Null model: Difference between revisions

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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]], thea '''null model''' is a graphtype whichof random object that matches one specific graphobject in some of its structural features, or more generally satisfies a collection of constraints, but which is otherwise taken to be an instanceunbiasedly of a [[random graph]]structure. The null model is used as a term of comparison, to verify whether the graphobject in question displays some featurenon-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,]] orin graphs. An appropriate null model behaves in accordance with a reasonable [[null hypothesis]] for the behavior of the system under notinvestigation.

One null model of utility in the study of [[complex networks]] is that proposed by [[Mark Newman|Newman]] and Girvan[[Michelle andGirvan|Girvan]], consistsconsisting of a randomized version of thean original graph <math>G</math>, whereproduced through edges arebeing 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|coauthors 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, in the study of statistical properties of [[Graph (mathematics)|graphs]], the '''null model''' is a graph which matches one specific graph in some of its structural features, but which is otherwise taken to be an instance of a [[random graph]]. The null model is used as a term of comparison, to verify whether the graph in question displays some feature, such as community structure, or not.

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).

One null model is that proposed by Newman and Girvan and consists of a randomized version of the original graph, where edges are 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|coauthors=M.Girvan|title=Finding and evaluating community structure in networks|journal=Phys. Rev. E|year=2004|volume=69|issue=2}}</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.

==See also==

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==References==

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