Locally densest subgraph discovery

Mining dense subgraphs from a large graph is a fundamental graph mining task and can be widely applied in a variety of application domains such as network science, biology, graph database, web mining, graph compression, and micro-blogging systems. Here a dense subgraph is defined as a subgraph with high density (#.edge/#.node). Existing studies of this problem either focus on finding the densest subgraph or identifying an optimal clique-like dense subgraph, and they adopt a simple greedy approach to find the top-k dense subgraphs. However, their identified subgraphs cannot be used to represent the dense regions of the graph. Intuitively, to represent a dense region, the subgraph identified should be the subgraph with highest density in its local region in the graph. However, it is non-trivial to formally model a locally densest subgraph. In this paper, we aim to discover top-k such representative locally densest subgraphs of a graph. We provide an elegant parameter-free definition of a locally densest subgraph. The definition not only fits well with the intuition, but is also associated with several nice structural properties. We show that the set of locally densest subgraphs in a graph can be computed in polynomial time. We further propose three novel pruning strategies to largely reduce the search space of the algorithm. In our experiments, we use several real datasets with various graph properties to evaluate the effectiveness of our model using four quality measures and a case study. We also test our algorithms on several real web-scale graphs, one of which contains 118.14 million nodes and 1.02 billion edges, to demonstrate the high efficiency of the proposed algorithms.