[Paper Review] Graph Similarity and Homomorphism Densities

Jan Böker|arXiv (Cornell University)|Apr 29, 2021

Graph theory and applications5 citations

TL;DR

This paper introduces the tree distance and path distance as polynomial-time computable graph similarity measures based on fractional isomorphism and homomorphism densities. It proves that two graphs are similar under these distances if and only if their tree or path homomorphism densities are close, extending Lovász-type correspondences from exact to approximate settings via graphon theory and functional analysis techniques.

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 fractional 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 Dvo\v{r}\'ak (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 densities" can be extended to a correspondence between the associated distance measures. Our result is inspired by a similar result due to Lov\'asz and Szegedy (2006) and Borgs, Chayes, Lov\'asz, S\'os, 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). We establish our main results by generalizing our definitions to graphons 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 Greb\'ik and Rocha (2021).

Motivation & Objective

• To establish a theoretical foundation for graph similarity measures based on homomorphism counts, addressing a gap in understanding machine learning embeddings for graphs.
• To resolve an open question by Grohe (2020) on whether matrix norm-based distances (like those from fractional isomorphism) correspond to homomorphism densities in an approximate sense.
• To generalize the correspondence between exact equivalence (fractional isomorphism) and homomorphism densities to approximate distances using graphons and functional analysis.
• To answer an open question by Grebík and Rocha (2021) on the closedness of certain graphon sets, showing the set of quotient graphons is dense in the space of graphons.

Proposed method

• Define the tree distance δT as a normalized variant of the matrix norm distance over doubly stochastic matrices, derived from the fractional isomorphism system.
• Introduce the path distance δP analogously for path homomorphism counts, generalizing results by Dell, Grohe, and Rattan (2018) to an approximate setting.
• Extend graphon pseudometrics to allow for functional-analytic techniques, proving that any 'reasonably defined' pseudometric on graphons with an exact homomorphism density correspondence must also satisfy an approximate one.
• Use compactness of the graphon space and inverse counting lemmas to derive quantitative bounds between pseudometric distances and homomorphism density differences.
• Prove a counting lemma for the color distance δC□, showing |t(T, G) − t(T, H)| ≤ |E(T)| · δC□(G, H) for trees T.
• Construct explicit graph sequences to demonstrate that δT□ and δC□ are topologically distinct, showing δT□ is strictly weaker than δC□.

Experimental results

Research questions

• RQ1 Can the correspondence between fractional isomorphism and equal tree homomorphism counts be extended from exact to approximate settings?
• RQ2 Is there a quantitative relationship between graph distance measures (like δT) and homomorphism densities for trees and paths?
• RQ3 Does the set {W/C(W) | W ∈ W₀} form a closed subset of the space of graphons, as posed by Grebík and Rocha?
• RQ4 Can the theory of graphons be used to generalize distance measures based on bounded treewidth or path-like substructures?
• RQ5 To what extent does the tree distance reflect structural properties like ε-equitable partitions or color refinement?

Key findings

• The tree distance δT satisfies the key correspondence: two graphs are close under δT if and only if their tree homomorphism densities are close.
• A quantitative counting lemma is established for the color distance: |t(T, G) − t(T, H)| ≤ |E(T)| · δC□(G, H), showing that small δC□ implies small density differences.
• The tree distance δT□ is strictly weaker than the color distance δC□, as demonstrated by a counterexample where δT□(Gn, K3) → 0 but δC□(Gn, K3) ≥ 1/9.
• The set {WG/CG∞ | G graph} is dense in the space of graphons W₀, answering negatively the question of Grebík and Rocha (2021) on closedness.
• For any graphon W and ε > 0, there exists a graph G on O(1/ε) vertices such that δ□(G/CG∞, W) ≤ 3·v(H)/n + 1/4·(v(H)/n)², showing effective approximation via quotient graphs.
• The paper proves a quantitative inverse counting lemma for paths, using the same factor e(F) as in the general graph counting lemma, but leaves the analogous result for trees open as a challenging direction.