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2.3.4 Joint degree distribution In addition to the degree distribution P(k), we also focus on the joint degree distri- bution (JDD) represented as J(k,m). The function J(k,m) represents what fraction of the links in the graph are between nodes of degree k and degree m. In the case of a directed network, J(k,m) represents the fraction of links that are from a node with outdegree k and to a node with indegree m. Thus, the JDD represents how often nodes of different degrees connect to each other. This property is also referred to as the 2K-distribution [101] or the mixing patterns [116]. The JDD provides many insights into the structural properties of networks. For example, networks where high-degree nodes tend to connect to other high-degree nodes are more likely to be subject to epidemics, as a single infected high-degree node will quickly infect other high-degree nodes. On the other hand, networks where high-degree nodes tend to connect to low-degree nodes show the opposite behavior; a single infected high-degree node will not spread an epidemic very fast. The JDD can be approximated by the degree correlation function knn, which is a mapping between outdegree and the average indegree of all nodes connected to nodes of that outdegree. Clearly, an increasing knn indicates a tendency of higher-degree nodes to connect to other high-degree nodes; a decreasing knn represents the opposite trend. 25PDF Image | Online Social Networks: Measurement, Analysis, and Applications to Distributed Information Systems
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