{"title":"Modularity with a more accurate baseline model.","authors":"Brian L Chang, Piet Van Mieghem","doi":"10.1103/PhysRevE.111.044317","DOIUrl":null,"url":null,"abstract":"<p><p>We derive an expression for the exact probability Pr[i∼j] of a link between a node i with degree d_{i} and a node j with degree d_{j} in a graph belonging to the class of Erdős-Rényi G(N,L) random graphs with N nodes and L links. The probability Pr[i∼j] is commonly approximated as d_{i}d_{j}/2L and appears in the formula of Newman's modularity, which plays a crucial rule in community detection in networks. We show that, when applied to graphs not belonging to the class of Erdős-Rényi random graphs, our formula for Pr[i∼j] is considerably more accurate than d_{i}d_{j}/2L and leads to the detection of different clusters or partitions than the original modularity formula.</p>","PeriodicalId":20085,"journal":{"name":"Physical review. E","volume":"111 4-1","pages":"044317"},"PeriodicalIF":2.4000,"publicationDate":"2025-04-01","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/PhysRevE.111.044317","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
We derive an expression for the exact probability Pr[i∼j] of a link between a node i with degree d_{i} and a node j with degree d_{j} in a graph belonging to the class of Erdős-Rényi G(N,L) random graphs with N nodes and L links. The probability Pr[i∼j] is commonly approximated as d_{i}d_{j}/2L and appears in the formula of Newman's modularity, which plays a crucial rule in community detection in networks. We show that, when applied to graphs not belonging to the class of Erdős-Rényi random graphs, our formula for Pr[i∼j] is considerably more accurate than d_{i}d_{j}/2L and leads to the detection of different clusters or partitions than the original modularity formula.
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.