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Showing papers by "Steven H. Strogatz published in 2011"


Journal ArticleDOI
TL;DR: A generalization of the Kuramoto model in which the oscillators are coupled to the mean field with random signs, which displays a traveling wave, in whichThe mean field oscillates at a frequency different from the population's mean natural frequency.
Abstract: We consider a generalization of the Kuramoto model in which the oscillators are coupled to the mean field with random signs. Oscillators with positive coupling are ``conformists''; they are attracted to the mean field and tend to synchronize with it. Oscillators with negative coupling are ``contrarians''; they are repelled by the mean field and prefer a phase diametrically opposed to it. The model is simple and exactly solvable, yet some of its behavior is surprising. Along with the stationary states one might have expected (a desynchronized state, and a partially-synchronized state, with conformists and contrarians locked in antiphase), it also displays a traveling wave, in which the mean field oscillates at a frequency different from the population's mean natural frequency.

336 citations


Journal ArticleDOI
TL;DR: This 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 determines whether they will end up in intractable conflict or global harmony.
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.

284 citations


Journal ArticleDOI
TL;DR: A variant of the Kuramoto model in which all the oscillators are now assumed to have the same natural frequency, but some of them are negatively coupled to the mean field, turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies.
Abstract: We consider a variant of the Kuramoto model in which all the oscillators are now assumed to have the same natural frequency, but some of them are negatively coupled to the mean field. These contrarian oscillators tend to align in antiphase with the mean field, whereas, the positively coupled conformist oscillators favor an in-phase relationship. The interplay between these effects can lead to rich dynamics. In addition to a splitting of the population into two diametrically opposed factions, the system can also display traveling waves, complete incoherence, and a blurred version of the two-faction state. Exact solutions for these states and their bifurcations are obtained by means of the Watanabe-Strogatz transformation and the Ott-Antonsen ansatz. Curiously, this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies.

133 citations


Journal ArticleDOI
TL;DR: Both the model and experiments show that individual macrophages can switch from the repair pathway to the inflammation pathway but that the reverse switch does not occur.

10 citations


Book ChapterDOI
31 Dec 2011

1 citations