Asymptotic analysis of network structures: degree-degree correlations and directed paths

Hoorn, Pim van der

doi:10.3990/1.9789036541794

“…In particular, Theorem 4.1 below states that Assumption 4.1 holds for D n = IID(D) whenever D has a finite (2 + η)-moment for some η > 0. The proof, which can be found in Section 9.2, is a straightforward extension of the proof of [15,Theorem 3.1].…”

Section: Sampled Degreesmentioning

confidence: 98%

“…Assumption 4.2 is satisfied for D n = IID(D) in several models, for example, the Configuration Model (see [15,Proposition 6.2]) and the Maximally Disassortative Graph Algorithm (see [19,Theorem 3.3]). …”

Section: Sampled Degreesmentioning

confidence: 99%

“…Since it generates graphs with neutral mixing (see e.g. [14,15]), it is also a crucial model for the analysis of degree-degree correlations. Given degree sequence D n , we assign D i stubs (half-edges) to each node i.…”

Section: Configuration Modelmentioning

confidence: 99%

“…To remedy this, new measures have been introduced [14,17] that are based on rank-correlations such as Spearman's rho and Kendall's tau. These measures are shown to converge to a proper limit when the size of the network tends to infinity [14,16], see also [15] for an overview.…”

Section: Introductionmentioning

confidence: 99%

“…To remedy this, new measures have been introduced [14,17] that are based on rank-correlations such as Spearman's rho and Kendall's tau. These measures are shown to converge to a proper limit when the size of the network tends to infinity [14,16], see also [15] for an overview.Both Pearson's correlation coefficient and rank-correlation measures represent the full degreedegree correlation structure of a network as one number and hence, are unable to capture local correlation structures. Instead, another common approach in the literature is to compute for a node of degree k, the average degree of it's neighbors and then take the average of these values over all nodes of degree k. This gives us a function of k called the Average Nearest Neighbor Degree (ANND) [8,25].…”

mentioning

confidence: 99%

See 3 more Smart Citations

Average nearest neighbor degrees in scale-free networks

Yao

¹

,

Hoorn²,

Litvak

³

2018

Internet Mathematics

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Limit theorems for assortativity and clustering in null models for scale-free networks

Hofstad

¹

,

Hoorn

²

,

Litvak

³

et al. 2020

Self Cite

View full text Add to dashboard Cite

An important problem in modeling networks is how to generate a randomly sampled graph with given degrees. A popular model is the configuration model, a network with assigned degrees and random connections. The erased configuration model is obtained when self-loops and multiple edges in the configuration model are removed. We prove an upper bound for the number of such erased edges for regularly-varying degree distributions with infinite variance, and use this result to prove central limit theorems for Pearson’s correlation coefficient and the clustering coefficient in the erased configuration model. Our results explain the structural correlations in the erased configuration model and show that removing edges leads to different scaling of the clustering coefficient. We prove that for the rank-1 inhomogeneous random graph, another null model that creates scale-free simple networks, the results for Pearson’s correlation coefficient as well as for the clustering coefficient are similar to the results for the erased configuration model.

show abstract

Average nearest neighbor degrees in scale-free networks

Yang¹,

Hoorn²,

Litvak³

2017

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Asymptotic analysis of network structures: degree-degree correlations and directed paths

Cited by 4 publications

References 74 publications

Average nearest neighbor degrees in scale-free networks

Average nearest neighbor degrees in scale-free networks

Limit theorems for assortativity and clustering in null models for scale-free networks

Average nearest neighbor degrees in scale-free networks

Contact Info

Product

Resources

About