Average nearest neighbor degrees in scale-free networks

Abstract

The average nearest neighbor degree (ANND) of a node of degree k is widely used to measure dependencies between degrees of neighbor nodes in a network. We formally analyze ANND in undirected random graphs when the graph size tends to infinity. The limiting behavior of ANND depends on the variance of the degree distribution. When the variance is finite, the ANND has a deterministic limit. When the variance is infinite, the ANND scales with the size of the graph, and we prove a corresponding central limit theorem in the configuration model (CM, a network with random connections). As ANND proved uninformative in the infinite variance scenario, we propose an alternative measure, the average nearest neighbor rank (ANNR). We prove that ANNR converges to a deterministic function whenever the degree distribution has finite mean. We then consider the erased configuration model (ECM), where self-loops and multiple edges are removed, and investigate the well-known 'structural negative correlations', or 'finite-size effects', that arise in simple graphs, such as ECM, because large nodes can only have a limited number of large neighbors. Interestingly, we prove that for any fixed k, ANNR in ECM converges to the same limit as in CM. However, numerical experiments show that finite-size effects occur when k scales with n. IntroductionThe goal of this paper is to analytically derive the limiting properties of the average nearest neighbor degree (ANND) in a general class of random graphs. The motivation for this analysis is that the ANND is one of the commonly accepted measures for dependencies between degrees of neighbor nodes. Such dependencies are called degree-degree correlations or network assortativity. A network is said to be assortative, if the correlation between degrees of neighbor nodes is positive and disassortative when it is negative. In assortative networks, nodes of high degree have a preference to connect to nodes of high degree. When the network is disassortative, nodes of high degree have a connection preference for nodes of low degree [23]. If there is no connection preference, the network is said to have neutral mixing.Currently, degree-degree correlations are part of the standard set of properties used to characterize the structure of networks. See [24] for a survey of the work on network assortativity. The effect of degree-degree correlations on disease spreading in networks has been extensively addressed in the literature, cf. [2,4,5,6]. For instance, it was shown that disassortative networks are easier to immunize and a disease takes longer to spread in assortative networks [11]. In the field of neuroscience, it was shown that assortative brain networks are better suited for signal processing [26], while assortative neural networks are more robust to random noise [12]. Under attacks, when edges or vertices are removed, assortative networks appear to be more resilient than disassortive networks [23,27]. On the other hand, when different networks interact, assortativity actually decreases the robust...

Degree correlations in scale-free random graph models

“…For k small, converges to a stable random variable, as was also shown in [45] for k fixed. Thus, for k small, different instances of the erased configuration model show wild fluctuations.…”

“…For large values of k , this is a rare event, by (3). Indeed, vertices of degree at most are present with high probability in the erased configuration model, whereas the probability that a vertex of degree is present tends to zero in the large network limit [45]. We avoid this problem by averaging over a small range of degrees.…”

“…There exists a vast array of papers, empirical, non-rigorous, and rigorous, on a ( k ) [2, 3, 9, 10, 15, 34, 37, 38, 42, 45]. The function describes the correlation between the degrees on the two sides of an edge, and classifies the network into one of the following three categories [35].…”

Generation of Scale-Free Assortative Networks via Newman Rewiring for Simulation of Diffusion Phenomena

“…Here, n is the maximum degree present in the network and P(h|k) is defined as the conditional probability, for a node of degree k, of being connected to a node of degree h. In general, networks with different correlations P(h|k) may have the same knn (k). Rigorous results about the convergence of the knn (k) function in random graphs with given joint degree distribution of neighbor nodes and in the configuration model have been given by [11].…”