Bootstrap Percolation on Complex Networks

TL;DR

This paper studies bootstrap percolation on complex networks, revealing phase transitions, the impact of degree distribution, and introducing a generalized process with arbitrary thresholds.

Contribution

It provides a detailed phase diagram for bootstrap percolation on uncorrelated networks, including novel insights into continuous and discontinuous transitions and effects of degree distribution.

Findings

Two phase transitions: continuous and discontinuous

Discontinuous transition involves diverging avalanches

Networks with divergent second moment show robust activation

Abstract

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.