2019
DOI: 10.48550/arxiv.1911.04350
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Recurrence versus Transience for Weight-Dependent Random Connection Models

Abstract: We investigate a large class of random graphs on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random weight and given weight and position of the points we form an edge between two points independently with a probability depending on the two weights and the distance of the points. In dimensions d ∈ {1, 2} we completely characterise recurrence vs transience of random walks on the infinite c… Show more

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Cited by 5 publications
(15 citation statements)
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“…The weight dependent random connection model is a class of graphs introduced in [17,18] as a general framework, which incorporates many (but not all) of our examples of spatial random graphs. In that context our assumptions roughly mean that the random graphs are stochastically dominated by the random connection model with preferential attachment kernel (Assumption 1.1) and dominate the random connection model with min kernel (Assumption 1.2).…”
Section: Frameworkmentioning
confidence: 99%
See 2 more Smart Citations
“…The weight dependent random connection model is a class of graphs introduced in [17,18] as a general framework, which incorporates many (but not all) of our examples of spatial random graphs. In that context our assumptions roughly mean that the random graphs are stochastically dominated by the random connection model with preferential attachment kernel (Assumption 1.1) and dominate the random connection model with min kernel (Assumption 1.2).…”
Section: Frameworkmentioning
confidence: 99%
“…What remains to be seen is that when the right hand side in (17) is denoted e T K (x, y) and summed over all T ∈ T c k−1 we obtain e K (x, y, k). This is clearly true when k = 1 and k = 2.…”
Section: The Ultrasmall Regimementioning
confidence: 99%
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“…Among the other few spatial models for which these properties have been proved to various extents, we mention the ultra-small scale-free geometric network (Yukich, 2006), the hyperbolic random graph (Gugelmann et al, 2012), the spatial preferential attachment model (Jacob and Mörters, 2015), the age-dependent random connection model (Gracar et al, 2019a) and the geometric inhomogeneous random graph (Bringmann et al, 2019). As noted in Gracar et al (2019b), most of these models can be thought as particular cases of the more general weight-dependent random connection model. To further confirm our original motivation, we point out that some of these random graphs have been proposed to model realworld networks such as the Internet (Papadopoulos et al, 2010), banking systems (Deprez et al, 2015) and livestock trades (Dalmau and Salvi, 2021).…”
Section: Introductionmentioning
confidence: 99%
“…One of the most basic and studied processes on a graph is the simple random walk, where at each time-step a particle moves from its current location to any neighboring vertex with equal probability. As far as we could check, the only available results for the simple random walk on spatial inhomogeneous random graphs are the analysis of transience and recurrence for weight-dependent random connection models in Gracar et al (2019b) and for SFP in Heydenreich et al (2017). The mixing time of the simple random walk is, roughly put, the time needed for the distribution of the chain to approach its invariant measure.…”
Section: Introductionmentioning
confidence: 99%