Colorful Exponential Random Graph Models
Abstract
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.
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BibTeX
@misc{bhattacharya2026colorful,
title = {{Colorful Exponential Random Graph Models}},
author = {Bhaswar B. Bhattacharya and Pierfrancesco Dionigi and Ankan Ganguly and Giulio Zucal},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.31130},
eprint = {2608.31130},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31130},
}
RIS
TY - UNPB
TI - Colorful Exponential Random Graph Models
AU - Bhaswar B. Bhattacharya
AU - Pierfrancesco Dionigi
AU - Ankan Ganguly
AU - Giulio Zucal
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.31130
UR - https://arxiv.org/abs/2608.31130
AB - 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.
ER -