Tensor Spectral Clustering for Partitioning Higher-order Network Structures

TL;DR

This paper introduces Tensor Spectral Clustering (TSC), a novel method that incorporates higher-order network structures like cycles and feed-forward loops into graph partitioning, improving upon traditional spectral methods.

Contribution

The paper presents a tensor-based spectral clustering algorithm that explicitly models and preserves higher-order network substructures during partitioning.

Findings

TSC effectively captures higher-order structures such as cycles and feed-forward loops.

TSC produces larger, more meaningful partitions with fewer feedback loops.

Application to synthetic networks demonstrates improved detection of layered flows and anomalies.

Abstract

Spectral graph theory-based methods represent an important class of tools for studying the structure of networks. Spectral methods are based on a first-order Markov chain derived from a random walk on the graph and thus they cannot take advantage of important higher-order network substructures such as triangles, cycles, and feed-forward loops. Here we propose a Tensor Spectral Clustering (TSC) algorithm that allows for modeling higher-order network structures in a graph partitioning framework. Our TSC algorithm allows the user to specify which higher-order network structures (cycles, feed-forward loops, etc.) should be preserved by the network clustering. Higher-order network structures of interest are represented using a tensor, which we then partition by developing a multilinear spectral method. Our framework can be applied to discovering layered flows in networks as well as graph anomaly detection, which we illustrate on synthetic networks. In directed networks, a higher-order structure of particular interest is the directed 3-cycle, which captures feedback loops in networks. We demonstrate that our TSC algorithm produces large partitions that cut fewer directed 3-cycles than standard spectral clustering algorithms.