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Register a new userPhase Transitions in the Quadratic Contact Process on Complex Networks
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The quadratic contact process (QCP) is a natural extension of the well studied linear contact process where infected (1) individuals infect susceptible (0) neighbors at rate $lambda$ and infected individuals recover ($1 longrightarrow 0$) at rate 1. In the QCP, a combination of two 1s is required to effect a $0 longrightarrow 1$ change. We extend the study of the QCP, which so far has been limited to lattices, to complex networks. comment{as a model for the change in a population through sexual reproduction and death.} We define t
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Power grids, road maps, and river streams are examples of infrastructural networks which are highly vulnerable to external perturbations. An abrupt local change of load (voltage, traffic density, or water level) might propagate in a cascading way and affect a significant fraction of the network. Almost discontinuous perturbations can be modeled by shock waves which can eventually interfere constructively and endanger the normal functionality of the infrastructure. We study their dynamics by solving the Burgers equation under random perturbations on several real and artificial directed graphs. Even for graphs with a narrow distribution of node properties (e.g., degree or betweenness), a steady state is reached exhibiting a heterogeneous load distribution, having a difference of one order of magnitude between the highest and average loads. Unexpectedly we find for the European power grid and for finite Watts-Strogatz networks a broad pronounced bimodal distribution for the loads. To identify the most vulnerable nodes, we introduce the concept of node-basin size, a purely topological property which we show to be strongly correlated to the average load of a node.
Community structure is one of the most relevant features encountered in numerous real-world applications of networked systems. Despite the tremendous effort of scientists working on this subject over the past few decades to characterize, model, and analyze communities, more investigations are needed to better understand the impact of community structure and its dynamics on networked systems. Here, we first focus on generative models of communities in complex networks and their role in developing strong foundation for community detection algorithms. We discuss modularity and the use of modularity maximization as the basis for community detection. Then, we overview the Stochastic Block Model, its different variants, and inference of community structures from such models. Next, we focus on time evolving networks, where existing nodes and links can disappear and/or new nodes and links may be introduced. The extraction of communities under such circumstances poses an interesting and non-trivial problem that has gained considerable interest over the last decade. We briefly discuss considerable advances made in this field recently. Finally, we focus on immunization strategies essential for targeting the influential spreaders of epidemics in modular networks. Their main goal is to select and immunize a small proportion of individuals from the whole network to control the diffusion process. Various strategies have emerged over the years suggesting different ways to immunize nodes in networks with overlapping and non-overlapping community structure. We first discuss stochastic strategies that require little or no information about the network topology at the expense of their performance. Then, we introduce deterministic strategies that have proven to be very efficient in controlling the epidemic outbreaks, but require complete knowledge of the network.
In the last two decades, network science has blossomed and influenced various fields, such as statistical physics, computer science, biology and sociology, from the perspective of the heterogeneous interaction patterns of components composing the complex systems. As a paradigm for random and semi-random connectivity, percolation model plays a key role in the development of network science and its applications. On the one hand, the concepts and analytical methods, such as the emergence of the giant cluster, the finite-size scaling, and the mean-field method, which are intimately related to the percolation theory, are employed to quantify and solve some core problems of networks. On the other hand, the insights into the percolation theory also facilitate the understanding of networked systems, such as robustness, epidemic spreading, vital node identification, and community detection. Meanwhile, network science also brings some new issues to the percolation theory itself, such as percolation of strong heterogeneous systems, topological transition of networks beyond pairwise interactions, and emergence of a giant cluster with mutual connections. So far, the percolation theory has already percolated into the researches of structure analysis and dynamic modeling in network science. Understanding the percolation theory should help the study of many fields in network science, including the still opening questions in the frontiers of networks, such as networks beyond pairwise interactions, temporal networks, and network of networks. The intention of this paper is to offer an overview of these applications, as well as the basic theory of percolation transition on network systems.
Complex networks have acquired a great popularity in recent years, since the graph representation of many natural, social and technological systems is often very helpful to characterize and model their phenomenology. Additionally, the mathematical tools of statistical physics have proven to be particularly suitable for studying and understanding complex networks. Nevertheless, an important obstacle to this theoretical approach is still represented by the difficulties to draw parallelisms between network science and more traditional aspects of statistical physics. In this paper, we explore the relation between complex networks and a well known topic of statistical physics: renormalization. A general method to analyze renormalization flows of complex networks is introduced. The method can be applied to study any suitable renormalization transformation. Finite-size scaling can be performed on computer-generated networks in order to classify them in universality classes. We also present applications of the method on real networks.
In recent years the research community has accumulated overwhelming evidence for the emergence of complex and heterogeneous connectivity patterns in a wide range of biological and sociotechnical systems. The complex properties of real-world networks have a profound impact on the behavior of equilibrium and nonequilibrium phenomena occurring in various systems, and the study of epidemic spreading is central to our understanding of the unfolding of dynamical processes in complex networks. The theoretical analysis of epidemic spreading in heterogeneous networks requires the development of novel analytical frameworks, and it has produced results of conceptual and practical relevance. A coherent and comprehensive review of the vast research activity concerning epidemic processes is presented, detailing the successful theoretical approaches as well as making their limits and assumptions clear. Physicists, mathematicians, epidemiologists, computer, and social scientists share a common interest in studying epidemic spreading and rely on similar models for the description of the diffusion of pathogens, knowledge, and innovation. For this reason, while focusing on the main results and the paradigmatic models in infectious disease modeling, the major results concerning generalized social contagion processes are also presented. Finally, the research activity at the forefront in the study of epidemic spreading in coevolving, coupled, and time-varying networks is reported.
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