Dynamic intimate contact social networks and epidemic interventions Page: 6
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6 C.D. Corley, A.R. Mikler, D.J. Cook and K. Singh
triangles; however, in the bipartite case, triangles among vertices of the same set
do not occur. The following descriptors will be used to analyze the topology of
the networks generated by our simulator. We define the clustering coefficient for
u V W
I * *
a b c d
Figure 1 The neighborhoods of vertices u and v intersect at vertices a and b. The
clustering coefficient between these two vertices is cc.(u, v) .= . However, there is no
overlap (clustering) between vertices u and w; thus the clustering coefficient of vertex u
remains the same (cc.(u) = ).
pairs of nodes, both in either T or I : cc. captures the overlap in neighborhoods of
vertices u and v. Whenever the neighborhood of vertices u and v do not overlap then
cc. (u, v) 0. Conversely, if vertices u and v are elements of the same neighborhood
then cc. (u, v) 1= . The equation for the neighborhood overlap is given in Eq. 5. The
cartoon in Fig.1 demonstrates clustering in a bipartite graph. The neighborhoods of
vertices u and v intersect at nodes a and b in the opposing subset, the corresponding
clustering coefficient is cc. (u, v) - 2; however, there is no neighborhood intersection
between vertices u and w and cc. (u, w) 0. To evaluate the clustering coefficient of
a particular node, the average over the subset is calculated for only those edge pairs
where an overlap in neighborhoods exist (Eq. 6). The graphs clustering coefficient
cc. (G) is the average of each bipartition subsets corresponding clustering coefficient
(cc. (T), cc. (I)) (Eq. 7). Considering complex networks with significant differences
between degrees of the vertices, the previously introduced clustering coefficient
may not provide a strong and informative analysis of the network topology. The
following two clustering coefficient flavors further describe neighborhood overlap.
Equation 8 describes a clustering coefficient lower bound and considers a setting
where a small neighborhood is encompassed by a large neighborhood. Equation 9
evaluates an upper bound on the clustering coefficient and evaluates occurrences
where small or large neighborhoods overlap. The following clustering coefficients
can be evaluated similarly to eqns. 6 and 7 : ccT(u), ccT(T), ccT(l), ccT(G),
ccT(u), cc1T(T), ccT(l), and ccT (G) (Latapy et al. 2006).
Ncc.((u)n N(v) (5)
IN(u) U N(v)l
( vCN(N(u)) cc.(u, v)
n1 cc. (T) + ni cc. (1)
nT + ni
cc1(U, v)= (8)
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Corley, Courtney; Mikler, Armin R.; Cook, Diane J., 1963- & Singh, Karan P. Dynamic intimate contact social networks and epidemic interventions, article, 2008; [Geneva, Switzerland]. (digital.library.unt.edu/ark:/67531/metadc132993/m1/6/: accessed October 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Engineering.