Many real-world networks are signed networks with positive and negative edge weights, such as social networks with positive (friend) or negative (foe) relationships between users, and gene interaction networks with positive (stimulatory) or negative (inhibitory) interactions between genes. A well-known data mining task in signed networks is to find groups of antagonistic communities, where the vertices in the same community have a strong positive relationship and the vertices in different communities have a strong negative relationship. Most existing methods find antagonistic communities by modelling a signed network as a static graph with constant positive and negative edge weights. However, since the relationship between vertices is often uncertain in many real-world networks, it is more practical and accurate to capture the uncertainty of the relationship in the network by a signed uncertain graph (SUG), where each edge is independently associated with a discrete probability distribution of signed edge weights. How to find groups of antagonistic communities in a SUG is a challenging data mining task that has not been systematically tackled before. In this paper, we propose a novel method to tackle this task. We first model a group of antagonistic communities by a set of subgraphs, where the vertices in the same subgraph have a large expectation of positive edge weights and the vertices in different subgraphs have a large expectation of negative edge weights. Then, we propose a method to efficiently find significant groups of antagonistic communities by restricting all the computations on small local subgraphs of the SUG. Extensive experiments on seven real-world datasets and a synthetic dataset demonstrate the outstanding effectiveness and efficiency of the proposed method.

Given a signed network where edges are weighted in real number, and positive weights indicate cohesion between vertices and negative weights indicate opposition, we are interested in finding k-Oppositive Cohesive Groups (k-OCG). Each k-OCG is a group of k subgraphs such that (1) the edges within each subgraph are dense and cohesive; and (2) the edges crossing different subgraphs are dense and oppositive. Finding k-OCGs is challenging since the subgraphs are often small, there are multiple k-OCGs in a large signed network, and many existing dense subgraph extraction methods cannot handle edges of two signs. We model k-OCG finding task as a quadratic optimization problem. However, the classical Proximal Gradient method is very costly since it has to use the entire adjacency matrix, which is huge on large networks. Thus, we develop FOCG, an algorithm that is two orders of magnitudes faster than the Proximal Gradient method. The main idea is to only search in small subgraphs and thus avoids using a major portion of the adjacency matrix. Our experimental results on synthetic and real data sets as well as a case study clearly demonstrate the effectiveness and efficiency of our method.