Graph partitioning using matrix differential equations

TL;DR

This paper introduces a novel approach to graph partitioning by formulating it as matrix nearness problems and solving them using constrained matrix differential equations, addressing minimum cut and robustness issues.

Contribution

It develops a matrix differential equation framework for graph partitioning, including constrained and robustness considerations, advancing spectral clustering methods.

Findings

Effective matrix differential equation models for graph partitioning.

Handling of constrained minimum cut problems.

Insights into robustness of spectral clustering algorithms.

Abstract

Given a connected undirected weighted graph, we are concerned with problems related to partitioning the graph. First of all we look for the closest disconnected graph (the minimum cut problem), here with respect to the Euclidean norm. We are interested in the case of constrained minimum cut problems, where constraints include cardinality or membership requirements, which leads to NP-hard combinatorial optimization problems. Furthermore, we are interested in ambiguity issues, that is in the robustness of clustering algorithms that are based on Fiedler spectral partitioning. The above-mentioned problems are restated as matrix nearness problems for the weight matrix of the graph. A key element in the solution of these matrix nearness problems is the use of a constrained gradient system of matrix differential equations.