Information Recovery in Shuffled Graphs via Graph Matching
This addresses the challenge of vertex alignment errors in graph analysis, which is crucial for applications in network science and machine learning, though it appears incremental by extending existing stochastic blockmodel frameworks.
The paper tackles the problem of errorfully observed vertex correspondences in multiple graph inference by establishing an information theoretic foundation for understanding the impact on subsequent inference and the capacity of graph matching to recover alignment. It demonstrates a phase transition for graph matchability in terms of graph correlation and shows practical effects on tasks like two-sample hypothesis testing and spectral clustering.
While many multiple graph inference methodologies operate under the implicit assumption that an explicit vertex correspondence is known across the vertex sets of the graphs, in practice these correspondences may only be partially or errorfully known. Herein, we provide an information theoretic foundation for understanding the practical impact that errorfully observed vertex correspondences can have on subsequent inference, and the capacity of graph matching methods to recover the lost vertex alignment and inferential performance. Working in the correlated stochastic blockmodel setting, we establish a duality between the loss of mutual information due to an errorfully observed vertex correspondence and the ability of graph matching algorithms to recover the true correspondence across graphs. In the process, we establish a phase transition for graph matchability in terms of the correlation across graphs, and we conjecture the analogous phase transition for the relative information loss due to shuffling vertex labels. We demonstrate the practical effect that graph shuffling---and matching---can have on subsequent inference, with examples from two sample graph hypothesis testing and joint spectral graph clustering.