Mixing patterns in networks

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.

Robustness, Evolvability, and Accessibility in Linear Genetic Programming

“…As these correlations (a.k.a. mixing patterns [21]) are not taken into account, the applicability of current phenotypic evolvability measures are left severely constrained, at least for this system.…”

Deterministic weighted scale-free small-world networks

“…Consequently, their recipes were adopted extensively to the weighted networks by the authors of Ref [38]. As shown in Ref [38], for correlated networks (assortative or disassortative mixing [39,40]), the average traffic through a link L ij during a time window can be represented as…”

“…In complex network, degree correlations [39,40,45,46,47], has attracted much attention, because it can give out a unique description of network structures, which could help researchers understand the characteristics of net-…”

“…When k nn (k) increases with k, it means that nodes have a tendency to connect to nodes with a similar or larger degree. In this case the network is defined as assortative [39,40]. In contrast, if k nn (k) is decreasing with k, which implies that nodes of large degree are likely to have near neighbors with small degree, then the network is said to be disassortative.…”

How people make friends in social networking sites—A microscopic perspective

“…Holme et al studied the structural evolution of Pussokram and found that its degree correlation coefficient is always negative over time, i.e. disassortative mixing [15], which is in stark contrast to the significant assortative mixing for real-world social networks [16]. Viswanath et al studied the structural evolution of the activity network of Facebook and found that the average degree, clustering coefficient, and average path length are all relatively stable over time [6].…”