Clustering in a hyperbolic model of complex networks
| Autor: | Tobias Müller, Nikolaos Fountoulakis, Pim van der Hoorn, Markus Schepers |
|---|---|
| Přispěvatelé: | Probability, ICMS Core, Stochastic Studies and Statistics |
| Rok vydání: | 2021 |
| Předmět: |
Statistics and Probability
Random graph Degree (graph theory) Hyperbolic geometry 05C80 Probability (math.PR) Zero (complex analysis) Hyperbolic random graph Function (mathematics) Clustering Combinatorics Bounded function FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) Statistics Probability and Uncertainty Constant (mathematics) Mathematics - Probability Random graphs Clustering coefficient Mathematics |
| Zdroj: | Electron. J. Probab. Electronic Journal of Probability, 26:13, 1-132. Institute of Mathematical Statistics Electronic journal of probability, 26:13, 1-132. UNIV WASHINGTON, DEPT MATHEMATICS |
| ISSN: | 1083-6489 |
| DOI: | 10.1214/21-ejp583 |
| Popis: | In this paper we consider the clustering coefficient and clustering function in a random graph model proposed by Krioukov et al.~in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been shown that this model has various properties associated with complex networks, e.g. power-law degree distribution, short distances and non-vanishing clustering coefficient. Here we show that the clustering coefficient tends in probability to a constant $\gamma$ that we give explicitly as a closed form expression in terms of $\alpha, \nu$ and certain special functions. This improves earlier work by Gugelmann et al., who proved that the clustering coefficient remains bounded away from zero with high probability, but left open the issue of convergence to a limiting constant. Similarly, we are able to show that $c(k)$, the average clustering coefficient over all vertices of degree exactly $k$, tends in probability to a limit $\gamma(k)$ which we give explicitly as a closed form expression in terms of $\alpha, \nu$ and certain special functions. We are able to extend this last result also to sequences $(k_n)_n$ where $k_n$ grows as a function of $n$. Our results show that $\gamma(k)$ scales differently, as $k$ grows, for different ranges of $\alpha$. More precisely, there exists constants $c_{\alpha,\nu}$ depending on $\alpha$ and $\nu$, such that as $k \to \infty$, $\gamma(k) \sim c_{\alpha,\nu} \cdot k^{2 - 4\alpha}$ if $\frac{1}{2} < \alpha < \frac{3}{4}$, $\gamma(k) \sim c_{\alpha,\nu} \cdot \log(k) \cdot k^{-1} $ if $\alpha=\frac{3}{4}$ and $\gamma(k) \sim c_{\alpha,\nu} \cdot k^{-1}$ when $\alpha > \frac{3}{4}$. These results contradict a claim of Krioukov et al., which stated that the limiting values $\gamma(k)$ should always scale with $k^{-1}$ as we let $k$ grow. Comment: 127 pages |
| Databáze: | OpenAIRE |
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