Graph Similarity and Homomorphism Densities

Abstract

We introduce the tree distance, a new distance measure on graphs. The tree distance can be computed in polynomial time with standard methods from convex optimization. It is based on the notion of isomorphism, a characterization based on a natural system of linear equations whose integer solutions correspond to graph isomorphism. By results of Tinhofer (1986, 1991) and Dvořak (2010), two graphs G and H are fractionally isomorphic if and only if, for every tree T, the number of homomorphisms from T to G equals the corresponding number from T to H, which means that the tree distance of G and H is zero. Our main result is that this correspondence between the equivalence relations fractional isomorphism and equal tree homomorphism can be extended to a correspondence between the associated distance measures. Our result is inspired by a similar result due to Lovasz and Szegedy (2006) and Borgs, Chayes, Lovasz, Sos, and Vesztergombi (2008) that connects the cut distance of graphs to their homomorphism densities (over all graphs), which is a fundamental theorem in the theory of graph limits. We also introduce the path distance of graphs and take the corresponding result of Dell, Grohe, and Rattan (2018) for exact path homomorphism counts to an approximate level. Our results answer an open question of Grohe (2020) and help to build a theoretical understanding of vector embeddings of graphs. The distance measures we define turn out be closely related to the cut distance. We establish our main results by generalizing our definitions to graphons, which are limit objects of sequences of graphs, as this allows us to apply techniques from functional analysis. We prove the fairly general statement that, for every reasonably defined graphon pseudometric, an exact correspondence to homomorphism densities can be turned into an approximate one. We also provide an example of a distance measure that violates this reasonableness condition. This incidentally answers an open question of Grebik and Rocha (2021).