The Influence of Social Network Graph Structure on Disease Dynamics in a Simulated Environment Page: 26
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2.4.2. Random Graphs, Ordered Lattices, Hypergraphs, and Small-World Graphs
In 1951, Solomonoff and Rapoport described structures referred to as random nets . In
1960, Erdos and Rinyi continued the investigation of random graphs , as shown in Figure 2.10.
Two distinct methods for building random graphs were described. One begins with a fixed number
of vertices, r n, and a fixed number of edges, m. The edges are randomly selected out of the (-1)
that are possible. Using this technique, there are 2 equiprobable random graphs that can
be constructed. The alternate definition is one in which the number of vertices are fixed, but the
edges are selected randomly with probability p. The number of edges using this technique is a
random variable. Therefore, to develop a graph with an average of mn edges, the value of p should
be set to -Q. For example, to construct random graph with an average of 5 edges in a graph with
10 nodes, the probability p that an edge exists between two vertices is9 as calculated below:
5 5 1
(10) 45 9
FIGURE 2.10. Random graph example
Contrary to a random graph, a completely ordered graph (also called a regular graph or a lattice)
is a graph in which each node is linked to k of its immediate neighbors. To visualize such a graph,
it is helpful to think of the vertices in aligned in a circular fashion as demonstrated in Figure 2.11.
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Johnson, Tina V. The Influence of Social Network Graph Structure on Disease Dynamics in a Simulated Environment, dissertation, December 2010; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc33173/m1/36/: accessed May 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; .