Graphon based Clustering and Testing of Networks: Algorithms and Theory

TL;DR

This paper introduces new graph clustering and testing methods based on graphon estimation, providing theoretical guarantees and demonstrating state-of-the-art performance on network data without vertex correspondence.

Contribution

The paper presents novel graph distance metrics and clustering algorithms inspired by graphon estimation, with proven consistency and applicability to graph two-sample testing.

Findings

Achieves state-of-the-art clustering results

Proves statistical consistency under Lipschitz conditions

Demonstrates effectiveness in graph two-sample testing

Abstract

Network-valued data are encountered in a wide range of applications and pose challenges in learning due to their complex structure and absence of vertex correspondence. Typical examples of such problems include classification or grouping of protein structures and social networks. Various methods, ranging from graph kernels to graph neural networks, have been proposed that achieve some success in graph classification problems. However, most methods have limited theoretical justification, and their applicability beyond classification remains unexplored. In this work, we propose methods for clustering multiple graphs, without vertex correspondence, that are inspired by the recent literature on estimating graphons -- symmetric functions corresponding to infinite vertex limit of graphs. We propose a novel graph distance based on sorting-and-smoothing graphon estimators. Using the proposed graph distance, we present two clustering algorithms and show that they achieve state-of-the-art results. We prove the statistical consistency of both algorithms under Lipschitz assumptions on the graph degrees. We further study the applicability of the proposed distance for graph two-sample testing problems.