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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.

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 degree distribution P(k) of a network is then defined to be the fraction of nodes in the network with degree k. Thus if there are n nodes in total in a network and nk of them have degree k, we have
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.

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:

(or Poisson in the limit of large n, if the average degree is held fixed). Most networks in the real world, however, have degree distributions very different from this. Most are highly right skewed, meaning that a large majority of nodes have low degree but a small number, known as "hubs", have high degree. Some networks, notably the Internet, the World Wide Web, and some social networks were argued to have degree distributions that approximately follow a power law: , where γ is a constant. Such networks are called scale free networks and have attracted particular attention for their structural and dynamical properties. However, a survey of a wide range of real world networks suggests that scale free networks are rare when assessed using statistically rigorous measures. Some researchers have disputed these findings arguing that the definitions used in the study are inappropriately strict, while others have argued that the precise functional form of the degree distribution is less important than knowing whether the degree distribution is fat tailed or not. The over interpretation of specific forms of the degree distribution has also been criticised for failing to consider how networks may evolve over time.