{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","department":[{"_id":"GradSch"}],"status":"public","language":[{"iso":"eng"}],"abstract":[{"text":"This paper focuses on Majority Dynamics in sparse graphs, in particular, as a\r\ntool to study internal cuts. It is known that, in Majority Dynamics on a finite\r\ngraph, each vertex eventually either comes to a fixed state, or oscillates with\r\nperiod two. The empirical evidence acquired by simulations suggests that for\r\nrandom odd-regular graphs, approximately half of the vertices end up\r\noscillating with high probability. We notice a local symmetry between\r\noscillating and non-oscillating vertices, that potentially can explain why the\r\nfraction of the oscillating vertices is concentrated around $\\frac{1}{2}$. In\r\nour simulations, we observe that the parts of random odd-regular graph under\r\nMajority Dynamics with high probability do not contain $\\lceil \\frac{d}{2}\r\n\\rceil$-cores at any timestep, and thus, one cannot use Majority Dynamics to\r\nprove that internal cuts exist in odd-regular graphs almost surely. However, we\r\nsuggest a modification of Majority Dynamics, that yields parts with desired\r\ncores with high probability.","lang":"eng"}],"oa":1,"year":"2024","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2406.07026"}],"author":[{"first_name":"Pavel","full_name":"Arkhipov, Pavel","last_name":"Arkhipov","id":"b25f2ab2-1fed-11ee-8599-fe02d211784f"}],"month":"06","date_updated":"2024-06-17T10:45:32Z","doi":"10.48550/arXiv.2406.07026","title":"Majority dynamics and internal partitions of random regular graphs: Experimental results","acknowledgement":"I am grateful to Matthew Kwan for setting the problem, providing useful literature,\r\nfruitful discussions, text review, mentorship, general encouragement and support.","_id":"17136","article_processing_charge":"No","external_id":{"arxiv":["2406.07026"]},"article_number":"2406.07026","date_published":"2024-06-11T00:00:00Z","date_created":"2024-06-12T07:01:52Z","day":"11","publication_status":"submitted","publication":"arXiv","citation":{"ista":"Arkhipov P. Majority dynamics and internal partitions of random regular graphs: Experimental results. arXiv, 2406.07026.","short":"P. Arkhipov, ArXiv (n.d.).","ama":"Arkhipov P. Majority dynamics and internal partitions of random regular graphs: Experimental results. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2406.07026\">10.48550/arXiv.2406.07026</a>","ieee":"P. Arkhipov, “Majority dynamics and internal partitions of random regular graphs: Experimental results,” <i>arXiv</i>. .","mla":"Arkhipov, Pavel. “Majority Dynamics and Internal Partitions of Random Regular Graphs: Experimental Results.” <i>ArXiv</i>, 2406.07026, doi:<a href=\"https://doi.org/10.48550/arXiv.2406.07026\">10.48550/arXiv.2406.07026</a>.","chicago":"Arkhipov, Pavel. “Majority Dynamics and Internal Partitions of Random Regular Graphs: Experimental Results.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2406.07026\">https://doi.org/10.48550/arXiv.2406.07026</a>.","apa":"Arkhipov, P. (n.d.). Majority dynamics and internal partitions of random regular graphs: Experimental results. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2406.07026\">https://doi.org/10.48550/arXiv.2406.07026</a>"},"type":"preprint"}