Correlation detection in trees for planted graph alignment

Abstract

Motivated by alignment of correlated sparse random graphs, we introduce a hypothesis testing problem of deciding whether or not two random trees are correlated. We obtain sufficient conditions under which this testing is impossible or feasible. We propose MPAlign, a message-passing algorithm for graph alignment inspired by the tree correlation detection problem. We prove MPAlign to succeed in polynomial time at partial alignment whenever tree detection is feasible. As a result our analysis of tree detection reveals new ranges of parameters for which partial alignment of sparse random graphs is feasible in polynomial time. We then conjecture that graph alignment is not feasible in polynomial time when the associated tree detection problem is impossible. If true, this conjecture together with our sufficient conditions on tree detection impossibility would imply the existence of a hard phase for graph alignment, i.e. a parameter range where alignment cannot be done in polynomial time even though it is known to be feasible in non-polynomial time.

38 pages, 9 figures

References in corpus (5)

• Information-theoretic thresholds for community detection in sparse networks
• Spectral Graph Matching and Regularized Quadratic Relaxations II: Erdős-Rényi Graphs and Universality
• From tree matching to sparse graph alignment
• Impossibility of Partial Recovery in the Graph Alignment Problem
• Aligning random graphs with a sub-tree similarity message-passing algorithm

Cited by in corpus (1)

• Faster algorithms for the alignment of sparse correlated Erdös-Rényi random graphs