Title: Two-walks degree assortativity in graphs and networks,
Abstract: Degree ssortativity is the tendency for nodes of high degree (resp.low
degree) in a graph to be connected to high degree nodes (resp. to low degree
ones). It is sually quantified by the Pearson correlation coefficient of the
degree-degree correlation. Here we extend this concept to account for the
effect of second neighbours to a given node in a graph. That is, we consider
the two-walks degree of a node as the sum of all the degrees of its adjacent
nodes. The two-walks degree assortativity of a graph is then the Pearson
correlation coefficient of the two-walks degree-degree correlation. We found
here analytical expression for this two-walks degree assortativity index as a
function of contributing subgraphs. We then study all the 261,000 connected
graphs with 9 nodes and observe the existence of assortative-assortative and
disassortative-disassortative graphs according to degree and two-walks degree,
respectively. More surprinsingly, we observe a class of graphs which are degree
disassortative and two-walks degree assortative. We explain the existence of
some of these graphs due to the presence of certain topological features, such
as a node of low-degree connected to high-degree ones. More importantly, we
study a series of 49 real-world networks, where we observe the existence of the
disassortative-assortative class in several of them. In particular, all
biological networks studied here were in this class. We also conclude that no
graphs/networks are possible with assortative-disassortative structure.