Metrics for sparse graphs

Abstract

Abstract Recently, Bollobas, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ( n ) edges. Roughly speaking, there is one model for each kernel, i.e. each symmetric measurable function from [0, 1] 2 to the non-negative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for large-scale real-world networks. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovasz, Sos, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ( n 2 ) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons , which are essentially bounded kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasi-random graphs are in a sense completely general: any sequence of dense graphs contains such a subsequence. Alternatively, their results show that certain natural models of dense inhomogeneous random graphs (one for each graphon) cover the space of dense graphs: there is one model for each point of the completion, producing graphs that converge to this point.