Percolation And Cascading Failures On Networks

In recent years, much researches have been carried out to explore the structural properties and dynamics of complex networks. Why so much focus on complex net-works? Firstly, networks are ubiquitous in almost every aspect of our lives, such as com-munication networks, Internet, relationship networks, neural networks and metabolic networks. Understanding how these networks work is highly meaningful for improving the qualities of our lives, that’s exactly what the scientists like to do. Secondly, in mod-ern physics and other related disciplines, networks with simple topologies can no longer meet the requirements of describing the structural relationships among the components of a system. Therefore, complex networks must be studied to give a new tool to deal with these problems.There are many networks of interest to scientists that are composed of individual parts or components linked together in some way. For example, the Internet is a col-lection of computers linked by data connections. As also noted, human societies are collections of people linked by acquaintance or social interaction. In a concept of com-plex networks, they all can be presented as a network, the components of the system being the network nodes and the connections or interactions the links. In this way, the systems can be studied mathematically. In another perspective, these networks are all many-body systems, which can be studied using statistical mechanics. Facing the gi-gantic network information, the physics approach may be the most advantageous for the understanding of the structures and dynamics of networks. Actually, many approaches used today in complex networks are a directed generalization of the classical methods in statistical mechanics, such as ensemble theory, phase transition theory, mean filed method, Ising model and percolation model.One of the important topics in complex networks is the robustness of networks. This study focuses on the vulnerability of a network after a fraction of nodes are re-moved. This helps us gain a deeper understanding of the vulnerability of the real net-works, such as the blackout that affected much of Italy on2003. On the other hand, this study can provide us some methods to make the networks more robust, which may avoid similar incidents happening again. Theoretically, we often use phase transition theory to explains the existence of the giant component of a network after a fraction of nodes are removed. In addition, sandpile model and percolation model are also used to study the cascading failures on networks.In recent studies, dependence links have been proposed to the percolation model and used to study the robustness of the networks with such links, which shows that the networks are more vulnerable than the classical networks containing only connectiv-ity links. This model usually demonstrates a first order phase transition, rather than the second order phase transition found in the classical network percolation. Consid-ering the real situation that the interdependent nodes are usually connected, we study the cascading dynamics of networks when the dependence links partially overlap with the connectivity links. We find that the percolation transitions are not always sharpened by making nodes interdependent. For a high fraction of overlapping, the network is robust for random failures, and the percolation transition is second order, while for a low fraction of overlapping, the percolation process shows a first order phase transition. This work demonstrates that the crossover between two types of transitions does not only depend on the density of the dependence links but also on the overlapping of the connectivity and dependence links. Using generating functional techniques, we present exact solutions for the size of the giant component and the critical point, which are in good agreement with the simulations.In addition, in order to properly model the dependence of nodes in real networks, we have studied the cascading failures on a network with asymmetric dependence. Instead of the unconditional dependence of two node connected by the dependence links, the de-pendence threshold is used to determine the dependence of nodes. In our model, a node tends to be dependent on a node with a larger degree. This could represent some real re-lations among interacting agents in a networked system. We find that when the networks contain only the asymmetric dependence links, the systems are robust and demonstrate a second order phase transition. Both simulation and analytical results reveal the exis-tence of the crossover between the first and second order percolation transitions in our model. We also develop an approach to study percolation on such networks, which is in agreement with the simulation results well.Besides, we will review the basic concepts of complex networks in this article, and give a summary of the theoretical approaches to study the percolation and cascading process on networks.