Improved achievability and converse bounds for erd?os-rényi graph matching

We consider the problem of perfectly recovering the vertex correspondence between two correlated Erd}os-Rényi (ER) graphs. For a pair of correlated graphs on the same ver-tex set, the correspondence between the vertices can be ob- scured 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 re- covery, i.e. the conditions under which the entire vertex correspondence can be correctly recovered given unbounded computational resources. Pedarsani and Grossglauser provided an achievability re-sult of this type. Their result establishes the scaling de-pendence 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 scal- ing 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 cor- related graphs. c 2016 Copyright held by the owner/author(s). Publication rights licensed to ACM.