Nested Graph Neural Networks

Abstract

Graph neural network (GNN)'s success in graph classification is closely related to the Weisfeiler-Lehman (1-WL) algorithm. By iteratively aggregating neighboring node features to a center node, both 1-WL and GNN obtain a node representation that encodes a rooted subtree around the center node. These rooted subtree representations are then pooled into a single representation to represent the whole graph. However, rooted subtrees are of limited expressiveness to represent a nontree graph. To address it, we propose Nested Graph Neural Networks (NGNNs). NGNN represents a graph with rooted subgraphs instead of rooted subtrees, so that two graphs sharing many identical subgraphs (rather than subtrees) tend to have similar representations. The key is to make each node representation encode a subgraph around it more than a subtree. To achieve this, NGNN extracts a local subgraph around each node and applies a base GNN to each subgraph to learn a subgraph representation. The whole-graph representation is then obtained by pooling these subgraph representations. We provide a rigorous theoretical analysis showing that NGNN is strictly more powerful than 1-WL. In particular, we proved that NGNN can discriminate almost all r-regular graphs, where 1-WL always fails. Moreover, unlike other more powerful GNNs, NGNN only introduces a constantfactor higher time complexity than standard GNNs. NGNN is a plug-and-play framework that can be combined with various base GNNs. We test NGNN with different base GNNs on several benchmark datasets. NGNN uniformly improves their performance and shows highly competitive performance on all datasets.

Motif Prediction with Graph Neural Networks

“…Such architectures capture substructures of an input graph to achieve expressivity beyond the 1-WL-test [256]. Some achieve this by learning subgraph representations [51], [148], [193], [204], [209], [230], [275], [282], which they subsequently transform to graph representations. Others predict the properties of subgraphs themselves.…”

“…Instead, the adjacency is usually very specific to a given scheme, we will refer collectively to it as the subgraph adjacency. Some architectures associate subgraphs with single nodes, subsequently defining subgraph adjacency via adjacency of the corresponding nodes [209], [275]. Another approach is to measure the number of nodes or edges that subgraphs share and define adjacencies between them based on these overlaps [230].…”

“…Finally, we turn to architectures for subgraph-collections. There is a large body of research on this topic and numerous models have been proposed [5], [51], [60], [86], [100], [193], [209], [230], [275], [282]. In general, the architectures based on subgraph-collections usually proceed as follows: 1) Construct a collection of subgraphs from an input graph.…”

“…In reconstructionbased schemes, one removes nodes or edges and learns representations for the resulting induced subgraphs [51], [86], [193], [204]. In ego-net schemes, subgraphs are constructed by selecting vertices according to some scheme, and then building subgraphs using multi-hop neighborhoods of these vertices [51], [209], [275], [282]. In general subgraph wiring, the approach is to pick specific subgraphs of interest [5], for example subgraphs that are isomorphic to chosen template graphs [230].…”

“…Finally, a GNN takes these new node features as input and learns a graph representation. Nested graph neural networks (NGNNs) [275] uses a similar approach. Initially, they build a subgraph-collection from the r-hop neighbourhood B r (v) of every node v ∈ V for some r ∈ Z >0 .…”