Mean-Field Theory for Heider Balance under Heterogeneous Social Temperatures

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.

💡 Research Summary

The paper extends the classic Heider structural balance model by assigning each edge of a complete graph its own “social temperature” T₍ᵢⱼ₎, thereby capturing heterogeneous volatility in interpersonal relations. The deterministic update rule xᵢⱼ(t+1)=sgn(∑ₖ xᵢₖ(t)xₖⱼ(t)) is replaced with a stochastic Boltzmann‑type rule: with probability pᵢⱼ = exp(βᵢⱼ ξᵢⱼ) /