CLÁUSULA SEXTA.- PRESENTACIÓN DE INFORMACIÓN Y ESTUDIOS

2.2 Directed Random Graphs

2.2.1 Denition Of A Directed Random Graph And C-Connectedness

A directed random graph is a mathematical structure from Graph Theory, which has its own standard notation and denitions. In this section we shall restrict our discussion to the directed graph only. In Section 2.2.2 corresponding terms for an epidemic process as dened in Section 1.2 will also be used. For the remainder of this chapter several epidemic and graph notations will be used interchangeably. For Chapters3 and 4the epidemic denitions will be preferred.

The denitions and theory presented in this section are sucient for this thesis. For a more detailed investigation of Graph Theory, including concepts not related to epidemic modelling, seeOre (1967) andBollobás (1998). AlsoHarary et al.(1965) discusses the theory and applications of directed graphs to structural models in the social sciences, though not with regard to epidemic modelling.

Standard Graph Theory is concerned with xed graphs, however we are interested in those of varying characteristics, so called Random Graphs. Again we refer the reader to more specic literature, Bollobás (1985) for example, to explore random graphs in more detail. There are two types of graph, directed and undirected, the correspondence to an epidemic requires the concept of one individual infecting another, i.e. a direction for the infection to occur, hence we consider the directed random graphs.

Dene a directed random graph, G, as a collection of N labelled vertices, 1, . . . , N (for

nite N). Set a subset of the vertices as roots, let there be R roots (1 ≤ R ≤ N).

Without loss of generality we may assign root vertices the labels 1, . . . , R and non-root vertices the labels R + 1, . . . , N. For each ordered pair of distinct vertices (i, j), where

2.2 Directed Random Graphs 37

1 ≤ i, j ≤ N, i 6= j, a directed edge from i to j occurs with a probability pi,j. If the edge (i, j) exists, we say there is a path from i to j. There are N(N − 1) possible directed edges between all pairs of distinct points, we do not consider parallel edges or loops beginning and ending at the same vertex. Thus a directed random graph G is a collection of N vertices and probabilities for the directed edges between all vertices.

The edge probabilities can take many forms. We shall initially consider independent edges, i.e. letting P[(i, j)] be the probability of the edge (i, j) being present, then P[(i, j), (i, k)] = P[(i, j)]P[(i, k)]for all 1 ≤ i, j, k ≤ N and i 6= j 6= k. More generally, random graphs are dened in terms of the out-degree distribution of each vertex. Let V_i be the out-degree distribution for vertex i. For independent edges, the out-degree distribution is multinomial, though we shall initially consider the simpler case of a binomial with parameters N − 1 and p, Vi∼ bin(N − 1, p).

A directed path is a sequence of directed edges, (v1, v₂), (v₂, v₃), . . . , (v_n−1, v_n). A non-root vertex i is said to be directionally connected from the non-root vertices if there exists a directed path from at least one root vertex to i, i.e. vn = i and 1 ≤ v1 ≤ R. The graph G, is said to be directionally connected if each non-root vertex is (directionally) connected to the root vertices. A random graph is said to be C-connected if exactly C non-root vertices are directionally connected to the root vertices. If C = N − R the graph is directionally connected.

The distance of vertex i to vertex j is equal to the number of edges in the shortest directed path from i to j. Let dij denote the distance of i to j. By convention, dii= 0 and if there is no directed path from i to j then dij =∞.

Denition 2.1

The rank of an individual is its minimal distance from a root vertex, i.e. rank(i) = min{dji:vertex j is a root}.

2.2 Directed Random Graphs 38

If vertex i is not connected to the root vertices, then it has an innite distance from all of them, hence an innite rank. For a given random directed graph, we can summarise the ranks of all vertices into a rank chain. The rank chain counts the number of vertices of a given rank. Dene Xi to be the number of vertices of rank i in the digraph for 0 ≤ i ≤ N − R + 1. The zeroth rank contains the root vertices, i.e. X0 = R for all digraphs, and hence the maximum nite rank for a vertex is N − R, i.e. one vertex of each rank. Thus we terminate the rank chain at rank N − R + 1 so that XN −R+1= 0 for all digraphs. Note that, X_∞= N− R − C, i.e. the number of vertices that are not connected to the roots and are at innite distance.

For t = 0, 1, . . . let Yt = P_t

i=0X_i, a cumulative total of the number of vertices and dene Zt= (X_t, Y_t). Then the rank chain for a digraph can be expressed as the vector Z = (Z0, Z1, . . . , Z_{N −R+1}).

The connectivity of a digraph can be written in terms of the ranks of its vertices, all vertices of nite non-zero rank are connected to the root vertices. The rank chain also encapsulates the connectivity of the digraph it corresponds to. For a given digraph G = g with the corresponding rank chain Z = z, its connectivity is

C =

N

X

i=1

I{0<rank(i)<∞}=

N −R+1

X

t=1

|{i : i ∈ g, rank(i) = t}| =

N −RX

t=1

x_t= y_{N −R}− y0.

Where I_{E} is the indicator function, equal to one if the condition E is true and zero otherwise.

We shall condition the random graph on its connectedness property in order to inves-tigate its behaviour in comparison to the unconditioned structure.

2.2 Directed Random Graphs 39

2.2.2 Epidemic Model And Its Relation To G

Recall the denition of the standard SIR (susceptible→infective→removed) stochastic epidemic model from Chapter 1. Consider a population of N individuals, of which R are initially (i.e. at t = 0) infective and N − R are susceptible. An infective individ-ual remains so for a period of time TI, the infectious period, a non-negative random variable. The infectious periods of dierent individuals are independent. For this chap-ter we shall initially let TI be a point mass distribution, i.e. TI = c for some c > 0.

While infectious an individual has potential contacts with other individuals within the population at times given by the points of a Poisson process of rate _N^λ > 0. Each such contact with another infective has no eect, while a contact with a susceptible individual immediately makes the susceptible an infective. At the end of its infec-tious period an infective no longer makes any contacts and is said to be removed, it is no longer involved in the epidemic. Let St and It be the number of susceptibles and infectives at time t ≥ 0, respectively. The epidemic continues until there are no more infectives remaining, so Iτ = 0 where τ is the stopping time of the epidemic, i.e. τ = inf{t ≥ 0 : It= 0}. The nal size of the epidemic is the number of susceptibles who became infected, S0− Sτ = N − R − Sτ.

The relation between the directed random graph G dened in Section 2.2.1 and the epidemic model is as follows (see, for example, Andersson and Britton (2000), chap-ter 7). The R root vertices correspond to the initial infective individuals, and the remaining vertices correspond to the initially susceptible individuals. By settting p = 1− exp(−_N^λT_I), an edge represents an infectious contact, since the probability of a susceptible avoiding infection from a single infective is exp(−_N^λT_I). The (random) set of vertices that are directionally connected to the root vertices has the same distri-bution as the set of individuals who become infected in the epidemic. Thus the number of directionally non-root connected vertices has the same distribution as the nal size