Assortativity in networks

TL;DR

This paper thoroughly analyzes the assortativity coefficient in networks, revealing its asymmetric range, characterizing networks with extreme values, and proposing methods to construct and identify neutral and assortative networks.

Contribution

It clarifies the range of the assortativity coefficient, characterizes extremal networks, and introduces new iterative methods for constructing neutral and assortative networks.

Findings

Assortativity coefficient ranges in [-1,1) rather than [-1,1].

Star networks uniquely achieve the lower bound of assortativity.

Constructed families of neutral and assortative networks using iterative operations.

Abstract

The degree-degree correlation is crucial in understanding the structural properties of and dynamics occurring upon network, and is often measured by the assortativity coefficient $r$. In this paper, we first study this measure in detail and conclude that $r$ belongs to an asymmetric range $[-1,1)$ rather than the widely-cited $[-1,1]$. Among which, we verify that star is the unique tree network that achieves the lower bound of index $r$. Next, we obtain that all the resultant networks based on several widely-used kinds of edge-based iterative operations are disassortative if seed model has negative $r$, and also generate a family of growing neutral networks. Then, we propose an edge-based iterative operation to construct growing assortative network when seed is assortative, and further extend it to work well in general setting. Lastly, we establish a sufficient condition for existence of neutral tree network, accordingly, not only find out a representative of any order neutral tree network for the first time, but also are the first to create growing neutral tree networks as well. Also, we obtain $8n/9$ neutral non-tree graphs of distinct order as $n\rightarrow\infty$.