Epidemic models often incorporate contact networks along which the disease can be passed. This study uses a recentering-restarting evolutionary algorithm to locate likely epidemic networks for six different epidemic profiles containing early peaks, late peaks, and multiple peaks in the number of infected individuals. This study demonstrates that the algorithm can fit a broad variety of epidemic profiles. The difficulty of finding a network likely to produce a given epidemic profile varies between profiles, but all six profiles are fitted well in at least some of the evolutionary runs. A pseudometric on pairs of networks based on diffusion characters is used to assess the networks distribution in the space of networks. Both the scatter of networks evolved to match a single epidemic profile and the between-profile distances are evaluated. The diffusion character based pseudometric separates the networks for some pairs of profiles neatly while others apparently overlap to some degree.