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".
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 degreedistribution, 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 degreedistribution, 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-lawdegreedistribution can be highly resistant to the random deletion of vertices, i.e., the vast majority of vertices remain connected together in a giant component.
where v is a vertex in the set of the graph's vertices V, and 
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