Correlation Detection in Trees for Planted Graph Alignment

Authors Luca Ganassali, Laurent Massoulié, Marc Lelarge

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Luca Ganassali
  • Inria, DI/ENS, PSL Research University, Paris, France
Laurent Massoulié
  • MSR-Inria Joint Centre, Inria, DI/ENS, PSL Research University, Paris, France
Marc Lelarge
  • Inria, DI/ENS, PSL Research University, Paris, France


The authors would like to thank Guilhem Semerjian for helpful discussions, and the reviewers for relevant comments.

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Luca Ganassali, Laurent Massoulié, and Marc Lelarge. Correlation Detection in Trees for Planted Graph Alignment. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 74:1-74:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Motivated by alignment of correlated sparse random graphs, we study a hypothesis problem of deciding whether two random trees are correlated or not. Based on this correlation detection problem, we propose MPAlign, a message-passing algorithm for graph alignment, which we prove 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 conjecture that the connection between partial graph alignment and tree detection runs deeper, and that the parameter range where tree detection is impossible, which we partially characterize, corresponds to a region where partial graph alignment is hard (not polytime feasible).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Hypothesis testing and confidence interval computation
  • Mathematics of computing → Max marginal computation
  • Mathematics of computing → Graph algorithms
  • inference on graphs
  • hypothesis testing
  • Erdős-Rényi random graphs


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