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

Degeneracy in sparse ERGMs with functions of degrees as sufficient statistics