Weak Balance in Random Signed Graphs

In this work we deal with questions on balance and weak balance in random graphs. The theory of balance goes back to Heider [12] who asserted that a social system is balanced if there is no tension and that unbalanced social structures exhibit a tension resulting in a tendency to change in the direction of balance. Since this first work of Heider, the notion of balance has been extensively studied by many mathematicians and psychologists [7, 17]. From a mathematical point of view, the most appropriate model for studying structural balance is that of signed graphs. Formally, a signed graph (G,σ) is a graph G = (V, E) together with a function σ : E → {+,−}, which associates each edge with the sign + or −. In such a signed graph, a subset H of E(G) is said to be positive if it contains an even number of negative edges; otherwise it is said to be negative. A signed graph G is balanced if each cycle of G is positive. Otherwise it is unbalanced. In 1956, Cartwright and Harary [4] obtained the following important result which states an equivalent definition of balance.