Reduction of frustration was the driving force in an approach to social balance as it was recently considered by Antal et al. [T. Antal, P. L. Krapivsky, and S. Redner, Phys. Rev. E 72, 036121 (2005)]. We generalize their triad dynamics to $k$-cycle dynamics for arbitrary integer $k$. We derive the phase structure, determine the stationary solutions, and calculate the time it takes to reach a frozen state. The main difference in the phase structure as a function of $k$ is related to $k$ being even or odd. As a second generalization we dilute the all-to-all coupling as considered by Antal et al. to a random network with connection probability $w<1$. Interestingly, this model can be mapped to a satisfiability problem of computer science. The phase of social balance in our original interpretation then becomes the phase of satisfaction of all logical clauses in the satisfiability problem. In common to the cases we study, the ideal solution without any frustration always exists, but the question actually is as to whether this solution can be found by means of a local stochastic algorithm within a finite time. The answer depends on the choice of parameters. After establishing the mapping between the two classes of models, we generalize the social-balance problem to a diluted network topology for which the satisfiability problem is usually studied. On the other hand, in connection with the satisfiability problem we generalize the random local algorithm to a $p$-random local algorithm, including a parameter $p$ that corresponds to the propensity parameter in the social balance problem. The qualitative effect of the inclusion of this parameter is a bias towards the optimal solution and a reduction of the needed simulation time.

• Received 1 August 2006

DOI:https://doi.org/10.1103/PhysRevE.75.026106