Introduction

The branching process model was introduced by Sir Francis Galton in 1873 to represent the genealogical descendance of individuals. More generally it provides a versatile model for the growth of a population of reproducing individuals in the absence of external limiting factors. It is an adequate starting point when studying epidemics since, as we shall see in Chapter 2, it describes accurately the early stages of an epidemic outbreak. In addition, our treatment of so-called dual branching processes paves the way for the analysis of the supercritical phase in Chapter 2. Finally, the present chapter gives an opportunity to introduce large deviations inequalities (and notably the celebrated Chernoff bound), which is instrumental throughout the book.

A Galton–Watson branching process can be represented by a tree in which each node represents an individual, and is linked to its parent as well as its children. The “root” of the tree corresponds to the “ancestor” of the whole population. An example of such a tree is depicted in Figure 1.1.

In the following we consider three distinct ways of exploring the so-called Galton–Watson tree, each well suited to establishing specific properties.

In the depth-first view, we start by exploring one child in the first generation, then explore using the same method recursively the subtree of its descendants, before we move to the next child of the first generation.

Introduction

In 1967, the sociologist Stanley Milgram published results of a letter-relaying experiment of his design. The now-famous experiment required a source individual to forward a letter to a destination individual, about whom was disclosed information such as address, name and profession. However, each source individual was forbidden to post the letter directly to the target person. Instead she was required to forward the letter to someone known on a first-name basis, who in turn was allowed to forward it only to such familiar contacts.

The outcome was that a significant fraction of letters reached their destinations. Moreover, they did so in at most six hops, justifying the term “six degrees of separation”. This observation is also often referred to as the “small-world phenomenon”. The problem, formulated in the social sciences, is highly relevant in many other settings, namely routing with limited information in communication networks and browsing behaviour on the World Wide Web.

Viewing the social world as a graph with edges between acquainted persons, if any individual can relay information to any other in a small number of hops as in Milgram's experiment, the corresponding graph must have a small diameter. As we saw in Chapter 4, the E-R graph does have a small diameter (logarithmic in the number of nodes).

Introduction

In Chapter 2 we saw that, when the average degree np of an Erdős–Rényi graph is of constant order λ > 1, the graph contains a giant component of size of order n with high probability. However, in that regime, this component's size is strictly less than n, so that the graph is disconnected.

In the present chapter we shall establish that connectivity appears when the product np is of order log n. We shall more precisely evaluate the probability of connectivity when np is asymptotic to log n + c for some constant c. In the framework of the Reed–Frost epidemic this corresponds to a regime known as atomic infection wherein all nodes are ultimately infected. In this regime we can analyse the time, in terms of the number of rounds, it takes the epidemic or the rumour to reach the whole population. This will be illustrated in Chapter 4 for the Reed–Frost epidemic and revisited in Chapter 6 when we introduce the small-world phenomenon.

The main mathematical tool required to prove the connectivity regime consists of Poisson approximation techniques, namely the Stein–Chen method. The Stein–Chen method provides bounds on how accurately a sum of {0, 1}-valued or Bernoulli random variables can be approximated by a Poisson distribution.

The idea of mimicking the propagation of biological epidemics to achieve diffusion of useful information was first proposed in the late 1980s, the decade that also saw the appearance of computer viruses. Back then, these viruses propagated by copies on floppy disks and caused much less harm than their contemporary versions. But it was already noticed that they evolved and survived much as biological viruses do, a fact that prompted the idea of putting these features to good (rather than evil) use. The first application to be considered was synchronisation of distributed databases.

Interest in this paradigm received new impetus with the advent of peer-to-peer systems, online social systems and wireless mobile ad hoc networks in the early 2000s. All these scenarios feature a complex network with potentially evolving connections. In such large-scale dynamic environments, epidemic diffusion of information is especially appealing: it is decentralised, and it relies on randomised decisions which can prove as efficient as carefully made decisions. Detailed accounts of epidemic algorithms can be found in papers by Birman et al. and Eugster et al. Their applications are manifold. They can be used to perform distributed computation of global statistics in a spatially extended environment (e.g. mean temperature seen by a collection of sensors), to perform real-time delivery of video data streams (e.g. to users receiving live TV via peer-to-peer systems over the internet) and to propagate updates of dynamic content (e.g. to mobile phone users whose phone operating system requires patching against vulnerabilities).