The IEEE standard 802.17 Resilient Packet Ring (RPR) aims at combining SONET/SDH's carrier-class functionalities of high availability, reliability, and profitable TDM service (voice) support with Ethernet's high bandwidth utilization, low equipment cost, and simplicity (Davik et al., 2004; Yuan et al., 2004; Spadaro et al., 2004). RPR is a ring-based architecture consisting of two counter directional optical fiber rings with up to 255 nodes. Similar to SONET/SDH, RPR is able to provide fast recovery from a single link or node failure within 50 ms, and carry legacy TDM traffic with a high level of quality of service (QoS). Similar to Ethernet, RPR provides advantages of low equipment cost and simplicity and exhibits an improved bandwidth utilization due to statistical multiplexing. The bandwidth utilization is further increased by means of spatial reuse. In RPR, packets are removed from the ring by the corresponding destination node (destination stripping). The destination stripping enables nodes in different ring segments to transmit simultaneously, resulting in spatial reuse of bandwidth and an increased bandwidth utilization. Furthermore, RPR provides fairness, as opposed to today's Ethernet, and allows the full ring bandwidth to be utilized under normal (failure-free) operation conditions, as opposed to today's SONET/SDH rings where 50% of the available bandwidth is reserved for protection. Current RPR networks are single-channel systems (i.e., each fiber ring carries a single wavelength channel) and are expected to be primarily deployed in metro edge and metro core areas.

In the following sections, we explain RPR in greater detail, paying particular attention to its architecture, access control, fairness control, and protection.

Optical burst switching (OBS) is one of the recently proposed optical switching techniques which probably received the greatest deal of attention (Chen et al., 2004). OBS may be viewed as a switching technique that combines the merits of optical circuit switching (OCS) and optical packet switching (OPS) while avoiding their respective shortcomings. The switching granularity at the burst rather than wavelength level allows for statistical multiplexing in OBS, which is not possible in OCS, while requiring a lower control overhead than OPS. More precisely, in OCS, the entire bandwidth of each lightpath is dedicated to one pair of source and destination nodes and unused bandwidth cannot be reclaimed by other nodes ready to send data. Thus, OCS does not allow for statistical multiplexing. On the other hand, in OCS networks no OEO conversion is needed at intermediate nodes. As a result, OCS networks provide all-optical circuits that are transparent in terms of bit rate, modulation scheme, and protocol. OCS is well suited for large data transmissions whose long connection holding time on the order of a few minutes, hours, days, weeks, or even months justify the involved twoway reservation overhead for setting up or releasing a lightpath, which may take a few hundred milliseconds. Since many applications require only subwavelength bandwidth and/or involve bursts that last only a few seconds or less, the coarse wavelength switching granularity of OCS becomes increasingly inefficient and impractical. Unlike OCS, OPS is able to provide a significant statistical multiplexing gain due to the fact that bandwidth is not dedicated to a single connection but may be shared by multiple data flows.

In this chapter we shift our attention from the existence of certain structures in random networks, to the ability of finding such structures. More precisely, we consider the problem of navigating towards a destination, using only local knowledge of the network at each node. This question has practical relevance in a number of different settings, ranging from decentralised routing in communication networks, to information retrieval in large databases, file sharing in peer-to-peer networks, and the modelling of the interaction of people in society.

The basic consideration is that there is a fundamental difference between the existence of network paths, and their algorithmic discovery. It is quite possible, for example, that paths of a certain length exist, but that they are extremely difficult, or even impossible to find without global knowledge of the network topology. It turns out that the structure of the random network plays an important role here, as there are some classes of random graphs that facilitate the algorithmic discovery of paths, while for some other classes this becomes very difficult.

Highway discovery

To illustrate the general motivation for the topics treated in this chapter, let us start with some practical considerations. We turn back to the routing protocol described in Chapter 5 to achieve the optimal scaling of the information flow in a random network. Recall from Section 5.3 that the protocol is based on a multi-hop strategy along percolation paths that arise w.h.p. inside rectangles of size m × κ log m that partition the entire network area.

One of the motivations to study random networks on the infinite plane has been the possibility of observing sharp transitions in their behaviour. We now discuss the asymptotic behaviour of sequences of finite random networks that grow larger in size. Of course, one expects that the sharp transitions that we observe on the infinite plane are a good indication of the limiting behaviour of such sequences, and we shall see to what extent this intuition is correct and can be made rigorous.

