# Two-walks degree assortativity in graphs and networks

## TL;DR

This paper introduces the two-walks degree assortativity measure, extending degree correlation analysis to second neighbors, and explores its properties and occurrences in all small graphs and real-world networks.

## Contribution

It provides an analytical expression for two-walks degree assortativity and reveals its structural implications and prevalence in various networks.

## Key findings

- Existence of graphs with degree disassortative and two-walks degree assortative properties.
- All biological networks studied are in the disassortative-assortative class.
- No networks exhibit assortative-disassortative structure.

## 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.