In mathematics, for example in the study of statistical properties of
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 the study of complex networks, a network is said to have community structure if the nodes of the network can be easily grouped into (potentially overlapping) sets of nodes such that each set of nodes is densely connected internally. In the par ...
in graphs. An appropriate null model behaves in accordance with a reasonable
null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
for the behavior of the system under investigation.
One null model of utility in the study of
complex networks
Complex Networks is an American media and entertainment company for youth culture, based in New York City. It was founded as a bi-monthly magazine, ''Complex'', by fashion designer Marc (Ecko) Milecofsky. Complex Networks reports on popular ...
is that proposed by Newman and
Girvan
Girvan ( gd, Inbhir Gharbhain, "mouth of the River Girvan") is a burgh and harbour town in Carrick, South Ayrshire, Scotland. Girvan is situated on the east coast of the Firth of Clyde, with a population of about 6,450. It lies south of Ayr, ...
, consisting of a randomized version of an original graph
, 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.
The null model is the basic concept behind the definition of
modularity
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
, a function which evaluates the goodness of partitions of a graph into clusters. In particular, given a graph
and a specific community partition
(an assignment of a community-index
(here taken as an integer from
to
) to each vertex
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
with that of a null model, which in this case is the
configuration model
In network science, the configuration model is a method for generating random networks from a given degree sequence. It is widely used as a reference model for real-life social networks, because it allows the modeler to incorporate arbitrary de ...
(the maximally random graph subject to a constraint on the degree of each vertex).
See also
*
Null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
References
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Graph theory