We prove that a graph G is asymptotically isomorphic to the ray if and only if G is uniformly spherically bounded and is of bounded local degrees. This problem arouse in combinatorics and was posed in [3] (Problem 10.1).
Sequence comparison and alignment has had an enormous impact on our understanding of evolution, biology, and disease. Comparison and alignment of biological networks will likely have a similar impact. Existing network alignments use information exter
nal to the networks, such as sequence, because no good algorithm for purely topological alignment has yet been devised. In this paper, we present a novel algorithm based solely on network topology, that can be used to align any two networks. We apply it to biological networks to produce by far the most complete topological alignments of biological networks to date. We demonstrate that both species phylogeny and detailed biological function of individual proteins can be extracted from our alignments. Topology-based alignments have the potential to provide a completely new, independent source of phylogenetic information. Our alignment of the protein-protein interaction networks of two very different species--yeast and human--indicate that even distant species share a surprising amount of network topology with each other, suggesting broad similarities in internal cellular wiring across all life on Earth.
