彩色指数随机图模型
Colorful Exponential Random Graph Models
- University of Pennsylvania（宾夕法尼亚大学）
- National University of Singapore（新加坡国立大学）
- Barcelona School of Economics（巴塞罗那经济管理学院）
- Boston University（波士顿大学）
- Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）
- Center for Systems Biology（系统生物学中心）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文研究彩色指数随机图模型（ERGMs），利用概率图子框架推导极限自由能变分表示，建立零温两层选择原理，结合数值实验分析诱导楔和彩虹三角形ERGMs的结构与对称性破缺。
中文摘要 AI 辅助
本文中，我们启动了彩色指数随机图模型（ERGMs）的研究，这是一类适用于具有多种边关系类型的网络的指数族模型。利用概率图子（graphons）框架，我们首先推导了极限自由能的变分表示，其最大化子决定了模型典型样本的渐近结构。随后，我们识别出几类具有复制对称性的通用彩色ERGMs，其中变分问题具有常数最大化子，且模型渐近集中于边独立的乘积着色。对于通用彩色ERGMs，我们推导了变分最大化子的欧拉-拉格朗日不动点方程，进而得到通用高温唯一性准则。在互补的零温区域，我们建立了两层选择原理：主导能量项决定基态，而低阶能量项结合熵作为决胜项，确定模型的渐近零温结构。我们通过诱导楔和彩虹三角形ERGMs说明该原理，这两类模型在多类型网络中具有自然解释，其零温极限呈现与极值组合学中知名结果相关的有趣结构。我们进一步为这两类模型建立了有限温对称性破缺，并通过数值实验补充了严格结果。
英文摘要
In this paper, we initiate the study of colored exponential random graph models (ERGMs), a class of exponential-family models for networks with multiple types of edge relations. Using the framework of probability graphons, we first derive a variational representation for the limiting free energy, whose maximizers determine the asymptotic structure of typical samples from the model. Then we identify several general families of colored ERGMs exhibiting replica symmetry, where the variational problem has constant maximizers and the model asymptotically concentrates on product colorings with independent edges. For general colored ERGMs, we derive Euler-Lagrange fixed-point equations for the variational maximizers, which in turn yield a general high-temperature uniqueness criterion. In the complementary zero-temperature regime, we establish a two-level selection principle: the leading energy term determines the ground states, while the lower-order energy terms, combined with entropy, act as a tie-breaker to determine the asymptotic zero-temperature structure of the model. We illustrate this principle through the induced wedge and rainbow triangle ERGMs. Both models have natural interpretations in multitype networks, and their zero-temperature limits exhibit interesting structures that connect to well-known results in extremal combinatorics. We further establish finite-temperature symmetry breaking for both these models and complement the rigorous results with numerical experiments.
