Limiting Behavior of Degree-Degree Metrics under Local Convergence in Probability

An important property that involves all networks is the degree-degree correlation, sometimes called network assortativity, which refers to the statistical relationship between the degrees of neighbouring nodes in a network. It quantifies how the degree of one node is related to the degrees of its adjacent nodes. For instance, if a network has a positive degree-degree correlation, it indicates that nodes of a high degree have a preference to connect to other high-degree nodes. In this case, the network is said to exhibit assortative mixing. Similarly, there exist disassortative networks, with negative degree-degree correlation, where nodes of high degree are mostly neighbours of nodes with small degrees. If the network is neither assortative nor disassortative, it is said to have neutral mixing.

Degree-degree correlations are an important property of networks. For example, network with assortative mixing might be more vulnerable to targeted attacks where the high-degree nodes are specifically removed [21]. In neuroscience assortative networks brain networks are shown to perform better in terms of signal processing [23]. Assortative networks are also more robust under edge or vertex removal. In financial networks, for example, assortativity may influence systemic risk, since highly interconnected banks are also highly connected to other similar entities [19]. Conversely, networks with disassortative mixing, where high-degree nodes are preferentially connected to low-degree nodes, can behave differently under stress or failure scenarios compared to assortative networks [20]. Disassortative networks do allow for easier immunization when considering epidemic spreading [7].

Given the impact of degree-degree correlations on the function of networks, it is important to properly measure and analyze these correlations. One natural way to do this is to study the asymptotic behaviour of degree-degree metrics in random graph models, as the number of nodes in the network grows large. There are many results available. Some concern specific models [22,2,25,18,16], while others prove limits for a given metric under certain assumptions on the random graph model [12,14,26]. While the latter have the potential to enable analysis of degree-degree correlation in general random graphs, the results are not always easy to directly apply, often resulting in a separate, and sometimes involved, analysis.

The recent development of the local convergence has opened up a powerful and general framework to study sparse random graphs, enabling a uniform way to analyze topologies of a wide variety of random graph models. Specifically, if (G n ) n≥1 is a sequence of finite graphs, local convergence means that the distribution of neighbourhoods around a uniformly sampled node converges to the distribution of neighbourhoods in an infinitely rooted random graph. This notion is particularly useful since it implies convergence of local properties of random graphs, while a wide range of random graph models have been shown to have a local limit [13]. The notion of local convergence is by now the setting to analyze random graph models and has lead to a wide variety of properties being studied. However, apart from some initial results, degree-degree metrics have not been extensively studied, even though most of them are local properties.

In this paper we address this gap by providing general limit results for a wide range of degreedegree metrics for random graphs with a local limit. This makes our results widely applicable to most of the current random graph models that are studied, including the popular Geometric Inhomogeneous Random Graphs [5], Weighted Random Connection Model [9] and the more general Spatial Inhomogeneous Random Graphs [13]. To showcase the usage of our results we analyze degree-degree correlations in rank-1 inhomogeneous random graphs and random geometric graphs. We provide a explicit expression for the Average Nearest Neighbor Degree in both models and for Pearson’s correlation coefficient in the latter one.

We organize the paper as follows. In Section 2 we provide basic preliminaries for local convergence and state our main convergence results for the different degree-degree metrics. The applications of our general results to rank-1 inhomogeneous random graphs and random geometric graphs is covered in Section 3. Sections 4 and 5 contain the proofs of the general results and the applications, under the assumption of a few technical lemmas. The proofs for these lemmas are included in the Appendix.

We start by briefly int