# Degree distribution

In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.

## Definition

The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes.

If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.

The same information is also sometimes presented in the form of a cumulative degree distribution, the fraction of nodes with degree smaller than k, or even the complementary cumulative degree distribution, the fraction of nodes with degree greater than or equal to k (1 - C) if one considers C as the cumulative degree distribution; i.e. the complement of C.

## Observed degree distributions

The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.

The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:

## Excess degree distribution

Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node.

In other words, it is the distribution of outgoing links from a node reached by following a link.

Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.

## The Generating Functions Method

And in general:

## Degree distribution for directed networks

Since every link in a directed network must leave some node and enter another, the net average number of links entering

a node is zero.

Therefore,

which implies that, the generation function must satisfy:

## See also

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Degree distribution.