Fast Rectangle Counting on Massive Networks

Rectangle has been recognized as an essential motif in a large number of real-world networks. Counting rectangles in a network plays an important role in network analysis. This paper comprehensively studies the rectangle counting problem on large networks. We propose a novel counting paradigm called the wedge-centric counting, where a wedge is a simple path consisting of three vertices. Unlike the traditional edge-centric counting, the wedge-centric counting uses wedges instead of edges as building blocks of rectangles. The main advantage of the wedge-centric counting is that it does not need to access two-hop neighbors. Based on this paradigm, we develop a collection of rectangle counting algorithms, including an in-memory algorithm with lower time complexity, an external-memory algorithm with the optimal I/O complexity, and two randomized algorithms with provable error bounds. The experimental results on a variety of real networks verify the effectiveness and the efficiency of the proposed wedge-centric rectangle counting algorithms.