Social Balance on Networks: The Dynamics of Friendship and Enmity

How do social networks evolve when both friendly and unfriendly relations exist? Here we propose a simple dynamics for social networks in which the sense of a relationship can change so as to eliminate imbalanced triads--relationship triangles that contains 1 or 3 unfriendly links. In this dynamics, a friendly link changes to unfriendly or vice versa in an imbalanced triad to make the triad balanced. Such networks undergo a dynamic phase transition from a steady state to "utopia"--all friendly links--as the amount of network friendliness is changed. Basic features of the long-time dynamics and the phase transition are discussed.