The Vertex-Attribute-Constrained Densest $k$-Subgraph Problem

TL;DR

This paper introduces a new variant of the densest k-subgraph problem that incorporates vertex attributes, providing a more meaningful community detection method that can be efficiently solved despite the problem's NP-hardness.

Contribution

We define the Vertex-Attribute-Constrained Densest k-Subgraph problem, prove its computational hardness, and develop an efficient continuous relaxation and algorithm for large-scale graph analysis.

Findings

The relaxation of VAC-DkS is tight and solvable with a Frank--Wolfe algorithm.

Experimental results show the method scales well and outperforms classical approaches.

Application to political networks reveals more balanced and meaningful communities.

Abstract

Dense subgraph mining is a fundamental technique in graph mining, commonly applied in fraud detection, community detection, product recommendation, and document summarization. In such applications, we are often interested in identifying communities, recommendations, or summaries that reflect different constituencies, styles or genres, and points of view. For this task, we introduce a new variant of the Densest $k$-Subgraph (D$k$S) problem that incorporates the attribute values of vertices. The proposed Vertex-Attribute-Constrained Densest $k$-Subgraph (VAC-D$k$S) problem retains the NP-hardness and inapproximability properties of the classical D$k$S. Nevertheless, we prove that a suitable continuous relaxation of VAC-D$k$S is tight and can be efficiently tackled using a projection-free Frank--Wolfe algorithm. We also present an insightful analysis of the optimization landscape of the relaxed problem. Extensive experimental results demonstrate the effectiveness of our proposed formulation and algorithm, and its ability to scale up to large graphs. We further elucidate the properties of VAC-D$k$S versus classical D$k$S in a political network mining application, where VAC-D$k$S identifies a balanced and more meaningful set of politicians representing different ideological camps, in contrast to the classical D$k$S solution which is unbalanced and rather mundane.