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.

Predicting the strength of a radio signal in the shadow of an obstacle is a vital function for propagation engineers. The mechanism by which a wave enters into the shadow of an obstacle is known as diffraction. Even the simplest of practical obstacles pose severe mathematical challenges. More easily solved approximations are adopted in order to estimate the strength of diffracted signals. The starting point for diffraction problems is the case where a receiver is in the shadow of a perfectly absorbing ‘knife-edge’ obstacle. This is then extended to encompass the situation where there are several such obstacles on the path. Many approximate multiple-knife-edge prediction methods exist and the most commonly used are analysed and compared. More accurate ‘near-exact’ methods are discussed. Although these methods usually make better predictions of the signal strength in the shadow of obstacles, they require significantly more computing time as well as being significantly more complicated to implement. Once an understanding of the properties of a diffracted signal has been obtained, it is possible to derive clearance requirements for a point-to-point path so that diffraction effects may be safely ignored. The insights gained by investigating the mechanism of diffraction into the shadow of an obstacle can be used to analyse two related phenomena: reflection from a finite surface and the formation of the radiation pattern of an aperture antenna.

Knife-edge diffraction

Diffraction is the name given to the mechanism by which waves enter into the shadow of an obstacle.

Point-to-area transmission is the generic name given to the way in which broadcasting transmitters or base stations for mobile communications provide coverage to a given area. A general overview is provided in order to deliver the most significant information as quickly as possible. Following that, the concept of electric field strength as an alternative to power density, prediction methods and the effect of frequency are explained in more detail. Digital mobile radio is selected for further study as a specific example of point-to-area communication. Path-loss-prediction methods specific to digital mobile radio are examined and various types of antenna that are used for the base stations of these systems are described. Finally, the effect that interference can have on the coverage range of a base station is explained.

Overview

The simplest form of antenna used is an ‘omni-directional’ antenna. These radiate equally in all directions in the horizontal plane. They often have a narrower beam (perhaps less than 20 degrees) in the vertical plane. Such antennas typically have a gain of 10 dBi. Two collinear wire elements that are fed at their junction with a signal form what is known as a dipole antenna. The basic omni-directional antenna is a dipole that is half a wavelength in height (a ‘half-wave dipole’ with each of the wire elements being a quarter of a wavelength in length). This has a wide vertical beam and has a gain of 2.1 dBi.

A further benefit from using the decibel scale becomes apparent when dealing with what is known as the link budget. Using decibels, a power budget on a radio link becomes as straightforward as a simple financial budget. Transmit power and gains can be thought of as equivalent to income, with losses and required margins being equivalent to expenditure.

For example, suppose that we transmit with a power of 30 dBm (1 watt). There are feeder losses, antenna gains, free-space loss, absorption loss, fading margin etc. The link budget is really a method of organising these parameters so as to make the calculation of the received signal level (under conditions of maximum fade, if the fade margin is considered) as straightforward as possible. The received signal level should be sufficient to deliver an acceptably low bit error ratio. Table A4.1 gives an example of such a link budget.