In our treatment of the structure and stability concepts for irreducible chains we have to this point considered only the dichotomy between transient and recurrent chains.

For transient chains there are many areas of theory that we shall not investigate further, despite the flourishing research that has taken place in both the mathematical development and the application of transient chains in recent years. Areas which are notable omissions from our treatment of Markovian models thus include the study of potential theory and boundary theory [326], as well as the study of renormalized models approximated by diffusions and the quasi-stationary theory of transient processes [108, 4].

Rather, we concentrate on recurrent chains which have stable properties without renormalization of any kind, and develop the consequences of the concept of recurrence. In this chapter we further divide recurrent chains into positive and null recurrent chains, and show here and in the next chapter that the former class provide stochastic stability of a far stronger kind than the latter.

For many purposes, the strongest possible form of stability that we might require in the presence of persistent variation is that the distribution of Φn does not change as n takes on different values. If this is the case, then by the Markov property it follows that the finite dimensional distributions of Φ are invariant under translation in time. Such considerations lead us to the consideration of invariant measures.

In applying the results and concepts of Part I in the domains of times series or systems theory, we have so far analyzed only linear models in any detail, albeit rather general and multidimensional ones. This chapter is intended as a relatively complete description of the way in which nonlinear models may be analyzed within the Markovian context developed thus far. We will consider both the general nonlinear state space model, and some specific applications which take on this particular form.

The pattern of this analysis is to consider first some particular structural or stability aspect of the associated deterministic control, or CM(F), model and then under appropriate choice of conditions on the disturbance or noise process (typically a density condition as in the linear models of Section 6.3.2) to verify a related structural or stability aspect of the stochastic nonlinear state space NSS(F) model.

Highlights of this duality are

1. (i) if the associated CM(F) model is forward accessible (a form of controllability), and the noise has an appropriate density, then the NSS(F) model is a T-chain (Section 7.1);

2. (ii) a form of irreducibility (the existence of a globally attracting state for the CM(F) model) is then equivalent to the associated NSS(F) model being a ψ-irreducible T-chain (Section 7.2);

3. (iii) the existence of periodic classes for the forward accessible CM(F) model is further equivalent to the associated NSS(F) model being a periodic Markov chain, with the periodic classes coinciding for the deterministic and the stochastic model (Section 7.3).

Books are individual and idiosyncratic. In trying to understand what makes a good book, there is a limited amount that one can learn from other books; but at least one can read their prefaces, in hope of help.

Our own research shows that authors use prefaces for many different reasons. Prefaces can be explanations of the role and the contents of the book, as in Chung [71] or Revuz [326] or Nummelin [303]; this can be combined with what is almost an apology for bothering the reader, as in Billingsley [37] or Çinlar [59]; prefaces can describe the mathematics, as in Orey [309], or the importance of the applications, as in Tong [388] or Asmussen [9], or the way in which the book works as a text, as in Brockwell and Davis [51] or Revuz [326]; they can be the only available outlet for thanking those who made the task of writing possible, as in almost all of the above (although we particularly like the familial gratitude of Resnick [325] and the dedication of Simmons [355]); they can combine all these roles, and many more.

This preface is no different. Let us begin with those we hope will use the book.

Who wants this stuff anyway?

This book is about Markov chains on general state spaces: sequences ψn evolving randomly in time which remember their past trajectory only through its most recent value. We develop their theoretical structure and we describe their application.

In this final section we collect together, for ease of reference, many of those mathematical results which we have used in developing our results on Markov chains and their applications: these come from probability and measure theory, topology, stochastic processes, the theory of probabilities on topological spaces, and even number theory.

We have tried to give results at a relevant level of generality for each of the types of use: for example, since we assume that the leap from countable to general spaces or topological spaces is one that this book should encourage, we have reviewed (even if briefly) the simple aspects of this theory; conversely, we assume that only a relatively sophisticated audience will wish to see details of sample path results, and the martingale background provided requires some such sophistication.

Readers who are unfamiliar with any particular concepts and who wish to delve further into them should consult the standard references cited, although in general a deep understanding of many of these results is not vital to follow the development in this book itself.

Some measure theory

We assume throughout this book that the reader has some familiarity with the elements of measure and probability theory. The following sketch of key concepts will serve only as a reminder of terms, and perhaps as an introduction to some non-elementary concepts; anyone who is unfamiliar with this section must take much in the general state space part of the book on trust, or delve into serious texts such as Billingsley [37], Chung [72] or Doob [99] for enlightenment.

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.

A representative of a major publishing house is on her way home from a conference in Singapore, excited about the possibility of a new book series. On the flight home to New York she opens her blackberry organizer, adding names of new contacts, and is disappointed to realize she may have caught the bug that was bothering her friend Alex at the cafè near the conference hotel. When she returns home she will send Alex an email to see how she's doing and to make sure this isn't a case of some new dangerous flu.

Of course, the publisher is aware that she is part of an interconnected network of other business men and women and their clients: Her value as an employee depends on these connections. She depends on the transportation network of taxis and airplanes to get her job done and is grateful for the most famous network today that allows her to contact her friend effortlessly even when separated by thousands of miles. Other networks of even greater importance escape her consciousness, even though consciousness itself depends on a highly interconnected fabric of neurons and vascular tissue. Communication networks are critical to support the air traffic controllers who manage the airspace around her. A supply chain of manufacturers makes her book business possible, as well as the existence of the airplane on which she is flying.