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DOI: 10.4230/LIPIcs.ITCS.2022.74

Correlation Detection in Trees for Planted Graph Alignment

Authors: Luca Ganassali, Laurent Massoulié, and Marc Lelarge

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract

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).

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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)

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@InProceedings{ganassali_et_al:LIPIcs.ITCS.2022.74,
  author =	{Ganassali, Luca and Massouli\'{e}, Laurent and Lelarge, Marc},
  title =	{{Correlation Detection in Trees for Planted Graph Alignment}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{74:1--74:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.74},
  URN =		{urn:nbn:de:0030-drops-156709},
  doi =		{10.4230/LIPIcs.ITCS.2022.74},
  annote =	{Keywords: inference on graphs, hypothesis testing, Erd\H{o}s-R\'{e}nyi random graphs}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2017.44

Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models

Authors: Lennart Gulikers, Marc Lelarge, and Laurent Massoulié

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract

Motivated by community detection, we characterise the spectrum of the non-backtracking matrix B in the Degree-Corrected Stochastic Block Model. Specifically, we consider a random graph on n vertices partitioned into two asymptotically equal-sized clusters. The vertices have i.i.d. weights {\phi_u}_{u=1}^n with second moment \PHItwo. The intra-cluster connection probability for vertices u and v is \frac{\phi_u \phi_v}{n}a and the inter-cluster connection probability is \frac{\phi_u \phi_v}{n}b. We show that with high probability, the following holds: The leading eigenvalue of the non-backtracking matrix B is asymptotic to \rho = \frac{a+b}{2} \PHItwo. The second eigenvalue is asymptotic to \mu_2 = \frac{a-b}{2} \PHItwo when \mu_2^2 > \rho, but asymptotically bounded by \sqrt{\rho} when \mu_2^2 \leq \rho. All the remaining eigenvalues are asymptotically bounded by \sqrt{\rho}. As a result, a clustering positively-correlated with the true communities can be obtained based on the second eigenvector of B in the regime where \mu_2^2 > \rho. In a previous work we obtained that detection is impossible when $\mu_2^2 \leq \rho,$ meaning that there occurs a phase-transition in the sparse regime of the Degree-Corrected Stochastic Block Model. As a corollary, we obtain that Degree-Corrected Erdös-Rényi graphs asymptotically satisfy the graph Riemann hypothesis, a quasi-Ramanujan property. A by-product of our proof is a weak law of large numbers for local-functionals on Degree-Corrected Stochastic Block Models, which could be of independent interest.

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Lennart Gulikers, Marc Lelarge, and Laurent Massoulié. Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 44:1-44:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gulikers_et_al:LIPIcs.ITCS.2017.44,
  author =	{Gulikers, Lennart and Lelarge, Marc and Massouli\'{e}, Laurent},
  title =	{{Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{44:1--44:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.44},
  URN =		{urn:nbn:de:0030-drops-81795},
  doi =		{10.4230/LIPIcs.ITCS.2017.44},
  annote =	{Keywords: Degree-Corrected Stochastic Block Model, Non-backtracking Matrix, Machine Learning, Social Networks}
}

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Documents authored by Lelarge, Marc

Correlation Detection in Trees for Planted Graph Alignment

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Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models

Abstract

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