{"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/efficient-random-graph-matching-via-degree","title":"Efficient random graph matching via degree profiles","arxiv_id":"1811.07821","date":"2018-11-19","proceeding":null,"authors":["Jian Ding","Zongming Ma","Yihong Wu","Jiaming Xu"],"abstract":"Random graph matching refers to recovering the underlying vertex\ncorrespondence between two random graphs with correlated edges; a prominent\nexample is when the two random graphs are given by Erd\\H{o}s-R\\'{e}nyi graphs\n$G(n,\\frac{d}{n})$. This can be viewed as an average-case and noisy version of\nthe graph isomorphism problem. Under this model, the maximum likelihood\nestimator is equivalent to solving the intractable quadratic assignment\nproblem. This work develops an $\\tilde{O}(n d^2+n^2)$-time algorithm which\nperfectly recovers the true vertex correspondence with high probability,\nprovided that the average degree is at least $d = \\Omega(\\log^2 n)$ and the two\ngraphs differ by at most $\\delta = O( \\log^{-2}(n) )$ fraction of edges. For\ndense graphs and sparse graphs, this can be improved to $\\delta = O(\n\\log^{-2/3}(n) )$ and $\\delta = O( \\log^{-2}(d) )$ respectively, both in\npolynomial time. The methodology is based on appropriately chosen distance\nstatistics of the degree profiles (empirical distribution of the degrees of\nneighbors). Before this work, the best known result achieves $\\delta=O(1)$ and\n$n^{o(1)} \\leq d \\leq n^c$ for some constant $c$ with an $n^{O(\\log n)}$-time\nalgorithm \\cite{barak2018nearly} and $\\delta=\\tilde O((d/n)^4)$ and $d =\n\\tilde{\\Omega}(n^{4/5})$ with a polynomial-time algorithm\n\\cite{dai2018performance}.","url_abs":"http://arxiv.org/abs/1811.07821v1","url_pdf":"http://arxiv.org/pdf/1811.07821v1.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":"efficient-random-graph-matching-via-degree","repo_url":"https://github.com/xjmoffside/degree_profile","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"graph-matching","task_name":"Graph Matching"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1811.07821","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}