Balance Maximization in Signed Networks via Edge Deletions

In signed networks, each edge is labeled as either positive or negative. The edge sign captures the polarity of a relationship. Balance of signed networks is a well-studied property in graph theory. In a balanced (sub)graph, the vertices can be partitioned into two subsets with negative edges present only across the partitions. Balanced portions of a graph have been shown to increase coherence among its members and lead to better performance. While existing works have focused primarily on finding the largest balanced subgraph inside a graph, we study the network design problem of maximizing balance of a target community (subgraph). In particular, given a budget b and a community of interest within the signed network, we aim to make the community as close to being balanced as possible by deleting up to b edges. Besides establishing NP-hardness, we also show that the problem is non-monotone and non-submodular. To overcome these computational challenges, we propose heuristics based on the spectral relation of balance with the Laplacian spectrum of the network. Since the spectral approach lacks approximation guarantees, we further design a greedy algorithm, and its randomized version, with provable bounds on the approximation quality. The bounds are derived by exploiting pseudo-submodularity of the balance maximization function. Empirical evaluation on eight real-world signed networks establishes that the proposed algorithms are effective, efficient, and scalable to graphs with millions of edges.