Exponential random graphs behave like mixtures of stochastic block models

Ronen Eldan*, Renan Gross

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We study the behavior of exponential random graphs in both the sparse and the dense regime. We show that exponential random graphs are approximate mixtures of graphs with independent edges whose probability matrices are critical points of an associated functional, thereby satisfying a certain matrix equation. In the dense regime, every solution to this equation is close to a block matrix, concluding that the exponential random graph behaves roughly like a mixture of stochastic block models. We also show existence and uniqueness of solutions to this equation for several families of exponential random graphs, including the case where the subgraphs are counted with positive weights and the case where all weights are small in absolute value. In particular, this generalizes some of the results in a paper by Chatterjee and Diaconis from the dense regime to the sparse regime and strengthens their bounds from the cut-metric to the one-metric.

Original languageEnglish
Pages (from-to)3698-3735
Number of pages38
JournalAnnals of Applied Probability
Issue number6
StatePublished - Dec 2018
Externally publishedYes


  • Exponential random graph models
  • Johnson-Lindenstrauss lemma
  • Mixture models
  • Random graph
  • Stochastic block models


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