Improving Subgraph-GNNs via Edge-Level Ego-Network Encodings

TL;DR

This work introduces ELENE, an edge-level ego-network encoding that augments MP-GNNs with edge-centric structural signals to boost expressivity, notably distinguishing SRGs beyond node-centric methods. It presents two learning variants, ELENE and ELENE-L, with Node-Centric (ND) and Edge-Centric (ED) encodings and embeddings, establishing that ED is strictly more expressive than ND and that ELENE-L can emulate Shortest Path Neural Networks and Graphormers. Empirical evaluation across expressivity benchmarks, h-Proximity, and real-world datasets demonstrates competitive performance and, in some settings, substantial memory savings (up to 18.1x) compared to sub-graph GNN baselines. The results position ELENE as a versatile bridge between MP-GNNs, SPNNs, and Graph Transformers, offering a scalable, interpretable edge-level encoding that enhances graph representation learning.

Abstract

We present a novel edge-level ego-network encoding for learning on graphs that can boost Message Passing Graph Neural Networks (MP-GNNs) by providing additional node and edge features or extending message-passing formats. The proposed encoding is sufficient to distinguish Strongly Regular Graphs, a family of challenging 3-WL equivalent graphs. We show theoretically that such encoding is more expressive than node-based sub-graph MP-GNNs. In an empirical evaluation on four benchmarks with 10 graph datasets, our results match or improve previous baselines on expressivity, graph classification, graph regression, and proximity tasks -- while reducing memory usage by 18.1x in certain real-world settings.

Figures (11)

• Figure 1: Expressive power is typically analyzed in terms of the families of non-isomorphic graphs that models fail to distinguish: $4\times 4$ Rook (a) and Shrikhande (b) graphs are indistinguishable by node-only sub-graph GNNs frasca2022understanding.
• Figure 2: $h$-Proximity binary classification task---A pair of positive (a) and negative (b) 1-Proximity graph examples. An $h$-Proximity graph is positive if all red nodes have at most 2 blue neighbors up to distance $h$, and negative otherwise.
• Figure 3: Example graph (right) and corresponding (left) degree triplets for nodes in the 2-hop ego-network rooted on the green node. The dashed blue node has one edge to the 0-hop root ($d^{(\texttt{-}1)}_{\mathcal{S}} = 1$), a degree of 4, and two edges 2-hops from the root ($d^{(\texttt{+}1)}_{\mathcal{S}} = 2$, red), so its degree triplet is (1, 4, 2).
• Figure 4: The $4\times 4$ Rook (a) and Shrikhande (b) graphs are indistinguishable by 3-WL as SRGs with parameters $\texttt{SRG}(16, 6, 2, 2)$arvind2020frasca2022understanding. Elene (ND, top sub-graphs) is also unable to distinguish the graphs, while Elene (ED, bottom sub-graphs) counts different numbers of edges.
• Figure 5: $k$-depth ego-network intersection following \ref{['eq:ELENE-L-EdgeMessage']} for the green edge. The ego-networks of $u$ and $v$ (yellow (left) and purple (center) respectively), intersect on five nodes around $\langle u, v \rangle$ (dotted, right). We show $\mathcal{V}^{k=2}_{\langle u, v\rangle} = \{\textcolor{green!60!black}{u, v}, \textcolor{blue!70!black}{w_1}, \textcolor{red!60!black}{w_2, w_3, w_4, w_5}\}$ (right), indicating nodes reachable in 0 or 1-hops, exactly 1-hop or 1 or 2-hops from $u$ and $v$.

Theorems & Definitions (17)

• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• Corollary 2
• Proposition 1
• Theorem 3
• proof
• Proposition 2