In general, asymptotic properties of networks are of interest because real systems are of finite size and one wants to discover the correct scaling laws that govern their behaviour. This means discovering how the system is likely to behave as its size increases.

We point out that there are two equivalent scalings that produce networks of a growing number of nodes: one can either keep the area where the network is observed fixed, and increase the density of the nodes to infinity; or one can keep the density constant and increase the area of interest to infinity. Although the two cases above can describe different practical scenarios, by appropriate scaling of the distance lengths, they can be viewed as the same network realisation, so that all results given in this chapter apply to both scenarios.

Preliminaries: modes of convergence and Poisson approximation

We make frequent use of a powerful tool, the Chen–Stein method, to estimate convergence to a Poisson distribution. This method is named after work of Chen (1975) and Stein (1978) and is the subject of the monograph by Barbour, Holst and Janson (1992).

What is this book about, and who is it written for? To start with the first question, this book introduces a subject placed at the interface between mathematics, physics, and information theory of systems. In doing so, it is not intended to be a comprehensive monograph and collect all the mathematical results available in the literature, but rather pursues the more ambitious goal of laying the foundations. We have tried to give emphasis to the relevant mathematical techniques that are the essential ingredients for anybody interested in the field of random networks. Dynamic coupling, renormalisation, ergodicity and deviations from the mean, correlation inequalities, Poisson approximation, as well as some other tricks and constructions that often arise in the proofs are not only applied, but also discussed with the objective of clarifying the philosophy behind their arguments. We have also tried to make available to a larger community the main mathematical results on random networks, and to place them into a new communication theory framework, trying not to sacrifice mathematical rigour. As a result, the choice of the topics was influenced by personal taste, by the willingness to keep the flow consistent, and by the desire to present a modern, communication-theoretic view of a topic that originated some fifty years ago and that has had an incredible impact in mathematics and statistical physics since then. Sometimes this has come at the price of sacrificing the presentation of results that either did not fit well in what we thought was the ideal flow of the book, or that could be obtained using the same basic ideas, but at the expense of highly technical complications.

In the words of Hungarian mathematician Alfréd Rényi, ‘the mathematical theory of information came into being when it was realised that the flow of information can be expressed numerically in the same way as distance, time, mass, temperature …’

In this chapter, we are interested in the dynamics of the information flow in a random network. To make precise statements about this, we first need to introduce some information-theoretic concepts to clarify – from a mathematical perspective – the notion of information itself and that of communication rate. We shall see that the communication rate between pairs of nodes in the network depends on their (random) positions and on their transmission strategies. We consider two scenarios: in the first one, only two nodes wish to communicate and all the others help by relaying information; in the second case, different pairs of nodes wish to communicate simultaneously. We compute upper and lower bounds on achievable rates in the two cases, by exploiting some structural properties of random graphs that we have studied earlier. We take a statistical physics approach, in the sense that we derive scaling limits of achievable rates for large network sizes.

Information-theoretic preliminaries

The topics of this section only scratch the surface of what is a large field of study; we only discuss those topics that are needed for our purposes. The interested reader may consult specific information-theory textbooks, such as McEliece (2004), and Cover and Thomas (2006), for a more in-depth study.

The act of communication can be interpreted as altering the state of the receiver due to a corresponding action of the transmitter.

One of the advantages of studying random network models on the infinite plane is that it is possible to observe sharp phase transitions. Informally, a phase transition is defined as a phenomenon by which a small change in the local parameters of a system results in an abrupt change of its global behaviour, which can be observed over an infinite domain. We shall see in subsequent chapters how these phenomena observed on the infinite plane are a useful indication of the behaviour in a finite domain. For now, however, we stick with the analysis on the infinite plane.

The random tree; infinite growth

We start by making a precise statement on the possibility that the branching process introduced in Chapter 1 grows forever. This is trivially true when the offspring distribution is such that P(Xi ≥ 1) = 1, i.e., when each node in the tree has at least one child. However, it is perhaps less trivial that for generic offspring distribution it is still possible to have an infinite growth if and only if E(Xi) = μ > 1.

Theorem 2.1.1When μ ≤ 1 the branching process does not grow forever with probability one, except when P(X = 1) = 1. When μ > 1, the branching process grows forever with positive probability.

The proof of Theorem 2.1.1 uses generating functions, so we start by saying a few words about these. Generating functions are a very convenient tool for all sorts of computations that would be difficult and tedious without them. These computations have to do with sums of random variables, expectations and variances.

