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summation in formula (2.6). The mathematical properties of the efficiency have been investigated in Ref. [32]. See Refs. [33–35] for some recent applications of the efficiency, and Refs. [32,36] for some extensions of formula (2.6). The communication of two non-adjacent nodes, say j and k, depends on the nodes belonging to the paths connecting j and k. Consequently, a measure of the relevance of a given node can be obtained by counting the number of geodesics going through it, and defining the so-called node betweenness. Together with the degree and the closeness of a node (defined as the inverse of the average distance from all other nodes), the betweenness is one of the standard measures of node centrality, originally introduced to quantify the importance of an individual in a social network [18,19,37]. More precisely, the betweenness bi of a node i, sometimes referred to also as load, is defined as [18,19,38,39]: bi =
j,k∈N,j =k
nj k (i) , nj k
(2.7)
where nj k is the number of shortest paths connecting j and k, while nj k (i) is the number of shortest paths connecting j and k and passing through i. See Refs. [20–22] for a description of the standard algorithms to find shortest paths (as the Dijkstra’s algorithm, or the breadth-first search method discussed in Section 7.2), and Refs. [40,41] for fast algorithms recently proposed to calculate the betweenness. Betweenness distributions have been investigated in Refs. [42–47]. Betweenness-betweenness correlations and betweenness-degree correlations have been studied respectively in Ref. [42] and in Refs. [48–50]. The concept of betweenness can be extended also to the edges. The edge betweenness is defined as the number of shortest paths between pairs of nodes that run through that edge [51]. This latter quantity will be used extensively in Section 5.3, as well as in Section 7.1.3. 2.1.3. Clustering Clustering, also known as transitivity, is a typical property of acquaintance networks, where two individuals with a common friend are likely to know each other [18]. In terms of a generic graph G, transitivity means the presence of a high number of triangles. This can be quantified by defining the transitivity T of the graph as the relative number of transitive triples, i.e. the fraction of connected triples of nodes (triads) which also form triangles [4,52,53]: T =
3 × # of triangles inG . # of connected triples of vertices inG
(2.8)
The factor 3 in the numerator compensates for the fact that each complete triangle of three nodes contributes three connected triples, one centered on each of the three nodes, and ensures that 0 T 1, with T = 1 for KN . An alternative possibility is to use the graph clustering coefficient C, a measure introduced by Watts and Strogatz in Ref. [28], and defined as follows. A quantity ci (the local clustering coefficient of node i) is first introduced, expressing how likely aj m = 1 for two neighbors j and m of node i. Its value is obtained by counting the actual number of edges (denoted by ei ) in Gi (the subgraph of neighbors of i defined as in Section 2.1). Notice that Gi can be, in some cases, unconnected. The local clustering coefficient is defined as the ratio between ei and ki (ki − 1)/2, the maximum possible number of edges in Gi [5,28]: 2ei j,m aij aj m ami ci = = . (2.9) ki (ki − 1) ki (ki − 1) Fast algorithms to compute ci are available in Ref. [54]. The clustering coefficient of the graph is then given by the average of ci over all the nodes in G: C = c =
1 ci . N
(2.10)
i∈N
By definition, 0 ci 1, and 0 C 1. The differences between C and T are illustrated in Refs. [4,31]. It is also useful to consider c(k), the clustering coefficient of a connectivity class k, which is defined as the average of ci taken over all nodes with a given degree k. Various higher-order clustering coefficients have been proposed over the years, among which we recall the kclustering coefficient, that accounts for k-neighbors [55,56], or other measures based on the internal structure of cycles