Mathematical modeling of local balance in signed networks and its applications to global international analysis

Alliances and conflicts in social, political and economic relations can be represented by positive and negative edges in signed networks. A cycle is said to be positive if the product of its edge signs is positive, otherwise it is negative. Then, a signed network is balanced if and only if all its cycles are positive. An index characterizing how much a signed network deviates from being balanced is known as a global balance index. Here we give a step forward in the characterization of signed networks by defining a local balance index, which characterizes how much a given vertex of a signed network contributes to its global balance. We analyze the mathematical foundations and unique structural properties of this index. Then, we apply this index to the study of the evolution of international relations in the globe for the period 1816–2014. In this way we detect and categorize major historic events based on balance fluctuations, helping our understanding towards new mixed approaches to history based on network theory.