Neural Trees for Learning on Graphs

Graph Neural Networks (GNNs) have emerged as a flexible and powerful approach for learning over graphs. Despite this success, existing GNNs are constrained by their local message-passing architecture and are provably limited in their expressive power. In this work, we propose a new GNN architecture -- the Neural Tree. The neural tree architecture does not perform message passing on the input graph but on a tree-structured graph, called the H-tree, that is constructed from the input graph. Nodes in the H-tree correspond to subgraphs in the input graph, and they are reorganized in a hierarchical manner such that a parent-node of a node in the H-tree always corresponds to a larger subgraph in the input graph. We show that the neural tree architecture can approximate any smooth probability distribution function over an undirected graph, as well as emulate the junction tree algorithm. We also prove that the number of parameters needed to achieve an $\epsilon$-approximation of the distribution function is exponential in the treewidth of the input graph, but linear in its size. We apply the neural tree to semi-supervised node classification in 3D scene graphs, and show that these theoretical properties translate into significant gains in prediction accuracy, over the more traditional GNN architectures.