ERGMs are Hard

TL;DR

This paper proves that computing the partition function for ERGMs, a key step in social network analysis, is computationally intractable, making exact inference and sampling extremely difficult.

Contribution

It establishes the P-hardness and inapproximability of the ERGM partition function, highlighting fundamental computational limitations in social network modeling.

Findings

Partition function computation is P-hard.

Exact sampling from ERGMs is computationally infeasible.

No efficient approximation algorithms exist under standard complexity assumptions.

Abstract

We investigate the computational complexity of the exponential random graph model (ERGM) commonly used in social network analysis. This model represents a probability distribution on graphs by setting the log-likelihood of generating a graph to be a weighted sum of feature counts. These log-likelihoods must be exponentiated and then normalized to produce probabilities, and the normalizing constant is called the \emph{partition function}. We show that the problem of computing the partition function is $\mathsf{\#P}$-hard, and inapproximable in polynomial time to within an exponential ratio, assuming $\mathsf{P} \neq \mathsf{NP}$. Furthermore, there is no randomized polynomial time algorithm for generating random graphs whose distribution is within total variation distance $1-o(1)$ of a given ERGM. Our proofs use standard feature types based on the sociological theories of assortative mixing and triadic closure.