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{{for|use in statistical testing|Statistical model}}
{{for|use in ecology|Theoretical ecology}}
{{one source |date=April 2024}}
In mathematics, for example in the study of statistical properties of [[Graph (discrete mathematics)|graphs]], a '''null model''' is a type of random object that matches one specific object in some of its features, or more generally satisfies a collection of constraints, but which is otherwise taken to be an unbiasedly random structure. The null model is used as a term of comparison, to verify whether the object in question displays some non-trivial features (properties that wouldn't be expected on the basis of chance alone or as a consequence of the constraints), such as [[community structure]] in graphs. An appropriate null model
One null model of utility in the study of [[complex networks]] is that proposed by [[Mark Newman|Newman]] and [[Michelle Girvan|Girvan]], consisting of a randomized version of an original graph <math>G</math>, produced through edges being rewired at random, under the constraint that the expected degree of each vertex matches the degree of the vertex in the original graph.<ref>{{cite journal|last=M.E.J|first=Newman| author-link=Mark Newman |author2=M.Girvan |author2-link= Michelle Girvan |title=Finding and evaluating community structure in networks|journal=Phys. Rev. E|year=2004|volume=69|issue=2|doi=10.1103/physreve.69.026113 |arxiv=cond-mat/0308217|bibcode=2004PhRvE..69b6113N|pmid=14995526|page=026113}}</ref>▼
▲In mathematics, for example in the study of statistical properties of [[Graph (discrete mathematics)|graphs]], a '''null model''' is type of random object that matches one specific object in some of its features, or more generally satisfies a collection of constraints, but which is otherwise taken to be an unbiasedly random structure. The null model is used as a term of comparison, to verify whether the object in question displays some non-trivial features (properties that wouldn't be expected on the basis of chance alone or as a consequence of the constraints), such as community structure in graphs. An appropriate null model exhibits behavior in accordance with a reasonable [[null hypothesis]] for the behavior of the system under investigation.
▲One null model of utility in the study of [[complex networks]] is that proposed by Newman and Girvan, consisting of a randomized version of an original graph <math>G</math>, produced through edges being rewired at random, under the constraint that the expected degree of each vertex matches the degree of the vertex in the original graph.<ref>{{cite journal|last=M.E.J|first=Newman|author2=M.Girvan |title=Finding and evaluating community structure in networks|journal=Phys. Rev. E|year=2004|volume=69|issue=2|doi=10.1103/physreve.69.026113 |arxiv=cond-mat/0308217|bibcode=2004PhRvE..69b6113N}}</ref>
The null model is the basic concept behind the definition of [[Modularity (networks)|modularity]], a function which evaluates the goodness of partitions of a graph into clusters. In particular, given a graph <math>G</math> and a specific community partition <math>\sigma:V(G)\rightarrow \{1,...,b\}</math> (an assignment of a community-index <math>\sigma(v)</math> (here taken as an integer from <math>1</math> to <math>b</math>) to each vertex <math>v\in V(G)</math> in the graph), the modularity measures the difference between the number of links from/to each pair of communities, from that expected in a graph that is completely random in all respects other than the set of degrees of each of the vertices (the [[degree sequence]]). In other words, the modularity contrasts the exhibited community structure in <math>G</math> with that of a null model, which in this case is the [[configuration model]] (the maximally random graph subject to a constraint on the degree of each vertex).
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==References==
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[[Category:Graph theory]]
[[Category:Statistical methods]]
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