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LA  Local AssortativityDefinition
Assortativity is the tendency in networks where nodes mostly connected with similar nodes based on degree of
nodes. In other words, highlyconnected nodes have connection with other highlyconnected nodes in the network,
and vice versa.
Local assortativity value for every node in the network can be defined as: where j, M and k are degree of v, number of links in the network and the average degree of node v’s neighbors, respectively; μ_{q} and σ_{q} are mean and standard deviation of the remaining degree distribution, q(k), respectively. q(k),is defined as: where p(k) is network degree distribution. Respectively, local assortativity for directed networks is a node's contribution to the directed assortativity of a network. A node's contribution to the assortativity of a directed network r_{d} is defined as, Where j_{out} is the outdegree of the node under consideration and j_{in} is the indegree, is the average indegree of its neighbors (to which node v} has an edge) and is the average outdegree of its neighbors (from which node v has an edge). , . By including the scaling terms σ^{in}_{q} and σ^{out}_{q} , we ensure that the equation for local assortativity for a directed network satisfies the condition . Further, based on whether the indegree or outdegree distribution is considered, it is possible to define local inassortativity and local outassortativity as the respective local assortativity measures in a directed network. Assortativity is high when highdegree nodes tend to connect to other highdegree nodes; it is low (i.e., negative) when highdegree nodes are linked to lowdegree nodes. Correlations between nodes of similar degree are often found in the mixing patterns of many observable networks. For instance, in social networks, highly connected nodes tend to be connected with other high degree nodes. This tendency is referred to as assortative mixing, or assortativity. On the other hand, technological and biological networks typically show disassortative mixing, or dissortativity, as high degree nodes tend to attach to low degree nodes. SoftwareReferences
