Correlation detection in trees for planted graph alignment
This work addresses graph alignment for sparse random graphs, with potential implications for network analysis, but it is incremental as it builds on existing methods and conjectures rather than proving a breakthrough.
The paper tackles the problem of detecting correlation between two random trees to inform graph alignment, showing conditions for feasibility and proposing MPAlign, a polynomial-time algorithm that succeeds when tree detection is feasible, revealing new parameter ranges for partial alignment of sparse random graphs.
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.