ERGMs are Hard

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 partition function. We show that the problem of computing the partition function is #P-hard, and inapproximable in polynomial time to within an exponential ratio, assuming P 6= 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. ar X iv :1 41 2. 17 87 v1 [ cs .D S] 4 D ec 2 01 4