Random networks arise when nodes are randomly deployed on the plane and randomly connected to each other. Depending on the specific rules used to construct them, they create structures that can resemble what is observed in real natural, as well as in artificial, complex systems. Thus, they provide simple models that allow us to use probability theory as a tool to explain the observable behaviour of real systems and to formally study and predict phenomena that are not amenable to analysis with a deterministic approach. This often leads to useful design guidelines for the development and optimal operation of real systems.

Historically, random networks has been a field of study in mathematics and statistical physics, although many models were inspired by practical questions of engineering interest. One of the early mathematical models appeared in a series of papers starting in 1959 by the two Hungarian mathematicians Paul Erdös and Alfréd Rényi. They investigated what a ‘typical’ graph of n vertices and m edges looks like, by connecting nodes at random. They showed that many properties of these graphs are almost always predictable, as they suddenly arise with very high probability when the model parameters are chosen appropriately. This peculiar property generated much interest among mathematicians, and their papers marked the starting point of the field of random graph theory. The graphs they considered, however, were abstract mathematical objects and there was no notion of geometric position of vertices and edges.

Mathematical models inspired by more practical questions appeared around the same time and relied on some notion of geometric locality of the random network connections.

In this chapter we examine the subcritical and the supercritical phase of a random network in more detail, with particular reference to bond percolation on the square lattice. The results presented lead to the exact determination of the critical probability of bond percolation on the square lattice, which equals 1/2, and to the discovery of additional properties that are important building blocks for the study of information networks that are examined later in the book.

One peculiar feature of the supercritical phase is that in almost all models of interest there is only one giant cluster that spans the whole space. This almost immediately implies that any two points in space are connected with positive probability, uniformly bounded below. Furthermore, the infinite cluster quickly becomes extremely rich in disjoint paths, as p becomes strictly greater than pc. So we can say, quite informally, that above criticality, there are many ways to percolate through the model. On the other hand, below criticality the cluster size distribution decays at least exponentially fast in all models of interest. This means that in this case, one can reach only up to a distance that is exponentially small.

To conclude the chapter we discuss an approximate form of phase transition that can be observed in networks of fixed size.

Preliminaries: Harris–FKG Inequality

We shall make frequent use of the Harris–FKG inequality, which is named after Harris (1960) and Fortuin, Kasteleyn and Ginibre (1971). This expresses positive correlations between increasing events.

Atmospheric effects add significant complicating factors to the job of the radio-system designer. Initially, the way in which multipath propagation can be established due to the atmospheric structure is explained together with the effect this can have on the received signal. Another phenomenon, ducting, whose existence depends upon the structure of the atmosphere, is then discussed. It is seen that ducting can lead to levels of long-distance interference rising. The way in which diversity techniques can reduce the effect of multipath fading is explained. Another form of fading (‘diffraction fading’) is then described. Diffraction fading occurs when the atmosphere causes the path of the radio wave to bend upwards as it travels, leading to no line of sight existing between points for which a clear line of sight would be expected. Next, it is shown that, in severe cases of multipath propagation, the delay between two paths can be significant. This leads to the received spectrum in large-bandwidth links becoming distorted. The amount of fading caused by rain is explained as another factor that must be considered when designing a microwave radio link. Further, the fact that, even if no fading occurs, the ever-present molecules present in the atmosphere will cause attenuation is described, together with an indication of the frequency dependence of this attenuation. Finally, the way in which atmospheric losses affect the noise performance of radio systems, particularly Earth–space systems, is analysed.

When a radio wave can reach a receiver by more than one route, we say that the receiver is in a multipath environment. The way in which a standing wave pattern is established when the received signal is the combination of both a direct and a reflected signal is explained. The characteristics of the standing wave are shown to depend upon the nature of the reflection as determined by its reflection coefficient. Further examples of propagation paths involving reflection include propagation over a flat plane and propagation over water, the latter having the additional complication of tidal variation often causing the position of the reflecting surface to change. The more complex situation that arises when there are many different routes from transmitter to receiver is analysed. It is seen that the nature of the standing wave depends on whether one of the contributing paths is dominant (the ‘Rician’ environment) or whether the strengths of all of the signals on all paths are about equal (the ‘Rayleigh’ environment). It is further shown that the reflected signal depends on whether the reflecting surface is smooth or rough and the difference in the nature of the reflected signal is analysed. A further possible propagation mechanism is that of penetration of materials. The amount of penetration is seen to be dependent upon the electrical characteristics of the material and the frequency of the electromagnetic wave.

Introduction

In practical situations, radio waves will reflect off walls and off the ground.