Balance in Random Signed Graphs

By extending Heider’s and Cartwright-Harary’s theory of balance in deterministic social structures, we study the problem of balance in social structures where relations between individuals are random. An appropriate model for representing such structures is the so called random signed graphs Gn,p,q defined as follows. Given a set of n vertices and fixed numbers p and q, 0 < p+ q < 1, then, between each pair of vertices, there exists either a positive edge with probability p, or a negative edge with probability q, or there is no edge with probability 1− p− q. We first show that, almost always (i.e. with probability tending to 1 as n −→ ∞), the random signed graph Gn,p,q is unbalanced. Subsequently we estimate the maximum order of a balanced induced subgraph in Gn,p,p, and show that its order achieves only a finite number of values. Next, we study the asymptotic behavior of the degree of balance and give upper and lower bounds for the line-index of balance. Finally, we study the threshold function of balance, e.g., a function p0(n) such that if p ≫ p0(n), then almost always the random signed graph Gn,p,p is unbalanced, else it is almost always balanced.