Mean-Field Theory for Heider Balance under Heterogeneous Social Temperatures

TL;DR

This work introduces a generalized Heider balance model on a complete graph in which each link is assigned its own social temperature and establishes universal bounds for the critical transition, where the homogeneous-temperature limit provides a universal lower bound for the critical mean of an inverse-temperature distribution governing the transition.

Abstract

Heider balance theory provides a fundamental framework for understanding the formation of friendly and hostile relations in social networks. Existing stochastic formulations typically assume a uniform social temperature, implying that all interpersonal relations fluctuate with the same intensity. However, studies show that social interactions are highly heterogeneous, with broad variability in stability, volatility, and susceptibility to change. In this work, we introduce a generalized Heider balance model on a complete graph in which each link is assigned its own social temperature. Within a mean-field formulation, we derive a distribution-dependent self-consistency condition for the collective opinion state and identify the criteria governing the transition between polarized and non-polarized configurations. This framework reveals how the entire distribution of interaction heterogeneity shapes the macroscopic behavior of the system. We show that the functional form of the inverse-temperature distribution, in particular whether it is light-tailed or heavy-tailed, leads to qualitatively distinct phase diagrams. We also establish universal bounds for the critical transition, where the homogeneous-temperature limit provides a universal lower bound for the critical mean of an inverse-temperature distribution governing the transition. Numerical simulations confirm the theoretical predictions and highlight the nontrivial effects introduced by heterogeneity. Our results provide a unified route to understanding structural balance in realistic social systems and lay the groundwork for extensions incorporating fluctuations beyond mean field, external fields, and network topologies beyond the complete graph.

Figures (6)

• Figure 1: Schematics of a complete graph for Heider balance. Each link is assigned with its own social temperature represented by different colors.
• Figure 2: Mean-field solutions of $\left<x\right>$ vs. $\mu^{-1}$ for $\sigma=1$ under the gamma distribution \ref{['eq:gamma']}. The solid curve and line represent the stable solutions, while the dotted line represents the unstable solution. The critical point is approximately given by $\mu_{\rm C}^{-1}\approx 0.478$. The plots of $\mu^{-1}\in [0, 0.075]$ are linearly continued according to analytical properties shown in Sec. II of SM. The red circles show the final 1000-step averages of 10000-step simulations of \ref{['eq:stochastic_rule']} on a complete graph with $N=100$, which agree with the mean-field results, giving the critical point as $\mu_{\rm C}^{-1}\approx 0.475$.
• Figure 3: Phase diagram with respect to $(\sigma,\mu^{-1})$ under the gamma distribution \ref{['eq:gamma']}. The diagram is separated by three fixed-points area and only one trivial fixed-point ($\left<x\right>=0$) area. The red-dotted curve and line show the upper and lower bounds for $\mu_{\rm C}$ in Eq. \ref{['eq:bounds']}, respectively, between which the critical curve separating the two areas is located. The magenta circles represent the simulation results of $\mu_{\rm C}$ on a complete graph with $N=100$.
• Figure 4: Phase diagram with respect to $(\sigma, \mu^{-1})$ under the Pareto distribution \ref{['eq:pareto']}. The diagram is separated by three fixed-points area and only one trivial fixed-point ($\left<x\right>=0$) area. The red-dotted curve and line show the upper and lower bounds for $\mu_{\rm C}$ in Eq. \ref{['eq:bounds']}, respectively, between which the critical curve separating the two areas is located. There is an asymptote (orange-dotted line) $\mu^{-1}\approx 0.437$ for the critical curve when $\sigma \to +\infty$. The magenta circles represent the simulation results of $\mu_{\rm C}$ on a complete graph with $N=100$.
• Figure S1: Mean-field solutions of $\left<x\right>$ vs. $\mu^{-1}$ for $\sigma=1$ under the Pareto distribution \ref{['eq:pareto']}. The solid curve and line represent the stable solutions, while the dotted line represents the unstable solution. The critical point is approximately given by $\mu_{\rm C}^{-1}\approx 0.524$. The plots of $\mu^{-1}\in [0, 0.01]$ are linearly continued according to analytical properties shown in Sec. II of SM. The red circles show the final 1000-step averages of 10000-step simulations on a complete graph with $N=100$, which agree with the mean-field results, giving the critical point as $\mu_{\rm C}^{-1}\approx 0.514$.