Weisfeiler and Leman Go Loopy: A New Hierarchy for Graph Representational Learning

TL;DR

Most notably, it is shown that r-loopy Weisfeiler-Leman can count homomorphisms of cactus graphs, which strictly extends classical 1-WL.

Abstract

We introduce $r$-loopy Weisfeiler-Leman ($r$-$\ell{}$WL), a novel hierarchy of graph isomorphism tests and a corresponding GNN framework, $r$-$\ell{}$MPNN, that can count cycles up to length $r + 2$. Most notably, we show that $r$-$\ell{}$WL can count homomorphisms of cactus graphs. This strictly extends classical 1-WL, which can only count homomorphisms of trees and, in fact, is incomparable to $k$-WL for any fixed $k$. We empirically validate the expressive and counting power of the proposed $r$-$\ell{}$MPNN on several synthetic datasets and present state-of-the-art predictive performance on various real-world datasets. The code is available at https://github.com/RPaolino/loopy

Figures (12)

• Figure 1: Visual depiction of $r$-$\ell{}$GIN: During preprocessing, we calculate the path neighborhoods $\mathcal{N}_r(v)$ for each node $v$ in the graph $G$. Paths of varying lengths are processed separately using simple GINs, and their embeddings are pooled to obtain the final graph embedding. The forward complexity scales linearly with the sizes of $\mathcal{N}_r(v)$, enabling efficient computation on sparse graphs.
• Figure 2: Example of $r$-neighborhoods.
• Figure 3: Indistinguishable pairs at initialization, symlog scale. For GRAPH8C and EXP_ISO, we report the proportion of indistinguished pairs: $2$ graphs are deemed indistinguishable if the L$^1$ distance of their embeddings is less than $10^{-3}$. For COSPECTRAL10 and SR16622, we report the L$^1$ distance between graph embeddings. We report the mean and standard deviation over $100$ seeds.
• Figure 4: Test accuracy on synthetic classification task: (left) shared and (right) non-shared weights.
• Figure 5: Examples of non-injective homomorphism (row 1), subgraph isomorphism (row 2), bijective homomorphism with non-homomorphic inverse (row 3), and isomorphism (row 4). For better clarity, the mappings $h:V(F)\to V(G)$ are visually represented with colors, where $F$ is consistently on the left, and $G$ is on the right in each row.

Theorems & Definitions (82)

• definition 1
• definition 2
• definition 3
• definition 4
• definition 5
• definition 6
• definition 7
• proposition 1
• theorem 1
• corollary 1