Null model: Difference between revisions

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{{for|use in statistical testing|Statistical model}}

The '''null model''' can be a graph which matches the original in some of its structural features, but which is otherwise a random graph. The null model is used as a term of comparison, to verify whether the graph at study displays community structure or not. The most popular null model is that proposed by Newman and Girvan and consisits 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|doi=026113}}</ref>

{{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 behaves in accordance with a reasonable [[null hypothesis]] for the behavior of the system under investigation.

TheOne '''null model''' canof be a graph which matches the originalutility in some of its structural features, but which is otherwise a random graph. The null model is used as a term of comparison, to verify whether the graph at study displaysof community[[complex structure or not. The most popular null modelnetworks]] is that proposed by [[Mark Newman|Newman]] and [[Michelle Girvan|Girvan]], and consisitsconsisting 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>

The null model is the basic concept behind the definition of modularity, a function which evaluates the goodness of partitions of a graph into clusters.

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).

==References==

==See also==

* [[Null hypothesis]]

==References==

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[[Category:Graph theory]]

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[[Category:Statistical methods]]

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