Some Thoughts on Graph Similarity

TL;DR

The paper surveys principled approaches to graph similarity, contrasting operational distances (edits, matrix-norm alignments, normalisation, and relaxations) with declarative distances (vector embeddings, graph kernels, homomorphism densities, sampling, and logic-based notions). It develops key connections, such as the equality of certain norms (e.g., the local edit distance) and the existence of blow-up limits for size-agnostic comparisons, and it links optimal transport formulations to fractional relaxations. It also discusses the complexity landscape, noting hardness for many operational metrics while highlighting tractable relaxations and kernel-based methods for practical similarity, and it explores topology and equivalence results that unify disparate viewpoints. The work provides a broad framework for choosing and combining graph similarity notions across applications, from chemistry to verification, while outlining many open questions about expressiveness and computability.

Abstract

We give an overview of different approaches to measuring the similarity of, or the distance between, two graphs, highlighting connections between these approaches. We also discuss the complexity of computing the distances.

Figures (4)

• Figure 1.1: Which of these three molecular graphs are most similar?
• Figure 3.1: A graph $G$ and its blow-up $G^{\odot 6}$
• Figure 3.2: The graphs of Example \ref{['exa:limit']}: (a) shows the graphs $G,H$, (b) shows their blow-ups $G^{\odot2},H^{\odot2}$ and the bijection $\pi$ between them
• Figure 4.1: Weisfeiler-Leman colours viewed as trees.

Theorems & Definitions (35)

• Proposition 3.1: Gervens and Grohe GervensG22
• proof : Proof of Proposition \ref{['prop:local-ed']}
• Example 3.3
• Theorem 3.4
• Lemma 3.5
• proof
• proof : Proof of Theorem \ref{['thm:limit']}
• Example 3.6: GoldreichKNR08
• Theorem 3.7: Borgs et al. BorgsCLSV08, Pikhurko Pikhurko10
• Proposition 3.8