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 socio-technical systems. The complex properties of real-world networks have a profound impact on the behavior of equilibrium and non-equilibrium 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. Here, we present a coherent and comprehensive review of the vast research activity concerning epidemic processes, 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 very similar models for the description of the diffusion of pathogens, knowledge and innovation. For this reason, while we focus on the main results and the paradigmatic models in infectious disease modeling, we also present the major results concerning generalized social contagion processes. Finally, we outline the research activity at the forefront in the study of epidemic spreading in co-evolving, coupled and time-varying networks.

The influence of the network characteristics on the virus spread is analyzed in a new-the -intertwined Markov chain-model, whose only approximation lies in the application of mean field theory. The mean field approximation is quantified in detail. The -intertwined model has been compared with the exact 2 -state Markov model and with previously proposed "homogeneous" or "local" models. The sharp epidemic threshold , which is a consequence of mean field theory, is rigorously shown to be equal to = 1 ( max ( )), where max ( ) is the largest eigenvalue-the spectral radius-of the adjacency matrix . A continued fraction expansion of the steady-state infection probability at node is presented as well as several upper bounds.Index Terms-Epidemic threshold, Markov theory, mean field theory, spectral radius, virus spread.

Analyzing the behavior of complex networks is an important element in the design of new man-made structures such as communication systems and biologically engineered molecules. Because any complex network can be represented by a graph, and therefore in turn by a matrix, graph theory has become a powerful tool in the investigation of network performance. This self-contained 2010 book provides a concise introduction to the theory of graph spectra and its applications to the study of complex networks. Covering a range of types of graphs and topics important to the analysis of complex systems, this guide provides the mathematical foundation needed to understand and apply spectral insight to real-world systems. In particular, the general properties of both the adjacency and Laplacian spectrum of graphs are derived and applied to complex networks. An ideal resource for researchers and students in communications networking as well as in physics and mathematics.

In recent years there has been a shift in focus from the study of local, mostly task-related activation to the exploration of the organization and functioning of large-scale structural and functional complex brain networks. Progress in the interdisciplinary field of modern network science has introduced many new concepts, analytical tools and models which allow a systematic interpretation of multivariate data obtained from structural and functional MRI, EEG and MEG. However, progress in this field has been hampered by the absence of a simple, unbiased method to represent the essential features of brain networks, and to compare these across different conditions, behavioural states and neuropsychiatric/neurological diseases. One promising solution to this problem is to represent brain networks by a minimum spanning tree (MST), a unique acyclic subgraph that connects all nodes and maximizes a property of interest such as synchronization between brain areas. We explain how the global and local properties of an MST can be characterized. We then review early and more recent applications of the MST to EEG and MEG in epilepsy, development, schizophrenia, brain tumours, multiple sclerosis and Parkinson's disease, and show how MST characterization performs compared to more conventional graph analysis. Finally, we illustrate how MST characterization allows representation of observed brain networks in a space of all possible tree configurations and discuss how this may simplify the construction of simple generative models of normal and abnormal brain network organization.

Serious epidemics, both in cyber space as well as in our real world, are expected to occur with high probability, which justifies investigations in virus spread models in (contact) networks. The N -intertwined virus spread model of the SIS-type is introduced as a promising and analytically tractable model of which the steady-state behavior is fairly completely determined. Compared to the exact SIS Markov model, the N -intertwined model makes only one approximation of a mean-field kind that results in upper bounding the exact model for finite network size N and improves in accuracy with N . We review many properties theoretically, thereby showing, besides the flexibility to extend the model into an entire heterogeneous setting, that much insight can be gained that is hidden in the exact Markov model.

Abstract-The underlying concepts of an exact QoS routing algorithm are explained. We show that these four concepts, namely 1) nonlinear definition of the path length; 2) a -shortest path approach; 3) nondominance; and 4) look-ahead, are fundamental building blocks of a multiconstrained routing algorithm. The main reasons to consider exact multiconstrained routing algorithms are as follows. First, the NP-complete behavior seems only to occur in specially constructed graphs, which are unlikely to occur in realistic communication networks. Second, there exist exact algorithms that are equally complex as heuristics in algorithmic structure and in running time on topologies that do not induce NP-complete behavior. Third, by simply restricting the number of paths explored during the path computation, the computational complexity can be decreased at the expense of possibly loosing exactness. The presented four concepts are incorporated in SAMCRA, a self-adaptive multiple constraints routing algorithm.

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This rigourous and self-contained book describes mathematical and, in particular, stochastic methods to assess the performance of networked systems. It consists of three parts. The first part is a review on probability theory. Part two covers the classical theory of stochastic processes (Poisson, renewal, Markov and queuing theory), which are considered to be the basic building blocks for performance evaluation studies. Part three focuses on the relatively new field of the physics of networks. This part deals with the recently obtained insights that many very different large complex networks - such as the Internet, World Wide Web, proteins, utility infrastructures, social networks - evolve and behave according to more general common scaling laws. This understanding is useful when assessing the end-to-end quality of communications services, for example, in Internet telephony, real-time video and interacting games. Containing problems and solutions, this book is ideal for graduate students taking courses in performance analysis.