Improved Achievability and Converse Bounds for Erdős-Rényi Graph Matching
This addresses a theoretical graph matching problem for researchers in information theory and network analysis, but is incremental as it refines prior bounds.
The paper tackles the problem of perfectly recovering vertex correspondence between two correlated Erdős-Rényi graphs, improving an existing achievability bound and providing a converse bound to establish scaling dependence on correlation, with bounds differing by a factor of two for sparse, correlated graphs.
We consider the problem of perfectly recovering the vertex correspondence between two correlated Erdős-Rényi (ER) graphs. For a pair of correlated graphs on the same vertex set, the correspondence between the vertices can be obscured by randomly permuting the vertex labels of one of the graphs. In some cases, the structural information in the graphs allow this correspondence to be recovered. We investigate the information-theoretic threshold for exact recovery, i.e. the conditions under which the entire vertex correspondence can be correctly recovered given unbounded computational resources. Pedarsani and Grossglauser provided an achievability result of this type. Their result establishes the scaling dependence of the threshold on the number of vertices. We improve on their achievability bound. We also provide a converse bound, establishing conditions under which exact recovery is impossible. Together, these establish the scaling dependence of the threshold on the level of correlation between the two graphs. The converse and achievability bounds differ by a factor of two for sparse, significantly correlated graphs.