Studies on the balancing, the minimal balancing, and the minimum balancing processes for social groups with planar and nonplanar graph structures

Abstract

Balancing processes of social groups which transform an unbalanced group to a balanced one by changing the interpersonal relations in the group are characterized in terms of signed graphs. The graphs are given by categorizing the (interpersonal) relations within groups as positive or negative. A balanced state of a group is defined as one in which all the circuits of the graph have a positive sign, and the other cases are called unbalanced states. First, we show that a balancing process for a signed graph (changing the signs of its lines) is decomposed into that for each of the block components of the graph. Second, we classify signed graphs as planar and nonplanar. For the planar cases, we define the dual graph of a planar signed graph, and the balancing processes are characterized by specified sets of lines of the graph. According to the characterization, effective algorithms to derive the minimal and the minimum balancing processes are proposed. For the nonplanar cases, the balancing processes are characterized in terms of adjacency matrices of nonplanar signed graphs by introducing the notion of sign vectors. The characterization results in effective algorithms to derive the minimal and the minimum balancing processes. These discussions are also extended to the case where interpersonal relations have their own directions.