Percolation in networks composed of connectivity and dependency links

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 Erdos-Renyi (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-infinity is given by P-infinity = p(s-1)[1 - exp(-kpP(infinity))](s), where 1 - p is the initial fraction of removed nodes. Our general result coincides with the known Erdos-Renyi equation for random networks for s = 1. For networks with Poissonian distribution of dependency links we find that P infinity is given by P-infinity = f(k,p)(P-infinity)e(({s}-1)[pfk,p(P infinity)-1]), where f(k,p)(P-infinity) 1 - exp (-kpP(infinity)) 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.