The Hierarchical Stochastic Block Model for Multiple Networks

The Hierarchical Stochastic Block Model for Multiple Networks | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article The Hierarchical Stochastic Block Model for Multiple Networks Marco Battiston, Clement Lee This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4601684/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract In many research fields, there is an increased availability of network data arising as multiple networks. However, most statistical models for network data in the literature are designed for a single network. Among these, the Stochastic Block Model is arguably the most popular model to perform vertex clustering and community detection. We propose the Hierarchical Stochastic Block Model, a generalization of the SBM to the setting of multiple networks. This model uses a Hierarchical Pitman-Yor prior for the block allocation vector of each graph. The proposed model has two main advantages: 1) it allows different networks to share the same latent blocks and the level of sharing is learnt from the data; 2) the number of blocks in each graph and the overall number of blocks are learnt from the data too, hence avoiding complicated model selection procedures. We derive both MCMC and Variational Inference algorithms. The former targets the correct posterior and is tuning-free, while the latter relies on an approximation of the posterior distribution, but is potentially more scalable than MCMC. We apply the HSBM to a co-authorship network and a brain connectomic network, to illustrate how the model is able to capture different levels of block sharing. Stochastic Block Model Multiple Random Networks Bayesian Nonparametrics Hierarchical Pitman–Yor Process Full Text Additional Declarations No competing interests reported. Supplementary Files Supplementarymaterial.pdf Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 11 Apr, 2025 Reviews received at journal 23 Aug, 2024 Reviewers agreed at journal 22 Jul, 2024 Reviewers agreed at journal 12 Jul, 2024 Reviewers invited by journal 12 Jul, 2024 Editor assigned by journal 19 Jun, 2024 Submission checks completed at journal 19 Jun, 2024 First submitted to journal 18 Jun, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4601684","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":319983925,"identity":"4ef1697d-de1b-42b4-bd5e-3f4e9fb6ca55","order_by":0,"name":"Marco Battiston","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABE0lEQVRIie2RMUvDQBSAXzi4LLFdr4v5BcIdgdBBzF95IXAuDo4BRW+qS6qrf8N/kHDQLsGsNzhUhM4tgtRFzHWRQtJ2dLhveA/e4+O9xwNwOP4lgQ3MBm+x2G3RfqVEZhPhuK2Qo5Rtouwo5UydVJ/rzThJ/Oksx9u3cPjUVB85XITAJHYpcTnIWLtYWgSv0uBsKZ5NRkQNmVBMlt1KwK2CAbuKDVKNYAgdKSAI7FL1KNGmVRKrXOOPxrDR/reC+31KbKd4RatAOtHIy4x6CtpxfYvpgRzXsr2lriOWPmrxYrJopPhcTIJl9/nzqTb5+V3iPxRitfrS4WlTva9VfhMOfcm7lL8n7ML7H+lwOByOw/wCZZhcfB/1bQsAAAAASUVORK5CYII=","orcid":"","institution":"Lancaster University","correspondingAuthor":true,"prefix":"","firstName":"Marco","middleName":"","lastName":"Battiston","suffix":""},{"id":319983926,"identity":"b24cb4d4-47da-465a-9cc3-0584e270705f","order_by":1,"name":"Clement Lee","email":"","orcid":"","institution":"Newcastle University","correspondingAuthor":false,"prefix":"","firstName":"Clement","middleName":"","lastName":"Lee","suffix":""}],"badges":[],"createdAt":"2024-06-18 18:31:38","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4601684/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4601684/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":59649305,"identity":"f6f161dd-4f5e-47ff-9936-e378dc9b37dd","added_by":"auto","created_at":"2024-07-04 09:24:10","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2817952,"visible":true,"origin":"","legend":"","description":"","filename":"Mainpaper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4601684/v1_covered_f7f27d47-130e-4e1f-afcc-b84eff0df96f.pdf"},{"id":59648631,"identity":"d5f96c40-d509-4328-b7b0-89faa25dc55d","added_by":"auto","created_at":"2024-07-04 09:16:07","extension":"pdf","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":244400,"visible":true,"origin":"","legend":"","description":"","filename":"Supplementarymaterial.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4601684/v1/386bea27dde637ac656360b0.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The Hierarchical Stochastic Block Model for Multiple Networks","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"statistics-and-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"stco","sideBox":"Learn more about [Statistics and Computing](http://link.springer.com/journal/11222)","snPcode":"11222","submissionUrl":"https://submission.nature.com/new-submission/11222/3","title":"Statistics and Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Stochastic Block Model, Multiple Random Networks, Bayesian Nonparametrics, Hierarchical Pitman–Yor Process","lastPublishedDoi":"10.21203/rs.3.rs-4601684/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4601684/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In many research fields, there is an increased availability of network data arising as multiple networks. 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