{"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/the-spectral-distribution-of-random-graphs","title":"The Spectral Distribution of Random Graphs with Given Degree Sequences","arxiv_id":"2412.02087","date":"2024-12-03","proceeding":null,"authors":["Shuyi Wang","Kevin Li","Jiaoyang Huang"],"abstract":"In this article, we study random graphs with a given degree sequence $d_1, d_2, \\cdots, d_n$ from the configuration model. We show that under mild assumptions of the degree sequence, the spectral distribution of the normalized Laplacian matrix of such random graph converges in distribution to the semicircle distribution as the number of vertices $n\\rightarrow \\infty$. This extends work by McKay (1981) and Tran, Vu and Wang (2013) which studied random regular graphs ($d_1=d_2=\\cdots=d_n=d$). Furthermore, we extend the assumption to show that a slightly more general condition is equivalent to the weak convergence to semicircle distribution. The equivalence is also illustrated by numerical simulations.","url_abs":"https://arxiv.org/abs/2412.02087v1","url_pdf":"https://arxiv.org/pdf/2412.02087v1.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-spectral-distribution-of-random-graphs","repo_url":"https://github.com/kevinmli/Spectral-Distribution-Given-Degree-Sequences","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}