Finding Densest Subgraphs with Edge-Color Constraints

Lutz Oettershagen, Honglian Wang, Aristides Gionis

Keywords: Density, Densest subgraph, Diversity, Social networks

Abstract: Finding densest subgraphs is a classical graph-mining problem that has many applications in Web-data analysis, such as identifying groups of related Web documents, finding communities of users, detecting fraudulent behavior, and more. In this paper, we consider a variant of the densest subgraph problem in networks with single or multiple edge attributes. For example, in a social network, the edge attributes may describe the type of relationship between users, such as friends, family, or acquaintances, or different types of communication between users. For conceptual simplicity, we view the attributes as \emph{edge colors}. The new problem we address is to find a \emph{diverse densest subgraph} that fulfills given requirements on the numbers of edges of specific colors. When searching for a dense social network community, our problem will enforce the requirement that the community is diverse according to criteria specified by the edge attributes. We show that the decision versions for finding \emph{exactly}, \emph{at most}, and \emph{at least} $\textbf{h}$ colored edges densest subgraph, where $\textbf{h}$ is a vector of color requirements, are \NP-complete, for already two colors. For the problem of finding a densest subgraph with \emph{at least} $\textbf{h}$ colored edges, we provide a linear-time constant-factor approximation algorithm when the input graph is sparse. On the way, we introduce the related at least $h$ (non-colored) edges densest subgraph problem, show its hardness, and also provide a linear-time constant-factor approximation. In our experiments, we demonstrate the efficacy and efficiency of our new algorithms.

Track: Graph Algorithms and Learning for the Web

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Submission Number: 1965