# The Influence of Social Network Graph Structure on Disease Dynamics in a Simulated Environment Page: 57

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i lies on one or more geodesic paths from s to t. If node i lies on every geodesic path from s to

t, the transmission centrality for node i is increased by one. If node i only lies on a portion of the

geodesic paths from s to t, the transmission centrality for node i is increased by that proportion.

The largest possible transmission centrality occurs when node i lies on every geodesic path

between every two nodes s and t, s Z i t. In terms of public health, if an individual is in a

position to have maximum transmission centrality, vaccinating this individual effectively protects

an entire segment of the population. For example, a person or group of people who bring supplies

to a remote village may have maximum transmission centrality to and from the village. Vaccination

prevents transfer of disease from the greater population to the village and likewise, transfer from

the village to the greater population.

Since there are n -r 1 nodes not equal to i and n -r 2 nodes not equal to i or s, the maximum

possible transmission centrality is one-half rn - 1 times rn - 2 as shown in Equation 24. Division

by 2 is necessary in an undirected graph since the path from s to t is equivalent to the path from

t to s. This maximum value is used to normalize transmission centralities. The normalized value

calculation for node i, Cf(i), is shown in Equation 25.

The formulas for transmissibility are identical to those of betweenness centrality. However,

there is a difference in the definition of the geodesic path between two points. In a non-weighted

graph, the path length between point s and point t is measured by the number of edges between the

two points. A geodesic path from s to t, therefore, is one that has the fewest edges between s and t.

In a weighted graph in which the weight indicates the number of contacts between two individuals,

it is reasonable to consider a path to be shorter along more heavily weighted edges. Thus, the

sum of the inverse of all edge weights along each path from s to t is calculated to determine the

geodesic path.

This calculation of the geodesic path makes a significant difference when considering disease

transmission. Consider a situation illustrated by Figure 4.4 in which the individual represented

by Node 0 is infectious and all other nodes are susceptible. Considering path length only, Node

3 is most likely to become infected via Node 4. However, if the frequency of contacts is taken57

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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/m1/67/: accessed May 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; .