Figure 1. Structurally balanced community.

(A): The community split into two factions such that members of the same faction are connected by friendly relationships (blue edges) and positively influence each other, while members of opposite factions are linked by adversary relationships (red edges) and negatively influence each other. All cycles and semicycles contain an even number of negative edges. (B): the gauge transformation, i.e., the switch of sign to all edges of the cut set (gray line), renders the signed graph completely blue. It corresponds to all individuals on one side of the cut set changing their mind simultaneously on their relationships with the other faction (in the drawing individuals are “flipped” for analogy with spins in Statistical Physics). (C): In this gauge transformation only the two individuals above the gray cut set switch side. The graph clearly remains bipartite. The three signed graphs in (A), (B) and (C) all are exactly structurally balanced.