  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 degree-degree 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.
