TY - JOUR

T1 - Percolation in networks composed of connectivity and dependency links

AU - Bashan, Amir

AU - Parshani, Roni

AU - Havlin, Shlomo

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/5/20

Y1 - 2011/5/20

N2 - Networks composed from both connectivity and dependency links were found to be more vulnerable compared to classical networks with only connectivity links. Their percolation transition is usually of a first order compared to the second-order transition found in classical networks. We analytically analyze the effect of different distributions of dependencies links on the robustness of networks. For a random Erdös-Rényi (ER) network with average degree k that is divided into dependency clusters of size s, the fraction of nodes that belong to the giant component P∞ is given by P∞=ps -1[1-exp(-kpP∞)]s, where 1-p is the initial fraction of removed nodes. Our general result coincides with the known Erdos-Rényi equation for random networks for s=1. For networks with Poissonian distribution of dependency links we find that P∞ is given by P∞=f k,p(P∞)e(s-1)[pfk,p(P∞) -1], where fk,p(P∞)1-exp(-kpP∞) and s is the mean value of the size of dependency clusters. For networks with Gaussian distribution of dependency links we show how the average and width of the distribution affect the robustness of the networks.

AB - Networks composed from both connectivity and dependency links were found to be more vulnerable compared to classical networks with only connectivity links. Their percolation transition is usually of a first order compared to the second-order transition found in classical networks. We analytically analyze the effect of different distributions of dependencies links on the robustness of networks. For a random Erdös-Rényi (ER) network with average degree k that is divided into dependency clusters of size s, the fraction of nodes that belong to the giant component P∞ is given by P∞=ps -1[1-exp(-kpP∞)]s, where 1-p is the initial fraction of removed nodes. Our general result coincides with the known Erdos-Rényi equation for random networks for s=1. For networks with Poissonian distribution of dependency links we find that P∞ is given by P∞=f k,p(P∞)e(s-1)[pfk,p(P∞) -1], where fk,p(P∞)1-exp(-kpP∞) and s is the mean value of the size of dependency clusters. For networks with Gaussian distribution of dependency links we show how the average and width of the distribution affect the robustness of the networks.

UR - http://www.scopus.com/inward/record.url?scp=79961074393&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.83.051127

DO - 10.1103/PhysRevE.83.051127

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C2 - 21728510

SN - 2470-0045

VL - 83

SP - 051127

JO - Physical Review E

JF - Physical Review E

IS - 5

M1 - 051127

ER -