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 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

1. .

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:

(or Poisson in the limit of large n 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.

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.

which is called the excess degree of that node. In the configuration model, which correlations between the nodes have been ignored and every node is assumed to be connected to any other nodes in the network with the same probability, the excess degree distribution can be found as:

is the mean-degree (average degree) of the model. It follows from that, that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:

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.

Generating functions method

Generating functions respectively, it is possible to write two power series in the following forms:

:

by differentiating:

and its derivatives:

And in general:

For Poisson-distributed random networks, such as the ER graph-th neighbors is generated by:

acting on itself.

Degree distribution for directed networks

then the generating function assigned to this joint probability distribution as:

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:

for the number leaving such a node:

.

Degree distribution for signed networks

denote negative degree distribution and positive degree distribution of the signed network.

See also

• Graph theory

• Complex network

• Scale-free network

• Random graph

• Structural cut-off

References