{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11270","name":"Modularity of Erd\u00f6s-R\u00e9nyi Random Graphs","abstract":"For a given graph G, modularity gives a score to each vertex partition, with higher values taken to indicate that the partition better captures community structure in G. The modularity q^*(G) (where 0 <= q^*(G)<= 1) of the graph G is defined to be the maximum over all vertex partitions of the modularity value. Given the prominence of modularity in community detection, it is an important graph parameter to understand mathematically.\nFor the Erd\u00f6s-R\u00e9nyi random graph G_{n,p} with n vertices and edge-probability p, the likely modularity has three distinct phases. For np <= 1+o(1) the modularity is 1+o(1) with high probability (whp), and for np --> infty the modularity is o(1) whp. Between these regions the modularity is non-trivial: for constants 1 < c_0 <= c_1 there exists delta>0 such that when c_0 <= np <= c_1 we have delta3.3.co;2-t","http:\/\/dx.doi.org\/10.1016\/j.endm.2013.07.063","http:\/\/dx.doi.org\/10.1093\/comnet\/cnx046","http:\/\/dx.doi.org\/10.1093\/acprof:oso\/9780199206650.001.0001","http:\/\/dx.doi.org\/10.1103\/physreve.69.026113","http:\/\/dx.doi.org\/10.1016\/j.endm.2017.07.058","http:\/\/dx.doi.org\/10.1016\/j.physd.2006.09.009","http:\/\/dx.doi.org\/10.1140\/epjb\/e2012-20898-3"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6313","volumeNumber":110,"name":"29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)","dateCreated":"2018-06-18","datePublished":"2018-06-18","editor":[{"@type":"Person","name":"Fill, James Allen","givenName":"James Allen","familyName":"Fill"},{"@type":"Person","name":"Ward, Mark Daniel","givenName":"Mark Daniel","familyName":"Ward"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11270","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6313"}}}