DISTINGUISHING VERTICES OF INHOMOGENEOUS RANDOM GRAPHS By

We explore under what conditions simple combinatorial attributes and algorithms such as the distance sequence and degree-based partitioning and refinement can be used to distinguish vertices of inhomogeneous random graphs. In the classical setting of Erdős-Renyi graphs and random regular graphs it has been proven that vertices can be distinguished in a constant number of rounds of degree-based refinement or the distance sequence at a logarithmic distance. This yields a high-probability canonical labeling algorithm, and hence an efficient high-probability isomorphism test. In this paper we analyze the same attributes in the context of random graphs that come from distributions that more closely model real-world networks. We first prove a technical result about the effects of one refinement step in the general setting were edges are chosen independently at random. This allows us to prove that an algorithm based on distinguishing vertices yields a canonical labeling for graphs with scale-free degree distribution where edges are added independently, and for Stochastic Kronecker Product Graphs with certain settings of the parameters. Along the way we prove results on the degree distribution, connectedness and diameter of Stochastic Kronecker Graphs with generating matrices of arbitrary size.