Discovering Cliques in Attribute Graphs Based on Proportional Fairness

Community detection is a fundamental problem and has been extensively studied. With the abundance of information in real-world networks, the discovery of communities in attribute graphs is increasingly valuable. However, numerous previous models in attribute graphs neglect the fairness concept, which plays an important role in ensuring that graph analysis is not biased toward specific groups. In this paper, we propose a novel model, named proportional fair clique (PFC). Specifically, given an attribute graph $G=(V,E,A)$, an integer $k$ and a threshold $\lambda \in [0,1/|A|]$, a subgraph $S$ of $G$ is a PFC if $(i)$ $S$ is a clique with size at least $k$ and $(ii)$ $|S_{a_{i}}|/|S| \geq \lambda$ for each attribute $a_{i}$ in $G$, where $S_{a_{i}}$ is the node set in $S$ associated with attribute $a_{i}$. We show that the problem of enumerating all the maximal proportional fair cliques (MPFC) is NP-hard. A reasonable baseline algorithm is first presented by extending the Bron-Kerbosch framework. To scale for large networks, we propose several optimization strategies to accelerate the computation. Finally, comprehensive experiments are conducted over 6 graphs to demonstrate the efficiency and effectiveness of the proposed techniques and model.