Piet Van Mieghem

Epidemic processes in complex 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.

Virus spread in networks

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

Graph Spectra for Complex Networks

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 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.

Concepts of exact QoS routing algorithms

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 /spl kappa/-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 /spl kappa/ 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.

Performance Analysis of Communications Networks and Systems

This rigorous and self-contained book describes mathematical and, in particular, stochastic methods to assess the performance of networked systems. It consists of three parts. Part one is a review of 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.

Link‐disjoint paths for reliable QoS routing

SUMMARY The problem of finding link/node-disjoint paths between a pair of nodes in a network has received much attention in the past. This problem is fairly well understood when the links in a network are only specified by a single link weight. However, in the context of quality of service routing, links are specified by multiple link weights and restricted by multiple constraints. Unfortunately, the problem of finding link/node disjoint paths in multiple dimensions faces different conceptual problems. This paper presents a first step to understanding these conceptual problems in link-disjoint quality of service routing and proposes a heuristic link-disjoint QoS algorithm that circumvents these problems. Copyright # 2003 John Wiley & Sons, Ltd.

On the efficiency of multicast

The average number of joint hops in a shortest-path multicast tree from a root to m arbitrary chosen group member nodes is studied. A general theory for all graphs, hence including the graph representation of the Internet, is presented which quantifies the multicast reduction in network links compared to m times unicast. For two special types of graphs, the random graph Gp(N) and the k-ary tree, exact and asymptotic results are derived. Comparing these explicit results with previously published Internet measurements [13] indicates that the number of routers in the Internet that can be reached from a root grows exponentially in the number of hops with an effective degree of approximately 3.2.

Generalized epidemic mean-field model for spreading processes over multilayer complex networks

Mean-field deterministic epidemic models have been successful in uncovering several important dynamic properties of stochastic epidemic spreading processes over complex networks. In particular, individual-based epidemic models isolate the impact of the network topology on spreading dynamics. In this paper, the existing models are generalized to develop a class of models that includes the spreading process in multilayer complex networks. We provide a detailed description of the stochastic process at the agent level where the agents interact through different layers, each represented by a graph. The set of differential equations that describes the time evolution of the state occupancy probabilities has an exponentially growing state-space size in terms of the number of the agents. Based on a mean-field type approximation, we developed a set of nonlinear differential equations that has linearly growing state-space size. We find that the latter system, referred to as the generalized epidemic mean-field (GEMF) model, has a simple structure characterized by the elements of the adjacency matrices of the network layers and the Laplacian matrices of the transition rate graphs. Finally, we present several examples of epidemic models, including spreading of virus and information in computer networks and spreading of multiple pathogens in a host population .

The N -intertwined SIS epidemic network model

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.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.

MAMCRA: a constrained-based multicast routing algorithm

Multicast routing algorithms that are capable of providing quality of service (QoS) to its members will play an important role in future communication networks. This paper discusses some fundamental properties of multicast routing subject to multiple QoS requirements. We will show that guaranteeing QoS and optimizing resource utilization are conflicting objectives and require a trade-off. We also present MAMCRA, a Multicast Adaptive Multiple Constraints Routing Algorithm, that guarantees QoS to the multicast members in an efficient, but not always optimal manner.