Assortativity in geometric and scale-free networks

The assortative behavior of a network is the tendency of similar (or dissimilar) nodes to connect to each other. This tendency can have an influence on various properties of the network, such as its robustness or the dynamics of spreading processes. In this paper, we study degree assortativity both in real-world networks and in several generative models for networks with heavy-tailed degree distribution based on latent spaces. In particular, we study Chung-Lu Graphs and Geometric Inhomogeneous Random Graphs (GIRGs). Previous research on assortativity has primarily focused on measuring the degree assortativity in real-world networks using the Pearson assortativity coefficient, despite reservations against this coefficient. We rigorously confirm these reservations by mathematically proving that the Pearson assortativity coefficient does not measure assortativity in any network with sufficiently heavy-tailed degree distributions, which is typical for real-world networks. Moreover, we find that other single-valued assortativity coefficients also do not sufficiently capture the wiring preferences of nodes, which often vary greatly by node degree. We therefore take a more fine-grained approach, analyzing a wide range of conditional and joint weight and degree distributions of connected nodes, both numerically in real-world networks and mathematically in the generative graph models. We provide several methods of visualizing the results. We show that the generative models are assortativity-neutral, while many real-world networks are not. Therefore, we also propose an extension of the GIRG model which retains the manifold desirable properties induced by the degree distribution and the latent space, but also exhibits tunable assortativity. We analyze the resulting model mathematically, and give a fine-grained quantification of its assortativity.Author summary: The degree of a node in a network, that is, the number of other nodes it is connected to, is a simple measure of “importance” within the network. Whether nodes connect predominantly to similarly important “peers”, or whether there exist hierarchies where less important nodes connect to much more important nodes, has far-reaching consequences for processes that occur within networks as well as for the underlying network structure. This property is often called “assortativity” and typically measured by computing a single numerical value for a network. A lot of information is lost in this process and, to make matters worse, the most widespread way of computing this value has severe statistical flaws. We provide new evidence of these flaws and instead propose a local approach to measuring assortativity, which studies how the degree distribution of a node’s neighbors changes with this node’s degree. We further propose a “tunable” model, which allows to adjust the wiring preferences of nodes based on the degrees of potential neighbors, while at the same time capturing many established structural properties of real networks. We evaluate our new assortativity measure both on real-world networks and theoretical network models including our new tunable models.