Bootstrap Percolation on Complex Networks

Details Bootstrap Percolation on Complex Networks Bootstrap Percolation on Complex Networks

2010-03-29

arxiv.org/abs/1003.5583v2

Phys. Rev. E 82, 011103 (2010)

Physics

Condensed Matter

Statistical Mechanics

9 pages, 3 figures

Scientific paper

10.1103/PhysRevE.82.011103

We consider bootstrap percolation on uncorrelated complex networks. We obtain the phase diagram for this process with respect to two parameters: $f$, the fraction of vertices initially activated, and $p$, the fraction of undamaged vertices in the graph. We observe two transitions: the giant active component appears continuously at a first threshold. There may also be a second, discontinuous, hybrid transition at a higher threshold. Avalanches of activations increase in size as this second critical point is approached, finally diverging at this threshold. We describe the existence of a special critical point at which this second transition first appears. In networks with degree distributions whose second moment diverges (but whose first moment does not), we find a qualitatively different behavior. In this case the giant active component appears for any $f>0$ and $p>0$, and the discontinuous transition is absent. This means that the giant active component is robust to damage, and also is very easily activated. We also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold.

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