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Encyclopedia > Degree distribution

In the mathematical field of graph theory the degree distribution of a graph is a function describing the total number of vertices in a graph with a given degree (number of connections to other vertices). Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... A graph diagram of a graph with 6 vertices and 7 edges. ... This article just presents the basic definitions. ... In the mathematical field of graph theory the degree or valency of a vertex v is the number of edges incident to v (with loops being counted twice). ...

Formally, the degree distribution is where v is a vertex in the set of the graph's vertices V, and deg(v) is the degree of vertex v.

This same information is often presented as the cumulative degree distribution, .

The degree distribution is a common way of classifying graphs into categories, such as Random graphs (Poisson distribution) and Scale-free networks (Power law distribution). In the science of mathematics, a random graph is a graph that is generated by some random process. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ... A scale-free network is a specific kind of complex network that has attracted attention since many real-world networks fall into this category. ... See Also: Watt In physics, a power law relationship between two scalar quantities x and y is any such that the relationship can be written as where a (the constant of proportionality) and k (the exponent of the power law) are constants. ...

## References

• Newman, Mark E.J. "The structure and function of complex networks".

Results from FactBites:

 Complex network - Wikipedia, the free encyclopedia (915 words) Most real world networks can be considered complex on account of their having several topological features that do not exist in simple networks, e.g., a heavy-tail in the degree distribution, a high clustering coefficient, assortativity or disassortativty among vertices, community structure at many scales and evidence of a hierarchical structure. A network is named scale-free if its degree distribution, i.e., the probability that a node selected uniformly at random has a certain number of links (degree), follows a particular mathematical function called a power law. Networks with a power-law degree distribution can be highly resistant to the random deletion of vertices, i.e., the vast majority of vertices remain connected together in a giant component.
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