Percolation in networks composed of connectivity and dependency links

Amir Bashan, Roni Parshani, Shlomo Havlin

Research output: Contribution to journalArticlepeer-review

102 Scopus citations


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.

Original languageEnglish
Article number051127
Pages (from-to)051127
JournalPhysical Review E
Issue number5
StatePublished - 20 May 2011


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