# Null model

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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 [graphs](/source/Graph_(discrete_mathematics)), 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](/source/community_structure) in graphs. An appropriate null model behaves in accordance with a reasonable [null hypothesis](/source/null_hypothesis) for the behavior of the system under investigation.

One null model of utility in the study of [complex networks](/source/complex_networks) is that proposed by [Newman](/source/Mark_Newman) and [Girvan](/source/Michelle_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|article-number=026113}}</ref>

The null model is the basic concept behind the definition of [modularity](/source/Modularity_(networks)), 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](/source/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](/source/configuration_model) (the maximally random graph subject to a constraint on the degree of each vertex).

==See also==
* [Null hypothesis](/source/Null_hypothesis)

==References==
<references />

Category:Graph theory
Category:Statistical methods

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