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.
Heider balance of a chain of actors as dependent on the interaction range and a thermal noise
在周期边界条件和值域$r$的邻域内，以$r$为参数，研究了一系列作用体中的Heider平衡.应用了两种模型动力学：确定性元胞自动机（Malarz等人，Physica D 411（2020）132556）和热浴算法，其中不平衡-平衡三元组的密度起能量的作用。其结果是在稳态和闪烁状态下的能量谱以及由热噪声驱动的平衡-不平衡网络转换。临界点$T_c$随着范围$r$而增加，并且它不依赖于系统大小。
On the dynamics of Social Balance on general networks (with an application to XOR-SAT)
我们研究离散动力系统的非确定性和概率版本（由于T。Antal，P. L. Krapivsky和S. Redner）的社会平衡理论的启发。我们研究了几类图的收敛时间的动态。我们的贡献包括：1.我们指出了三重动力学和广义的零化走超图之间的连接。特别是，这种连接使我们能够完全表征图中的递归状态，其中每条边最多属于两个三角形。2.我们还解决了超图不包含由一个或两个顶点组成的边的情况。3.我们证明了在所谓的“三元圈”图上，收敛时间是线性的。4.我们得到了2-正则三元单形G的收敛时间的三次上界。这个界可以进一步改进到一个依赖于G的Cheeger常数的量。特别是这提供了一些严格的对应以前的实验观察。我们还指出了一个应用程序的随机游走算法的分析在某些情况下的3-异或-SAT问题。
