  We propose and analyse a novel nonparametric goodness of fit testing
procedure for exchangeable exponential random graph models (ERGMs) when a
single network realisation is observed. The test determines how likely it is
that the observation is generated from a target unnormalised ERGM density. Our
test statistics are derived from a kernel Stein discrepancy, a divergence
constructed via Steins method using functions in a reproducing kernel Hilbert
space, combined with a discrete Stein operator for ERGMs. The test is a Monte
Carlo test based on simulated networks from the target ERGM. We show
theoretical properties for the testing procedure for a class of ERGMs.
Simulation studies and real network applications are presented.
