{"title":"Large deviation properties of minimum spanning trees for random graphs.","authors":"Mahdi Sarikhani, Alexander K Hartmann","doi":"10.1103/xc5g-t9dj","DOIUrl":null,"url":null,"abstract":"<p><p>We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with N nodes. First, we consider complete graphs. Second, we study Erdős-Rényi (ER) random graphs with edge probability p=c/N conditioned on being connected. By using large-deviation Markov-chain sampling, we can obtain the distribution P(W) of the spanning-tree weight W down to probability densities as small as 10^{-300}. For the complete graph, we confirm analytical predictions with respect to the expectation value. For both ensembles, the large-deviation principle is fulfilled. For the connected ER graphs, we observe a remarkable change of the distributions at the value of c=1, which is the percolation threshold for the original ER ensemble.</p>","PeriodicalId":48698,"journal":{"name":"Physical Review E","volume":"114 2-1","pages":"024314"},"PeriodicalIF":2.5000,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review E","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/xc5g-t9dj","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, FLUIDS & PLASMAS","Score":null,"Total":0}
引用次数: 0
Abstract
We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with N nodes. First, we consider complete graphs. Second, we study Erdős-Rényi (ER) random graphs with edge probability p=c/N conditioned on being connected. By using large-deviation Markov-chain sampling, we can obtain the distribution P(W) of the spanning-tree weight W down to probability densities as small as 10^{-300}. For the complete graph, we confirm analytical predictions with respect to the expectation value. For both ensembles, the large-deviation principle is fulfilled. For the connected ER graphs, we observe a remarkable change of the distributions at the value of c=1, which is the percolation threshold for the original ER ensemble.
期刊介绍：
Physical Review E (PRE), broad and interdisciplinary in scope, focuses on collective phenomena of many-body systems, with statistical physics and nonlinear dynamics as the central themes of the journal. Physical Review E publishes recent developments in biological and soft matter physics including granular materials, colloids, complex fluids, liquid crystals, and polymers. The journal covers fluid dynamics and plasma physics and includes sections on computational and interdisciplinary physics, for example, complex networks.
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