3 years ago

# Insights into Bootstrap Percolation: Its Equivalence with k-core Percolation and the Giant Component.

M. A. Di Muro, L. D. Valdez, S. V. Buldyrev, H. E. Stanley, L. A. Braunstein

K-core and bootstrap percolation are widely studied models that have been used to represent and understand diverse deactivation and activation processes in natural and social systems. Since these models are considerably similar, it has been suggested in recent years that they could be complementary, however, this statement has not yet been proven mathematically. In this manuscript we provide a rigorous analysis that shows that the heterogeneous bootstrap percolation is the complement of the heterogeneous k-core percolation in complex networks.

On the other hand, a problem that has been addressed in bootstrap percolation is the theoretical estimation of the relative size of the active nodes belonging to the giant component, $P_{\infty,b}$. However, we perform several stochastic simulations and find that they do not totally match these theoretical estimations. Here we present equations that describe accurately the fraction of nodes belonging to the giant component in the process of bootstrap percolation. We show that when using the branching theory to compute the size of the giant component, we must consider two different types of links, which are related to distinct spanning branches of active nodes. We show that our simulations exhibit a great agreement with our theoretical approach.

Publisher URL: http://arxiv.org/abs/1811.03995

DOI: arXiv:1811.03995v1

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