This problem in graph theory is known as the graph center problem. So here's a cop-out solution using Graph.pm:

I'm sure there should be a more efficient algorithm for graph center than computing all pairs shortest paths (although it makes a difference if the bottleneck is computing Levenshtein distance or dealing with the big graph), but this is a starting point. Especially interesting would be an algorithm that explored the graph in an "on-line" fashion to avoid precomputing all the pairwise distances.

BTW, Graph.pm does offer a center_vertices method for the APSP object, but it appears to be broken (at least in my version of the module).