Percolation of Networks Composed of Connectivity and Dependency Relations


  Amir Bashan  ,  Roni Parshani  ,  Shlomo Havlin  
Department of Physics, Bar-Ilan University

Networks composed from connectivity and dependency links are found to be more vulnerable than regular networks and their percolation transition is of first order instead the regular second order. We study networks having both relations, and find that for a random network of average degree k organized in dependency clusters of size s, the fraction of nodes belong to the giant component, Pāˆž, is given by        Pāˆž=ps-1[1-exp(-kpPāˆž)]s where 1-p is the initial fraction of removed nodes. Our general result coincides for s=1 with the known  Erdos-Renyi second-order phase transition for a network without dependency relations. For any sā‰„2 cascading failures occur and the transition becomes a first-order percolation transition. We also study the cases of Gaussian and Poissonian distributions of dependency clusters.