Weak Convergence of Laws of Finite Graphs

Abstract

The law of a finite graph is a probability measure induced by the orbits of the graph under its automorphism group. Every law satisfies the intrinsic mass transport principle, which is also known as unimodularity. We discuss the convergence of sequences of laws of finite graphs. Of particular importance is a conjecture proposed by Aldous and Lyons that claims every unimodular measure is a limit of a sequence of laws. Aside from this open problem, other directions of research are also mentioned. We work out in detail a number of results and examples, some of which are new, and others that have been previously stated without proofs. These results include a new characterization of laws of finite connected graphs, a description of the topological space of paths, and a proof that the compact space of weak limits of laws is convex.