When Can Two Unlabeled Networks Be Aligned Under Partial Overlap?

Network alignment refers to the problem of matching the vertex sets of two unlabeled graphs, which can be viewed as a generalization of the classic graph isomorphism problem. Network alignment has applications in several fields, including social network analysis, privacy, pattern recognition, computer vision, and computational biology. A number of heuristic algorithms have been proposed in these fields. Recent progress in the analysis of network alignment over stochastic models sheds light on the interplay between network parameters and matchability. In this paper, we consider the alignment problem when the two networks overlap only partially, i.e., there exist vertices in one network that have no counterpart in the other. We define a random bigraph model that generates two correlated graphs $G_{1,2}$; it is parameterized by the expected node overlap $t^2$ and by the expected edge overlap $s^2$. We define a cost function for structural mismatch under a particular alignment, and we identify a threshold for perfect matchability: if the average node degrees of $G_{1,2}$ grow as $\omega\left( (s^{-2}t^{-1} \log(n) \right)$, then minimization of the proposed cost function results in an alignment which (i) is over exactly the set of shared nodes between $G_1$ and $G_2$, and (ii) agrees with the true matching between these shared nodes. Our result shows that network alignment is fundamentally robust to partial edge and node overlaps.