Dynamical Processes On Complex Networks:Cascades And Its Interaction With Epidemics

Networks are very common in our daily life. In some cases, the constituents are physically networked, such as Internet, while in others they are not, but network is a natural language to de-scribe the interaction pattern of them, e.g. social networks. These two factors make network a very powerful modeling tool in the study of complex systems. Complex network science (some-times referred to as network science) thus emerges. It is an interdisciplinary subject. Mathematics, physics, biology, social and information science are all relevant.In this paper, by considering scenarios similar to communication networks, we study cascades and the interaction between epidemics and cascades on networks. To be more specific, the main contents of this paper can be summarized as follows:1. We study how network load is changed due to node removal. Cascading failure, a dynamic process widely found in many real-world networks, has been intensively investigated in com-plex networks during the past decade. But because of its non-linearity, theoretical study are difficult to perform. The interaction between the underlying topology and cascades has not been systematically studied. This issue is widely concerned, but still open. We haven’t tried to resolve it directly. Instead, we study how topology change, specifically, node removal, in-fluences the load of the network. We consider a scenario similar to the data communication networks. The removal operation can model attacks and errors in networks, or the planned control of network topology. The load of a node is measured by its betweenness central-ity, and the load of a network is measured by the sum of betweenness of constituent nodes. By analysis and simulations, we show that when a single node is removed, the change of the remaining network’s load is positively correlated with the degree of the removed node. In multiple-node removal, by comparing several node removal schemes, we show in detail how significantly different the change of the remaining network’s load will be between start-ing the removal from small degree/betweenness nodes and from large degree/betweenness nodes. Moreover, when starting the removal from small degree/betweenness nodes, we not only observe that the remaining network’s load decreases, which is consistent with previous studies, but also find that the load of hubs keeps decreasing. These results help us to make a deeper understanding about the dynamics after topology changes, and are useful in planned control of network topology.2. We study the interaction between epidemics and cascades. Recently, it is realized that in complex systems, not only can constituents interact with each other, but also can subsystems. Correspondingly, coupled networks becomes a hot topic in the study of complex networks. This inspires researchers to study also coupled dynamics on networks, e.g., study the inter-action between epidemics and rumors about the epidemic. Coupling of networks reflects the complexity of interactions among subsystems, while coupling of dynamics reflects the com-plexity of behavior of complex systems. The study of the latter is just in its starting stage. Under this background, we study the interaction between epidemics and cascades. Tradi-tionally, epidemics and cascades are independently studied, but in practice, there are many cases where these two dynamics interact with each other and neither of their effects can be ignored. For example, consider that a digital virus is spreading in a communication network, which is transferring data in the meantime. We build a model based on the epidemiological SIR model and a local load sharing cascading failure model to study the interplay between these two dynamics. We observe several new phenomena:when the capacity of nodes is very small, the network is fragile against cascades; when their capacity is very large, the network is also fragile, because the epidemic is more likely to be able to spread out. In the equilibrium, a giant component, constitutes by nodes both uninfected and un-failed, emerges only if the tolerance parameter a, capturing the capacity of nodes, is within some limited in-terval (αl, αu). After analyzing the cause of these phenomena, we then present a theoretical solution of the key values of αl and αu in random networks and uncorrelated networks. This work deepens our understanding of the interaction between these two dynamics and extends the study of coupled dynamics on networks. In some applications, the model is more realistic comparing to its counterpart in traditional study and the result is thus more convincing.3. We study the epidemic outbreak threshold in uncorrelated networks when the load and ca-pacity of nodes are considered. There is a tendency in the research of epidemics, which is to study more realistic and thus more complex spreading models. We thus study the epidemic outbreak threshold in a model which takes the load and capacity of nodes into account. In communication networks, nodes manipulates load and bare limited capacity. The study of epidemics on these networks better considers the effects of these factors. In conventional studies of epidemics, the outbreak threshold, with the infectivity under which the epidemic will finally dies out, is one of the most important issues. If the capacity of nodes is finite, some nodes may fail due to overloading, or there may even be a cascading failure. The cas-cading failure will suppress the spreading of epidemics. So the threshold is different from traditional studies. We first calculate respectively the fraction of failed nodes and removed nodes (due to infection) in the equilibrium. Then we present the epidemic threshold criteria with methods borrowed from dynamical systems. This criteria involves two parameters, the infection probability, capturing the infection rate, and the tolerance parameter, capturing the capacity of nodes. When the infection probability is fixed, the tolerance parameter α must be larger than a critical value to fulfill the criteria, and the fraction of nodes both uninfected and un-failed in the steady state is the largest at this critical point, indicating a most desirable state in the real-world application, i.e. communication networks. So the the presentation of the epidemic threshold criteria is of significance.