Despite the fact that complex networks are the driving force behind the investigation of the spectra of graphs, it is not the purpose of this book to dwell on complex networks. A generally accepted, all-encompassing definition of a complex network does not seem to be available. Instead, complex networks are understood by instantiation: the Internet, transportation (car, train, airplane) and infrastructural (electricity, gas, water, sewer) networks, biological molecules, the human brain network, social networks, software dependency networks, are examples of complex networks. By now, there is such a large literature about complex networks, predominantly in the physics community, that providing a detailed survey is a daunting task. We content ourselves here with referring to some review articles by Strogatz (2001); Newman et al. (2001); Albert and Barabasi (2002); Newman (2003b), and to books in the field by Watts (1999); Barabasi (2002); Dorogovtsev and Mendes (2003); Barrat et al. (2008); Dehmer and Emmert-Streib (2009); Newman (2010), and to references in these works. Application of spectral graph theory to chemistry and physics are found in Cvetković et al. (1995, Chapter 8).

Complex networks can be represented by a graph, denoted by G, consisting of a set N of N nodes connected by a set ℒ of L links. Sometimes, nodes and links are called vertices and edges, respectively, and are correspondingly denoted by the set V and E.

During the first years of the third millennium, considerable interest arose in complex networks such as the Internet, the world-wide web, biological networks, utility infrastructures (for transport of energy, waste, water, trains, cars and aircrafts), social networks, human brain networks, and so on. It was realized that complex networks are omnipresent and of crucial importance to humanity, whose still augmenting living standards increasingly depend on complex networks. Around the beginning of the new era, general laws such as “preferential attachment” and the “power law of the degree” were observed in many, totally different complex networks. This fascinating coincidence gave birth to an area of new research that is still continuing today. But, as is often the case in science, deeper investigations lead to more questions and to the conclusion that so little is understood of (large) networks. For example, the rather simple but highly relevant question “What is a robust network?” seems beyond the realm of present understanding. The most natural way to embark on solving the question consists of proposing a set of metrics that tend to specify and quantify “robustness”. Soon one discovers that there is no universal set of metrics, and that the metrics of any set are dependent on each other and on the structure of the network.

Any complex network can be represented by a graph. Any graph can be represented by an adjacency matrix, from which other matrices such as the Laplacian are derived.

Recent years have seen the great success of OFDM (orthogonal frequency division multiplexing) and DMT (discrete multitone) transceivers in many applications. The OFDM system has found many applications in wireless communications. It has been adopted in IEEE 802.11 for wireless local area networks, DAB for digital audio broadcasting, and DVB for digital video broadcasting. The DMT system is the enabling technology for high-speed transmission over digital subscriber lines. It is used in ADSL (asymmetric digital subscriber lines) and VDSL (very-high-speed digital subscriber lines). The OFDM and DMT systems are both examples of DFT transceivers that employ redundant guard intervals for equalization. Having a guard interval can greatly simplify the task of equalization at the receiver and it is now one of the most effective approaches for channel equalization. In this book we will study the OFDM and DMT under the framework of filter bank transceivers. Under such a framework, there are numerous possible extensions. The freedom in the filter bank transceivers can be exploited to better the systems for various design criteria. For example, transceivers can be optimized for minimum bit error rate, for minimum transmission power, or for higher spectral efficiency. We will explore all these possible optimization problems in this book.

The first three chapters describe the major building blocks relevant for the discussion of signal processing for communication and give the tools useful for solving problems in this area. Chapters 4–5 introduce the multirate building blocks and filter bank transceivers, and the basic idea of guard intervals for channel equalization. Chapter 6 gives a detailed discussion of OFDM and DMT systems. Chapters 7–10 consider the design of filter bank transceivers for different criteria and channel environments. A detailed outline is given at the end of Chapter 1. This book has been used as a textbook for a first-year graduate course at National Chiao Tung University, Taiwan, and at National Taiwan University. Most of the chapters can be covered in 16-18 weeks. Homework problems are given for Chapters 2–10.

In earlier chapters we saw that the use of redundancy in block transceivers allows us to remove ISI completely without using IIR filters. When the number of redundant samples per block v is more than the channel order L, there is no IBI, and we can further achieve zero ISI using a constant receiving matrix. The most notable example is the OFDM system studied in Chapter 6. But the use of redundant samples also decreases the transmission rate. For every M input symbols, the transmitter sends out N = M + ν samples. The actual transmission rate is decreased by a factor of N/M. There are ν redundant samples in every N samples transmitted. Reducing redundancy leads to a higher transmission rate and hence better bandwidth efficiency. At the same time, we would like the redundancy to be large enough so that the zero-forcing condition can still be satisfied without using IIR filters. A natural question to ask is: for a given channel and N, what is the smallest redundancy such that FIR transceivers exist? In other words, if we are to use an FIR transceiver that achieves zero ISI, what is the largest number of symbols that can be transmitted out of every N samples? This chapter aims to answer the question of minimum redundancy for the existence of FIR zero-forcing transceivers.

We will consider general FIR transceivers (Fig. 10.1) in which the filters are not constrained to be DFT filters as in the OFDM system. Moreover the length of the filters can be longer than the block size N. In this case the transmitting and receiving matrices are allowed to have memories, rather than constant matrices as in the OFDM case. We will see that the minimum redundancy depends on the underlying channel C(z), and it can be easily determined from the location of the zeros of the channel C(z) directly by inspection. The topic of minimum redundancy for FIR transceivers was first addressed in [182].

In this chapter we present a joint analysis of cooperation stimulation and security in autonomous mobile ad hoc networks under a game-theoretic framework. We first investigate a simple yet illuminating two-player packet-forwarding game, and derive the optimal and cheat-proof packet-forwarding strategies. We then investigate the secure-routing and packet-forwarding game for autonomous ad hoc networks in noisy and hostile environments, and derive a set of reputation-based cheat-proof and attackresistant cooperation-stimulation strategies. When analyzing the cooperation strategies, besides Nash equilibrium, other optimality criteria, such as Pareto optimality, subgame perfection, fairness, and cheat-proofing, are also considered. Both analysis and simulation studies show that the strategies discussed here can effectively stimulate cooperation among selfish nodes in autonomous mobile ad hoc networks under noise and attacks, and that the damage that can be caused by attackers is bounded and limited.

Introduction

Node cooperation is a very important issue in order for ad hoc networks to be successfully deployed in an autonomous way. In addition to many schemes that have been studied to stimulate node cooperation in ad hoc networks, ARCS was considered in the previous chapter to simultaneously stimulate cooperation among selfish nodes and defend against various attacks.