Degeneracy in sparse ERGMs with functions of degrees as sufficient statistics

Abstract

A sufficient criterion for "non-degeneracy" is given for Exponential Random Graph Models on sparse graphs with sufficient statistics which are functions of the degree sequence. This criterion explains why statistics such as alternating $k$-star are non-degenerate, whereas subgraph counts are degenerate. It is further shown that this criterion is "almost" tight. Existence of consistent estimates is then proved for non-degenerate Exponential Random Graph Models.

arXiv commentMajor changes both in results and presentation. Theorems 1.9 and 1.11 are new, and previous results have been proved with less conditions