page_title: Degree distribution
section_title: Degree distribution for directed networks
hierarchical_section_title: Degree distribution / Degree distribution for directed networks
caption_attribution_description: English: Wikipedia's hyperlink graph degree distribution; a figure from "Wikipedia as an ant-hill" http://no-gritzko-here.livejournal.com/22900.html
context_page_description: 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.
context_section_description: In a directed network, each node has some in-degree  and some out-degree  which are the number of links which have run into and out of that node respectfully. If  is the probability that a randomly chosen node has in-degree  and out-degree  then the generating function assigned to this joint probability distribution can be written with two valuables  and  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:
  where  is the mean degree (both in and out) of the nodes in the network; 
  Using the function , we can again find the generation function for the in/out-degree distribution and in/out-excess degree distribution, as before.  can be defined as generating functions for the number of arriving links at a randomly chosen node, and can be defined as the number of arriving links at a node reached by following a randomly chosen link. We can also define generating functions  and  for the number leaving such a node:
  Here, the average number of 1st neighbors, , or as previously introduced as , is  and the average number of 2nd neighbors reachable from a randomly chosen node is given by: . These are also the numbers of 1st and 2nd neighbors from which a random node can be reached, since these equations are manifestly symmetric in  and .
caption: In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales)
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