Phase transitions for scaling of structural correlations in directed networks

Hoorn, Pim van der; Litvak, Nelly

doi:10.1103/physreve.92.022803

Cited by 8 publications

(11 citation statements)

References 37 publications

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“…Since the rank correlations are not affected by high dispersion in the values of the degrees, these finite-size effects can only be explained by simplicity of the graph. This is in agreement with previous work [18], where we observed structural correlations in another rank-based dependency measure -Spearman's rho. We see that for ANNR, these effects appear only when k is very large, say, greater than some k critical (n).…”

Section: Annd and Annrsupporting

confidence: 83%

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Average nearest neighbor degrees in scale-free networks

Yao

Hoorn²,

Litvak

2018

Internet Mathematics

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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...

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Section: Annd and Annrsupporting

confidence: 83%

“…It is explained by the fact that the graph is simple, therefore, nodes of large degrees are forced to connect to nodes of smaller degrees. Structural correlations for the rank-based correlation measure Spearman's rho have been observed before [18]. Here we see that this phenomenon holds for the ANNR as well.…”

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confidence: 62%

Average nearest neighbor degrees in scale-free networks

Yao

Hoorn²,

Litvak

2018

Internet Mathematics

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“…Based on the empirical results found by us in [16], we conjecture that the bounds we obtained are tight, up to some slowly varying functions. Therefore, as a next step one could try to prove Central Limit Theorems for the number of erased edges, using the bounds from Theorem 1 as the correct scaling factors.…”

Section: Discussionmentioning

confidence: 54%

“…Hence, random graph models that produce simple graphs are of primary interest from the application point of view. One well established model for generating a graph with given degree distribution is the configuration model [5,19,21], which has been studied extensively in the literature [6,12,15,16]. In this model, each node first receives a certain number of half-edges, or stubs, and then the stubs are connected to each other at random.…”

Section: Introductionmentioning

confidence: 99%

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Upper Bounds for Number of Removed Edges in the Erased Configuration Model

Hoorn

Litvak

2015

Lecture Notes in Computer Science

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Abstract. Models for generating simple graphs are important in the study of real-world complex networks. A well established example of such a model is the erased configuration model, where each node receives a number of half-edges that are connected to half-edges of other nodes at random, and then self-loops are removed and multiple edges are concatenated to make the graph simple. Although asymptotic results for many properties of this model, such as the limiting degree distribution, are known, the exact speed of convergence in terms of the graph sizes remains an open question. We provide a first answer by analyzing the size dependence of the average number of removed edges in the erased configuration model. By combining known upper bounds with a Tauberian Theorem we obtain upper bounds for the number of removed edges, in terms of the size of the graph. Remarkably, when the degree distribution follows a power-law, we observe three scaling regimes, depending on the power law exponent. Our results provide a strong theoretical basis for evaluating finite-size effects in networks.

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

et al. 2020

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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.

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Phase transitions for scaling of structural correlations in directed networks

Cited by 8 publications

References 37 publications

Average nearest neighbor degrees in scale-free networks

Average nearest neighbor degrees in scale-free networks

Upper Bounds for Number of Removed Edges in the Erased Configuration Model

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

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