Fast and scalable likelihood maximization for Exponential Random Graph Models with local constraints

Exponential Random Graph Models (ERGMs) have gained increasing popularity over the years. Rooted into statistical physics, ERGMs framework has been successfully employed for reconstructing networks, detecting statistically significant patterns in graphs, counting networked configurations with given properties. From a technical point of view, workflow is defined by two subsequent optimization steps: first one concerns maximization Shannon entropy and leads to identify functional form ensemble probability distribution that maximally non-committal respect missing information; second likelihood function induced this its numerical determination. This step translates resolution system $O(N)$ non-linear, coupled equations (with $N$ being total number nodes network under analysis), problem affected three main issues, i.e. accuracy, speed scalability. The present paper aims at addressing these problems comparing performance algorithms (i.e. Newton's method, quasi-Newton method recently-proposed fixed-point recipe) solving several ERGMs, binary weighted constraints both directed an undirected fashion. While performs best relatively little recipe be preferred when large are considered, as it ensures convergence solution within seconds networks hundreds thousands (e.g. Internet, Bitcoin). We attach Python code implementing aforementioned on all considered work.