New algebraic models of social systems (Q1075279)

This highly theoretical paper deals with a generalization of the concept of balance in a social system. The system of individuals and their interactions is represented by a group network (D,M,f,g), where \(D=(V,A)\) is a digraph, M an abstract algebraic group with identity e, f:A\(\to M\) an ''M-weighting'' and \(g: V\to M\) an ''M-marking''. The substantial generalization concerns so called ''voltage graphs'' (\(\vec G,M,s,g)\) with \(\vec G=\{(u,v)\in V\times V| uv\in E(G)\}\), where g is an identity function \(g(u)=e\) for all \(u\in V\) and where \(s(u,v)s(v,u)=e\) for all v,u\(\in V\). Such graphs are shown to meet Kirchhoff's voltage law and enable a better concept of balance than that by \textit{E. Sampathkumar} and \textit{V. N. Bhave} [J. Karnatak Univ. 18, 325-328 (1973; Zbl 0284.05113)]; the network is now called voltage balanced, iff every cycle Z is balanced - \(s(Z)=e.\) The advantage of the new conception is that a set of important criteria of balance remain valid even for non-Abelian M. The authors have proven four of them, that are in a sense parallel to those used with former concepts of balance; a path criterion, a partition one, a vertices- oriented criterion for block graphs and an M-marking criterion.