{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9398","name":"Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models","abstract":"Motivated by community detection, we characterise the spectrum of the non-backtracking matrix B in the Degree-Corrected Stochastic Block Model.\r\n\r\nSpecifically, 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. \r\n\r\nWe 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.\r\n\r\nIn 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. \r\n\r\nAs a corollary, we obtain that Degree-Corrected Erd\u00f6s-R\u00e9nyi graphs asymptotically satisfy the graph Riemann hypothesis, a quasi-Ramanujan property.\r\n\r\nA 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.","keywords":["Degree-Corrected Stochastic Block Model","Non-backtracking Matrix","Machine Learning","Social Networks"],"author":[{"@type":"Person","name":"Gulikers, Lennart","givenName":"Lennart","familyName":"Gulikers"},{"@type":"Person","name":"Lelarge, Marc","givenName":"Marc","familyName":"Lelarge"},{"@type":"Person","name":"Massouli\u00e9, Laurent","givenName":"Laurent","familyName":"Massouli\u00e9"}],"position":44,"pageStart":"44:1","pageEnd":"44:27","dateCreated":"2017-11-28","datePublished":"2017-11-28","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Gulikers, Lennart","givenName":"Lennart","familyName":"Gulikers"},{"@type":"Person","name":"Lelarge, Marc","givenName":"Marc","familyName":"Lelarge"},{"@type":"Person","name":"Massouli\u00e9, Laurent","givenName":"Laurent","familyName":"Massouli\u00e9"}],"copyrightYear":"2017","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ITCS.2017.44","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6270","volumeNumber":67,"name":"8th Innovations in Theoretical Computer Science Conference (ITCS 2017)","dateCreated":"2017-11-28","datePublished":"2017-11-28","editor":{"@type":"Person","name":"Papadimitriou, Christos H.","givenName":"Christos H.","familyName":"Papadimitriou"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9398","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6270"}}}