Exact Representation of Sparse Networks with Symmetric Nonnegative Embeddings

TL;DR

This paper introduces a symmetric, nonnegative embedding model for undirected graphs that can exactly represent sparse networks, improving interpretability and applicability to real-world power-law degree distributions.

Contribution

The authors prove a new bound for exact graph representation based on arboricity and propose a symmetric, nonnegative embedding model that extends LPCA's capabilities to real-world sparse networks.

Findings

Model effectively predicts links and communities in real-world graphs.

Bound based on arboricity increases applicability to power-law networks.

Symmetric embeddings enhance interpretability of node relationships.

Abstract

Many models for undirected graphs are based on factorizing the graph's adjacency matrix; these models find a vector representation of each node such that the predicted probability of a link between two nodes increases with the similarity (dot product) of their associated vectors. Recent work has shown that these models are unable to capture key structures in real-world graphs, particularly heterophilous structures, wherein links occur between dissimilar nodes. In contrast, a factorization with two vectors per node, based on logistic principal components analysis (LPCA), has been proven not only to represent such structures, but also to provide exact low-rank factorization of any graph with bounded max degree. However, this bound has limited applicability to real-world networks, which often have power law degree distributions with high max degree. Further, the LPCA model lacks interpretability since its asymmetric factorization does not reflect the undirectedness of the graph. We address these issues in two ways. First, we prove a new bound for the LPCA model in terms of arboricity rather than max degree; this greatly increases the bound's applicability to many sparse real-world networks. Second, we propose an alternative graph model whose factorization is symmetric and nonnegative, which allows for link predictions to be interpreted in terms of node clusters. We show that the bounds for exact representation in the LPCA model extend to our new model. On the empirical side, our model is optimized effectively on real-world graphs with gradient descent on a cross-entropy loss. We demonstrate its effectiveness on a variety of foundational tasks, such as community detection and link prediction.