BaCon: Band-Concentrated Exponential Random Graph Models
Exponential-family random graph models ERGMs provide a parametric interpretable framework for modeling real graphs including the Web and social networks Fitted ERGMs may suffer from degeneracy matching observed graph statistics in expectation does not guarantee that the observed graph and similar ones are highly probable under the fitted distribution We introduce Band-Concentrated ERGMs BaCon a framework that avoids degeneracy by imposing a probabilistic constraint requiring at least $1-δ$ of the distribution probability mass to lie on the set of graphs with statistics within an $\varepsilon$-tolerance band around the observed statistics BaCons maximumentropy distribution still belongs to the exponential family while including a factor checking for inclusion in the tolerance band We derive the exact Metropolis-Hastings acceptance rule for this target and use a practical stochastic approximation procedure with a smoothed band membership during fitting Experiments on social networks across increasingly challenging model specifications show that BaCon substantially increases the probability mass assigned to graphs whose statistics are close to the observed ones while remaining estimable for specifications where standard ERGM estimation fails These results demonstrate that probability mass constraints are a principled way to mitigate degeneracy while retaining the benefits of exponential-family graph models