Higher order assortativity for directed weighted networks and Markov chains

Assortativity is a global indicator that provides meaningful insights about the network structure. In the classical definition, the assortativity is a global measure based on the Pearson correlation between the degrees of nodes. This definition can be extended into two different directions. On the one side, one can consider other quantitative attributes of the nodes different from the degree; on the other side, one can move from the adjacency of the nodes – which is the basis of Newman’s degree-degree assortativity – and propose more general ways to connect them. We provide a generalized concept of the assortativity measure for directed and weighted networks, moving beyond the adjacency relations in both directions. The proposed concept is formulated on a node attribute that is not necessarily the degree or strength, and nodes are connected through walks or paths. In this way, we totally extend the assortativity definitions provided in the literature until now. We provide an empirical application of these measures for the paradigmatic case of the trade network. Interestingly, this interpretation of the higher-order assortativity measure allows stating a natural bridge between complex networks and stochastic processes. In so doing, we are able to move from the information content of the higher order assortativity of a network to the dynamical properties of the underlying Markov chain. Specifically, the temporal dimension of the network and the regularities captured by the autocorrelation – which are hidden in the network structure – become clear in moving to the Markov chain theory.