  Network epidemiology has become a vital tool in understanding the effects of
high-degree vertices, geographic and demographic communities, and other
inhomogeneities in social structure on the spread of disease. However, many
networks derived from modern datasets are quite dense, such as mobility
networks where each location has links to a large number of potential
destinations. One way to reduce the computational effort of simulating
epidemics on these networks is sparsification, where we select a representative
subset of edges based on some measure of their importance. Recently an approach
was proposed using an algorithm based on the effective resistance of the edges.
We explore how effective resistance is correlated with the probability that an
edge transmits disease in the SI model. We find that in some cases these two
notions of edge importance are well correlated, making effective resistance a
computationally efficient proxy for the importance of an edge to epidemic
spread. In other cases, the correlation is weaker, and we discuss situations in
which effective resistance is not a good proxy for epidemic importance.
