Mining Dual Networks: Models, Algorithms, and Applications.

Finding the densest subgraph in a single graph is a fundamental problem that has been extensively studied. In many emerging applications, there exist dual networks. For example, in genetics, it is important to use protein interactions to interpret genetic interactions. In this application, one network represents physical interactions among nodes, for example, protein--protein interactions, and another network represents conceptual interactions, for example, genetic interactions. Edges in the conceptual network are usually derived based on certain correlation measure or statistical test measuring the strength of the interaction. Two nodes with strong conceptual interaction may not have direct physical interaction. In this article, we propose the novel dual-network model and investigate the problem of finding the densest connected subgraph (DCS), which has the largest density in the conceptual network and is also connected in the physical network. Density in the conceptual network represents the average strength of the measured interacting signals among the set of nodes. Connectivity in the physical network shows how they interact physically. Such pattern cannot be identified using the existing algorithms for a single network. We show that even though finding the densest subgraph in a single network is polynomial time solvable, the DCS problem is NP-hard. We develop a two-step approach to solve the DCS problem. In the first step, we effectively prune the dual networks, while guarantee that the optimal solution is contained in the remaining networks. For the second step, we develop two efficient greedy methods based on different search strategies to find the DCS. Different variations of the DCS problem are also studied. We perform extensive experiments on a variety of real and synthetic dual networks to evaluate the effectiveness and efficiency of the developed methods.