Assortativity in networks

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 $n\rightarrow\infty$.