Percolation in networks composed of connectivity and dependency links

Abstract

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 Erd$\ddot{o}$s-R$\acute{e}$nyi (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, $P_{\infty }$, is given by $P_{\infty }=p^{s-1}[1-\exp (-kp{P}_{\infty }){]}^s$ where $1-p$ is the initial fraction of removed nodes. Our general result coincides with the known Erd$\ddot{o}$s-R$\acute{e}$nyi equation for random networks for $s=1$ and with the result of Parshani et al (PNAS, in press, 2011) for $s=2$. For networks with Poissonian distribution of dependency links we find that $P_{\infty }$ is given by $P_{\infty }=f_{k,p}(P_{\infty })e^{(<s>-1)(p{f}_{k,p}(P_{\infty })-1)}$ where $f_{k,p}(P_{\infty })\equiv 1-\exp (-kp{P}_{\infty })$ 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.

8 pages, 7 figures

References in corpus (2)

• Cascade control and defense in complex networks
• The critical effect of dependency groups on the function of networks

Cited by in corpus (23)

• Multilayer Networks
• The structure and dynamics of multilayer networks
• Robustness of interdependent networks under targeted attack
• Percolation on complex networks: Theory and application
• The extreme vulnerability of interdependent spatially embedded networks
• Spatially localized attacks on interdependent networks: the existence of a finite critical attack size
• Percolation of localized attack on complex networks
• Evolution of robust network topologies: Emergence of central backbones
• Resilience of Networks Formed of Interdependent Modular Networks
• Robustness of a Network Formed of Spatially Embedded Networks
• The combined effect of connectivity and dependency links on percolation of networks
• Cascading failures in coupled networks with both inner-dependency and inter-dependency links
• Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and SOS
• Towards a Realistic Model for Failure Propagation in Interdependent Networks
• Color-avoiding percolation
• Optimal redundancy against disjoint vulnerabilities in networks
• The effect of interdependence on the percolation of interdependent networks
• Percolation of networks with directed dependency links
• Enhanced robustness of single-layer networks with redundant dependencies
• Percolation on Networks with Antagonistic and Dependent Interactions
• Network-Growth Rule Dependence of Fractal Dimension of Percolation Cluster on Square Lattice
• Percolation on the institute-enterprise R&D collaboration networks
• The impact of analytical outage modeling on expansion planning problems in the area of power systems