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, defined as the fraction of friendly links rho, is changed. Basic features of the long-time dynamics and the phase transition are discussed. (c) 2006 Elsevier B.V. All rights reserved.