Graph matching the matchable nodes when some nodes are unmatchable

Often in inference across multiple graphs there is a latent correspondence between the nodes across networks, and graph matching---the problem of finding an alignment between the nodes of two graphs that best preserves common structure across graphs---seeks to uncover this latent correspondence in the presence of unknown or errorfully known vertex labels. However, in many realistic applications only a core subset of the nodes are matchable in each graph with the remaining nodes being junk vertices, only participating in one of the two networks. Under a statistical model for this situation, we show that correctly matching the core nodes is still possible provided the number of junk nodes does not grow too rapidly, allowing for asymptotically perfect core recovery in the presence of nearly linear (in core size) junk. We also consider the setting of junk present in only one network, showing that a smaller core network can be asymptotically perfectly matched to the corresponding core in a network with exponentially many (in core size) junk vertices present. These theoretical results are further corroborated in both simulated and real data experiments. Although it is possible to correctly match the core vertices across networks, many graph matching algorithms do not identify these proper matches (versus junk--to--junk or core--to--junk matches). As such, we present a procedure for a posteriori detecting the matched core--to--core vertices after the networks have been matched, and demonstrate the effectiveness of our core detection procedure across both simulated and real data.