Consider the random graph G(n, p) with mean degree c. Show that in the limit of large n the expected number of triangles in the network is 1/z c^3 This means that the number of triangles is constant, neither growing nor vanishing in the limit of large n. Show that the expected number of connected triples in the network (as defined on page 200) is 1/2 nc^2. Hence calculate the clustering coefficient C, as defined in Eq. (7.41), and confirm that it agrees for large n with the value given in Eq. (12.11). A very simple quantity to calculate for the Poisson random graph is the clustering coefficient. Recall that the clustering coefficient C is a measure of the transitivity in a network (Section 7.9) and is defined as the probability that two network neighbors of a vertex are also neighbors of each other. In a random graph the probability that any two vertices are neighbors is exactly the same- all such probabilities are equal to p = c/(n-1). Hence c = c/n-1. This is one of several respects in which the random graph differs sharply from most from real- world networks, many of which have quite high clustering coefficients-see Table 8.1-while Eq. (12.11) tends to zero in the limit n rightarrow infinity if the mean degree c stays fixed. This discrepancy is discussed further in Section 12.8. Yet another way to write the clustering coefficient would be to note that if we have a path of length two, uvw, then it is also true to say that vertices u and w have a common neighbor in v- they share a mutual acquaintance in social network terms. If the triad uvw is closed then u and w are themselves acquainted, so the clustering coefficient can be thought of also as the fraction of pairs of people with a common friend who are themselves friends or equivalently as the mean probability that two people with a common friend are themselves friends. This is perhaps the most common way of defining the clustering coefficient. In mathematical notation: C= (number of triangles) times 3/(number of connected triples) Here a “connected triple” means three vertices uvw with edges (u, v) and (v, w). (The edge (u, w) can be present or not.) The factor of three in the numerator arises because each triangle gets counted three times when we count the connected triples in the network. The triangle uvw for instance contains the triples uvw, vwu, and wuv. In the older social networks literature the clustering coefficient is sometimes referred to as the “fraction of transitive triples, ” which is a reference to this definition of the coefficient. The textbook is network: an introduction

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