Dynamic intimate contact social networks and epidemic interventions Page: 5
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Dynamic intimate contact social networks and epidemic interventions 5
tribution p(k) and clustering coefficients. Degree distribution gives the probability
of degrees in a network and has become an integral descriptive of the topology of
complex networks. The degree distribution function p(k) describes the total num-
ber of vertices in a graph with a given degree (Eq. 1). This same information is also
described by the cumulative degree distribution (Erdos & Renyi 1959). (Eq. 2).
p(k)= v (1)
EvEVjdg=k
Pk = k (2)
k' =k
The second graph descriptor is the clustering coefficient. It has been empirically
shown that many social networks have a higher neighborhood transitivity than that
of other random networks such as Internet topology (Newman & Park 2003, Watts
& Strogatz 393). Much of the analysis of the networks generated by our simulator
is evaluated using clustering coefficients. This statistic describes the overlap in
the network topology. The clustering coefficient Cv is the probability that any
two nodes are linked together if they have a neighbor in common. In an undirected
graph e(u, v) and e(v, u) are the same link. Hence, if vertex v has k neighbors k(k-1)
edges could exist in the neighborhood. Eq 3 defines the clustering coefficient for
undirected graphs. The clustering coefficient for the entire graph is the average of
each vertices's clustering coefficient over the graph (Eq. 4) (Watts & Strogatz 393).
C. N(v)(IN(v)-l) 2 {do(do 1) (3)
: y,u N(v),e(y,u) EE
Cc. C. (4)
i=1
3.2 Bipartite graph statistics
Many of the bipartite graph statistics relate to their classical counterparts. Some
of these descriptors are redefined while others are dual components of their classical
property. A recent technical paper by Latapy et al. describes the following bipartite
graph statistics in greater detail (Latapy, Magnien & Del Vecchio 2006).
Consider a bipartite graph G = (T, I, E). The size of the graph is now divided
into the size of the top portion nT = I T and the size of the bottom subset n = 1,
these are the number of nodes in the top vertex set and the bottom set, respectively.
The size of the edge set remains the same as for classical graphs m = IE. The
average degree is now separated for each bipartition subset; the top subsets average
degree is k = -" and the bottom subset kI = -. The average degree of the
graph G* (TU I, E) is now k - k ~ T+ The bipartite density is
thus 6(G) = m and 6(G*) ~ 2m with 6(G*) < 6(G).
Clustering coefficients are evaluated much differently in the bipartite setting.
In the classical graphs, the overlap among vertices is measured by the number of
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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/m1/5/: accessed December 10, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Engineering.