First Order Logic of Sparse Graphs with Given Degree Sequences

TL;DR

This paper investigates the limit probabilities of first-order properties in random graphs with specified degree sequences, characterizing when these probabilities form a full interval and analyzing cycle distributions and graph fragments.

Contribution

It provides a complete description of the limit probability set, characterizes degree sequences with full interval closure, and corrects previous proofs on limit probabilities.

Findings

Limit probabilities form a finite union of closed intervals under mild conditions.

Characterization of degree sequences where the limit probability set is [0,1].

Full description of cycle distribution and graph fragments in the subcritical regime.

Abstract

We consider limit probabilities of first order properties in random graphs with a given degree sequence. Under mild conditions on the degree sequence, we show that the closure set of limit probabilities is a finite union of closed intervals. Moreover, we characterize the degree sequences for which this closure set is the interval $[0,1]$, a property that is intimately related with the probability that the random graph is acyclic. As a side result, we compile a full description of the cycle distribution of random graphs and study their fragment (disjoint union of unicyclic components) in the subcritical regime. Finally, we amend the proof of the existence of limit probabilities for first order properties in random graphs with a given degree sequence; this result was already claimed by Lynch~[IEEE LICS 2003] but his proof contained some inaccuracies.