  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.

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