Partitioning Signed Networks

Abstract

Signed networks appear naturally in contexts where conflict or animosity is apparent. In this book chapter we review some of the literature on signed networks, especially in the context of partitioning. Most of the work is founded in what is known as structural balance theory. We cover the basic mathematical principles of structural balance theory. The theory yields a natural formulation for partitioning. We briefly compare this to other partitioning approaches based on community detection. Finally, we analyse an international network of alliances and conflicts and discuss the implications of our findings for structural balance theory.We are concerned with signed networks, where each link is associated with either a positive (+) or negative sign (−). More generally, weights w ij could be used. Although weights are often assumed to be positive, we explicitly allow them also to be negative. For simplicity, we deal primarily with non-weighted networks, but most concepts used here can be adapted easily to the weighted case. I. NOTATIONWhile we try to be as consistent as possible with the general notation used throughout this book, we require some additional notation because signed networks have signs for arcs and edges. We denote a directed signed network by G = (V, A − , A + ) where A − ⊆ V × V are the negative links and A + ⊆ V × V the positive links. We assume that A − ∩A + = ∅, so that no link is both positive and negative. We exclude loops on nodes. Many studied signed networks are directed. Some are not, including the network we study here. Similarly, an undirected signed network is denoted by G = (V, E − , E + ) where E − ⊆ V ×V are the negative links and E + ⊆ V ×V the positive links. As for the directed case, E − ∩ E + = ∅.We present our initial discussion in terms of directed signed networks. However, if we restrict ourselves to undirected graphs, then (i, j) ∈ E ± is identical to (j, i) ∈ E ± . Also, we assume that there are no self-loops, i.e. no (i, i) exists. For edges, the signs on them are symmetrical by definition.We define the adjacency matrices A + and A − . We set A + ij = 1 whenever (i, j) ∈ A + and A + ij = 0 otherwise. Similarly, A − ij = 1 whenever (i, j) ∈ A − and A − ij = 0 otherwise. We denote the signed adjacency matrix A = arXiv:1803.02082v1 [physics.soc-ph]

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