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Assortativity was first introduced by Newman and has been extensively
studied and applied to many real world networked systems since
then. Assortativity is a graph metrics and describes the tendency of high
degree nodes to be directly connected to high degree nodes and low degree
nodes to low degree nodes. It can be interpreted as a first order measure of
the connection between nodes, i.e. the first autocorrelation of the degreedegree
vector. Even though assortativity has been used so extensively,
to the author’s knowledge, no attempt has been made to extend it theoretically.
This is the scope of our paper. We will introduce higher order
assortativity by extending the Newman index based on a suitable choice
of the matrix driving the connections. Higher order assortativity will be
defined for paths, shortest paths, random walks of a given time length,
connecting any couple of nodes. The Newman assortativity is achieved
for each of these measures when the matrix is the adjacency matrix, or,
in other words, the correlation is of order 1. Our higher order assortativity
indexes can be used for describing a variety of real networks, help
discriminating networks having the same Newman index and may reveal
new topological network features
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