Overlapping community detection in labeled graphs

Abstract

We present a new approach for the problem of finding overlapping communities in graphs and social networks. Our approach consists of a novel problem definition and three accompanying algorithms. We are particularly interested in graphs that have labels on their vertices, although our methods are also applicable to graphs with no labels. Our goal is to find k communities so that the total edge density over all k communities is maximized. In the case of labeled graphs, we require that each community is succinctly described by a set of labels. This requirement provides a better understanding for the discovered communities. The proposed problem formulation leads to the discovery of vertex-overlapping and dense communities that cover as many graph edges as possible. We capture these properties with a simple objective function, which we solve by adapting efficient approximation algorithms for the generalized maximum-coverage problem and the densest-subgraph problem. Our proposed algorithm is a generic greedy scheme. We experiment with three variants of the scheme, obtained by varying the greedy step of finding a dense subgraph. We validate our algorithms by comparing with other state-of-the-art community-detection methods on a variety of performance measures. Our experiments confirm that our algorithms achieve results of high quality in terms of the reported measures, and are practical in terms of performance.

Appendix: Residual dense subgraph is NP-hard

Let us first define the problem of discovering a graph with high residual density.

Problem 5

(ResDenseGraph) Let $G = (V, E, w)$ be a graph with weighted edges. Find a subgraph $H = (X, R)$ such that

$$\begin{aligned} d \mathopen {}\left( H\right) - \sum _{e \in R} w(e) \end{aligned}$$

is maximized.

Proposition 1

ResDenseGraph is NP-hard.

Proof

We will prove hardness by reducing the clique problem. Assume that we are given a graph $G = (V, E)$ and a size of a clique k. Define the weights to be $w(e) = 2 / (2K - 1)$.

Let us assume that $G$ contains a clique of size k, say $H = (X, R)$. We will first show that $H$ has the highest density. To see this let $H' = (X', R')$. Let $N = {\left| X'\right| }$. If $N < K$, then the profit of $H'$ is genuinely smaller than the profit of $H$. If $N = K$, then the profit of $H$ is larger or equal to the profit of $H'$. If the profits are equal, then $H'$ has to be a clique as well. Assume that $N > K$. Then we can upper-bound the profit by

$$\begin{aligned} (N - 1) - N(N - 1)/(2K + 1) = -\frac{(N - 1)(N - 2K + 1)}{2K - 1}. \end{aligned}$$

This bound is a parabola, obtaining its apex at k. This shows that the profit of $H'$ is genuinely lower than the profit of $H$.

We have shown that $G = (V, E)$ has a k-clique if and only if the optimal answer for ResDenseGraph is a clique of size k.