Fairness-Aware Maximal Clique in Large Graphs: Concepts and Algorithms

Abstract

Cohesive subgraph mining on attributed graphs is a fundamental problem in graph data analysis. Existing cohesive subgraph mining algorithms on attributed graphs do not consider the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fairness</i> of attributes in the subgraph. In this paper, we, for the first time, introduce fairness into the widely-used clique model to mine fairness-aware cohesive subgraphs. In particular, we propose three novel fairness-aware maximal clique models on attributed graphs, called weak fair clique, strong fair clique and relative fair clique, respectively. To enumerate all weak fair cliques, we develop an efficient backtracking algorithm called WFCEnum equipped with a novel colorful <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core based pruning technique. We also propose an efficient enumeration algorithm called SFCEnum to find all strong fair cliques based on a new attribute-alternatively-selection search technique. To further improve the efficiency, we also present several non-trivial ordering techniques for both weak and strong fair clique enumerations. To enumerate all relative fair cliques, we design an enhanced colorful <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core based pruning technique for 2D attributes, and develop two efficient search algorithms: RFCRefineEnum and RFCAlterEnum for arbitrary dimension attributes. The results of extensive experiments on four real-world graphs demonstrate the efficiency, scalability and effectiveness of the proposed algorithms.