{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-semidefinite-relaxations-for-the-block","title":"On semidefinite relaxations for the block model","arxiv_id":"1406.5647","date":"2014-06-21","proceeding":null,"authors":["Arash A. Amini","Elizaveta Levina"],"abstract":"The stochastic block model (SBM) is a popular tool for community detection in\nnetworks, but fitting it by maximum likelihood (MLE) involves a computationally\ninfeasible optimization problem. We propose a new semidefinite programming\n(SDP) solution to the problem of fitting the SBM, derived as a relaxation of\nthe MLE. We put ours and previously proposed SDPs in a unified framework, as\nrelaxations of the MLE over various sub-classes of the SBM, revealing a\nconnection to sparse PCA. Our main relaxation, which we call SDP-1, is tighter\nthan other recently proposed SDP relaxations, and thus previously established\ntheoretical guarantees carry over. However, we show that SDP-1 exactly recovers\ntrue communities over a wider class of SBMs than those covered by current\nresults. In particular, the assumption of strong assortativity of the SBM,\nimplicit in consistency conditions for previously proposed SDPs, can be relaxed\nto weak assortativity for our approach, thus significantly broadening the class\nof SBMs covered by the consistency results. We also show that strong\nassortativity is indeed a necessary condition for exact recovery for previously\nproposed SDP approaches and not an artifact of the proofs. Our analysis of SDPs\nis based on primal-dual witness constructions, which provides some insight into\nthe nature of the solutions of various SDPs. We show how to combine features\nfrom SDP-1 and already available SDPs to achieve the most flexibility in terms\nof both assortativity and block-size constraints, as our relaxation has the\ntendency to produce communities of similar sizes. This tendency makes it the\nideal tool for fitting network histograms, a method gaining popularity in the\ngraphon estimation literature, as we illustrate on an example of a social\nnetworks of dolphins. We also provide empirical evidence that SDPs outperform\nspectral methods for fitting SBMs with a large number of blocks.","url_abs":"http://arxiv.org/abs/1406.5647v3","url_pdf":"http://arxiv.org/pdf/1406.5647v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"on-semidefinite-relaxations-for-the-block","repo_url":"https://github.com/aaamini/SBM-SDP","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"community-detection","task_name":"Community Detection"},{"task_slug":"graphon-estimation","task_name":"Graphon Estimation"},{"task_slug":"stochastic-block-model","task_name":"Stochastic Block Model"},{"task_slug":"model","task_name":"model"}],"methods":[{"method_slug":"pca","method_name":"PCA"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1406.5647","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}