Densest Periodic Subgraph Mining on Large Temporal Graphs

Abstract

Densest subgraphs are often interpreted as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">communities</i> , based on a basic assumption that the connections inside a community are much denser than those between communities. In a graph with temporal information, a densest periodic subgraph is the most densely connected periodic behavior which needs to be captured. Unfortunately, the existing work do not model the densest periodic subgraph in temporal graphs, and the current algorithms for mining the densest subgraph cannot be applied to detect the densest periodic subgraph in the temporal networks. To tackle this problem, we propose a novel model, called the densest <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -periodic subgraph, which presents the densest periodic subgraph whose period size is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> . We prove that finding the densest <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -periodic subgraph can be solved in polynomial time, but it is still challenging because the naive algorithm needs to repeatedly invoke a maximum flow algorithm for many periodic subgraphs. To compute the densest <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -periodic subgraph efficiently, we first develop an effective pruning technique based on the degeneracy of the graph to significantly prune the number of the periodic subgraphs. Then, we present a more efficient algorithm that can reduce the computations for the degeneracy and maximum flow. Next, we develop a greedy algorithm that can compute the approximate densest <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -periodic subgraph and achieve an approximation ratio of 1/2. Finally, the results of extensive experiments on several real-life datasets demonstrate the efficiency, scalability, and effectiveness of our algorithms.