypergraphs. The lines are solutions of the recursion relation \ref{['eq:recur']} (solid) and the leading-order solution (\ref{['eq:tau_nodup']}) (dotted).