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Edited by Ronald L. Graham, University of California, San Diego, La Jolla, CA, and approved November 30, 2010 (received for review September 3, 2010)
Abstract
It is not uncommon for certain social networks to divide into two opposing camps in response to stress. This happens, for example, in networks of political parties during winner-takes-all elections, in networks of companies competing to establish technical standards, and in networks of nations faced with mounting threats of war. A simple model for these two-sided separations is the dynamical system dX/dt = X2, where X is a matrix of the friendliness or unfriendliness between pairs of nodes in the network. Previous simulations suggested that only two types of behavior were possible for this system: Either all relationships become friendly or two hostile factions emerge. Here we prove that for generic initial conditions, these are indeed the only possible outcomes. Our analysis yields a closed-form expression for faction membership as a function of the initial conditions and implies that the initial amount of friendliness in large social networks (started from random initial conditions) determines whether they will end up in intractable conflict or global harmony.
Footnotes
- 1To whom correspondence should be addressed. E-mail: kleinber{at}cs.cornell.edu.
Author contributions: S.A.M., J.K., R.D.K., and S.H.S. designed research; S.A.M., J.K., R.D.K., and S.H.S. performed research; S.A.M. analyzed data; and S.A.M., J.K., R.D.K., and S.H.S. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1013213108/-/DCSupplemental.
Freely available online through the PNAS open access option.
Continuous-time model of structural balance
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