Social Balance on Networks: The Dynamics of Friendship and Hatred

We study the evolution of social networks that contain both friendly and unfriendly links between individual nodes. The network is endowed with dynamics in which the sense of a link in an imbalanced (frustrated) triad—a triangular loop with 1 or 3 unfriendly links—is reversed to make the triad balanced. Thus a balanced triad fulfills the adage: “friend of my friend is my friend; an enemy of my friend is my enemy; a friend of my enemy is my enemy; an enemy of my enemy is my friend.” With this frustration-reducing dynamics, an infinite network undergoes a dynamics phase transition from a stead state to “utopia”—all links are friendly—as the propensity for friendly links in an update event passes through 1/2. A finite network always falls into an socially-balanced absorbing state where no imbalanced triads remain. One example of the trend to social balance was the evolution of treaties between various European countries between approximately 1880 1910 that ultimately led to the alliances that comprised the protagonists of World War I.