  Network science has increasingly become central to the field of epidemiology
and our ability to respond to infectious disease threats. However, many
networks derived from modern datasets are not just large, but dense, with a
high ratio of edges to nodes. This includes human mobility networks where most
locations have a large number of links to many other locations. Simulating
large-scale epidemics requires substantial computational resources and in many
cases is practically infeasible. One way to reduce the computational cost of
simulating epidemics on these networks is sparsification, where a
representative subset of edges is selected based on some measure of their
importance. We test several sparsification strategies, ranging from naive
thresholding to random sampling of edges, on mobility data from the U.S.
Following recent work in computer science, we find that the most accurate
approach uses the effective resistances of edges, which prioritizes edges that
are the only efficient way to travel between their endpoints. The resulting
sparse network preserves many aspects of the behavior of an SIR model,
including both global quantities, like the epidemic size, and local details of
stochastic events, including the probability each node becomes infected and its
distribution of arrival times. This holds even when the sparse network
preserves fewer than $10\%$ of the edges of the original network. In addition
to its practical utility, this method helps illuminate which links of a
weighted, undirected network are most important to disease spread.
