Georgia Institute of Technology College of Sciences

Graph alignment refers to recovering the underlying vertex correspondence between two random graphs with correlated edges. This problem can be viewed as an average-case and noisy version of the well-known graph isomorphism problem. For correlated Erdős-Rényi random graphs, we will give insights on the fundamental limits for the planted formulation of this problem, establishing statistical thresholds for partial recovery. From the computational point of view, we are interested in designing and analyzing efficient (polynomial-time) algorithms to recover efficiently the underlying alignment: in a sparse regime, we exhibit an local rephrasing of the planted alignment problem as the correlation detection problem in trees. Analyzing this related problem enables to derive a message-passing algorithm for our initial task and gives insights on the existence of a hard phase.

Based on joint works with Laurent Massoulié and Marc Lelarge: 

https://arxiv.org/abs/2002.01258

https://arxiv.org/abs/2102.02685

https://arxiv.org/abs/2107.07623