We study the mutual percolation of a system composed of two interdependent random regular networks. We introduce a notion of distance to explore the effects of the proximity of interdependent nodes on the cascade of failures after an initial attack. We find a nontrivial relation between the nature of the transition through which the networks disintegrate and the parameters of the system, which are the degree of the nodes and the maximum distance between interdependent nodes. We explain this relation by solving the problem analytically for the relevant set of cases. In the process, we solve a variant of Rényi's parking problem on treelike graphs.
[1]
Feller William,et al.
An Introduction To Probability Theory And Its Applications
,
1950
.
[2]
Mark Newman,et al.
Networks: An Introduction
,
2010
.
[3]
R. Rosenfeld.
Nature
,
2009,
Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.
[4]
Reuven Cohen,et al.
Complex Networks: Structure, Robustness and Function
,
2010
.
[5]
宁北芳,et al.
疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A
,
2005
.