The structure or interconnection pattern of a network can be represented by a graph. This chapter mainly focuses on general properties of graphs that are of interest to the modeling of complex networks.

Around 2000, several remarkable, quite universal phenomena observed in many different complex networks gave birth to a new discipline network science, which encompasses parts of physics, mathematics, engineering, biology, medical sciences, social sciences, and even finance. The most important universal complex network characteristics are (1) a power-law or scale-free degree distribution (first reported via Internet measurements by Faloutsos et al. (1999)), (2) small-world structure proposed by Watts and Strogatz (1998), (3) preferential attachment as a simple driver to explain the power-law degree distribution of scale-free graphs (Barabási and Albert, 1999), (4) clustering and community structure since most complex networks are networks of networks and (5) a high robustness against random failures, but a vulnerability against targeted attacks of mainly the hubs (high-degree nodes).

After the book of Watts (1999), many articles and books on complex networks have followed (see e.g. Strogatz (2001); Barabási (2002); Dorogovtsev and Mendes (2003); Barrat et al. (2008); Dehmer and Emmert-Streib (2009); Newman (2010); Cohen and Havlin (2010); Estrada (2012), and references in these books). Those books overview the current state of the art in network science and present many applications to, for example, the Internet, the World Wide Web, and brain, financial, social and biological networks.

The spread of information in communications and on-line social networks is similarly described as the virus spread in a biological population. In epidemiology (see e.g. Bailey (1975); Anderson and May (1991); Daley and Gani (1999); Diekmann et al. (2012)), two simple “compartment” models form the basis on which many variants exist: the Susceptible-Infected-Susceptible (SIS) and the Susceptible-Infected-Removed (SIR) models. Both models start with infected items that can infect their direct, healthy neighbors that are susceptible to the disease. In the SIS model, each infected item can cure and become healthy, but susceptible again after recovering from the disease, while in the SIR model, the infected item that recovers (or dies) is removed from the population and from the infection process. Just as in queuing theory where the Kendall notation (Section 13.1.4) specifies the queueing system, epidemic models mainly differ by the number and type of compartments. Typical compartment names are healthy, but susceptible S, infected I, recovered or removed R, exposed to the virus E, alert A, and so on. Before concentrating on epidemic spread on networks, we briefly sketch important modelling aspects in classical epidemiology in Section 17.1.

This chapter mainly focuses on the SIS model, in continuous-time, applied to networks: the epidemic spreads over the links that interconnect the nodes (computers or living beings) of a network.

The aim of this first chapter is to motivate why stochastic processes, probability theory and graph theory are useful to solve problems in network science.

In any system or node in a network, there is always a non-zero probability of failure or of error penetration. A lot of problems in quantifying the failure rate, bit error rate or the computation of redundancy to recover from hazards are successfully treated by probability theory. Often we deal in communications with a large variety of signals, calls, source-destination pairs, messages, the number of customers per region, and so on. Often, precise information at any time is not available or, if it is available, deterministic studies or simulations are simply not feasible due to the large number of different parameters involved. For such problems, a stochastic approach is often a powerful vehicle, as has been demonstrated in the field of statistical physics or thermodynamics. Failure or attacks at the network level have reestablished the interest in network robustness analyses in relation to network security. In spite of the intuitively easy concept, a globally accepted definition as well as a framework to compute the robustness of a network is still lacking. Graph and probability theory are essential to address questions like: “Which are the vulnerable nodes?”, “Is this network robust?”, “Where do we need to add, remove or rewire links at minimum cost in order to maximize the network robustness?”

This book is a development of Performance Analysis of Communications Networks and Systems of 2006. Its current incarnation has a broader scope, extending to complex networks, the broad term covering all types of real-world networks, that encompass communications networks. Apart from the correction of numerous errors in the earlier book, nearly all chapters have been extended. Chapter 17 on Epidemics in networks has been added, while the appendix on algebraic graph theory has been deleted, because our book Graph Spectra for Complex Networks amply replaces this chapter. Similar to Graph Spectra for Complex Networks, art. x has been used to refer to article x in Appendix A. The number of problems, together with their solutions in Appendix B, has been doubled at least.

Performance analysis belongs to the domain of applied mathematics. In particular, the branches of mathematics as probability theory, stochastic processes and graph theory are exploited, besides analysis (calculus) and linear algebra that are omnipresent nearly everywhere. The major aim of this book is to offer several mathematical methods to address challenges in network science, the rapidly growing field of complex networking. The link with technology is kept shallow, on purpose, because most technical advances in micro-electronics, in communications protocols and services, standards, etc. have a more limited lifetime and a narrower scope compared to mathematical concepts.

This book aims to present methods rigorously, hence mathematically, with minimal resorting to intuition. It is my belief that intuition is often gained after the result is known and rarely before the problem is solved, unless the problem is simple.

The shortest path problem asks for the computation of the path from a source to a destination node that minimizes the sum of the positive weights of its constituent links. The related shortest path tree (SPT) is the union of the shortest paths from a source node to a set of m other nodes in the graph with N nodes. If m = N — 1, the SPT connects all nodes and is termed a spanning tree. The SPT belongs to the fundamentals of graph theory and has many applications. Moreover, powerful shortest path algorithms like that of Dijkstra exist.

Routers in the Internet forward IP packets to the next hop router, which is found by routing protocols (such as OSPF and BGP). Intra-domain routing as OSPF is based on the Dijkstra shortest path algorithm, while inter-domain routing with BGP is policy-based, which implies that BGP does not minimize a length criterion. Nevertheless, end-to-end paths in the Internet are shortest paths in roughly 70% of the cases. Therefore, we consider the shortest path between two arbitrary nodes because (a) the IP address does not reflect a precise geographical location and (b) uniformly distributed world wide communication, especially, on the web seems natural since the information stored in servers can be located in places unexpected and unknown to browsing users. The Internet type of communication is different from classical telephony because (a) telephone numbers have a direct binding with a physical location and (b) the intensity of average human interaction rapidly decreases with distance.