  We consider general Exponential Random Graph Models (ERGMs) where the
sufficient statistics are functions of homomorphism counts for a fixed
collection of simple graphs $F_k$. Whereas previous work has shown a degeneracy
phenomenon in dense ERGMs, we show this can be cured by raising the sufficient
statistics to a fractional power. We rigorously establish the na\"ive
mean-field approximation for the partition function of the corresponding Gibbs
measures, and in case of "ferromagnetic" models with vanishing edge density
show that typical samples resemble a typical Erd\H{o}s--R\'enyi graph with a
planted clique and/or a planted complete bipartite graph of appropriate sizes.
We establish such behavior also for the conditional structure of the
Erd\H{o}s--R\'enyi graph in the large deviations regime for excess
$F_k$-homomorphism counts. These structural results are obtained by combining
quantitative large deviation principles, established in previous works, with a
novel stability form of a result of [5] on the asymptotic solution for the
associated entropic variational problem. A technical ingredient of independent
interest is a stability form of Finner's generalized H\"older inequality.
