Mixing patterns in networks

TLDR

Assortative mixing in networks is examined, focusing on discrete traits such as language or race and scalar traits like age, with particular attention to degree‑based mixing where highly connected nodes tend to link with similarly connected nodes. The paper aims to develop and apply quantitative measures of assortative mixing across different mixing types, and to construct analytic and numerical models of assortatively mixed networks. The authors introduce measures of assortativity, construct analytic generating‑function and Monte‑Carlo models, and use these to explore how varying assortativity affects network properties. They find that assortative mixing is widespread across real networks, and that degree‑based assortativity strongly influences network connectivity and resilience to node removal.

Abstract

We study assortative mixing in networks, the tendency for vertices in networks to be connected to other vertices that are like (or unlike) them in some way. We consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age. As a special example of the latter we consider mixing according to vertex degree, i.e., according to the number of connections vertices have to other vertices: do gregarious people tend to associate with other gregarious people? We propose a number of measures of assortative mixing appropriate to the various mixing types, and apply them to a variety of real-world networks, showing that assortative mixing is a pervasive phenomenon found in many networks. We also propose several models of assortatively mixed networks, both analytic ones based on generating function methods, and numerical ones based on Monte Carlo graph generation techniques. We use these models to probe the properties of networks as their level of assortativity is varied. In the particular case of mixing by degree, we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices.