Scale-free percolation in continuous space: quenched degree and clustering coefficient

Dalmau, Joseba; Salvi, Michele

doi:10.1017/jpr.2020.76

Cited by 6 publications

(9 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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: 71%

“…A good deal of effort is being put in finding the precise order of these distances, see Deprez et al (2015), Hao and Heydenreich (2021), van der Hofstad and Komjáthy (2017). Finally, in Dalmau and Salvi (2021) the authors show the positivity of the clustering coefficient for continuous SFP.…”

Section: Introductionmentioning

confidence: 92%

“…In particular, the scale-free property has been proved for the discrete-space model, where the nodes lay on Z d , in Deijfen et al (2013a). In its continuum counterpart, where the position of the nodes is given by a Poisson point process, the graph has been shown to be scale-free in an annealed sense in Deprez and Wüthrich (2019) and in a quenched sense in Dalmau and Salvi (2021). The convergence in distribution for the maximum of the degrees on a growing observation window has been studied in Bhattacharjee and Schulte (2019).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Scale-free percolation mixing time

Cipriani¹,

Salvi²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Assign to each vertex of the one-dimensional torus i.i.d. weights with a heavy-tail of index τ − 1 > 0. Connect then each couple of vertices with probability roughly proportional to the product of their weights and that decays polynomially with exponent α > 0 in their distance. The resulting graph is called scale-free percolation. The goal of this work is to study the mixing time of the simple random walk on this structure. We depict a rich phase diagram in α and τ. In particular we prove that the presence of hubs can speed up the mixing of the chain. We use different techniques for each phase, the most interesting of which is a bootstrap procedure to reduce the model from a phase where the degrees have bounded averages to a setting with unbounded averages.

show abstract

Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Scale-free percolation mixing time

Cipriani¹,

Salvi²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Scale-free percolation is an inhomogeneous version of the longrange percolation model. It was originally introduced with nodes placed on the lattice Z n in [Deijfen et al(2013)Deijfen, van der Hofstad, and Hooghiemstra] and then also studied on a Poisson point process in [Deprez and Wüthrich(2018)] and [Dalmau and Salvi(2021)]. Under measure P, we let V be the realization of a Poisson point process of parameter γ > 0 and assign independently to each vertex x ∈ V a random weight W x ∈ [1, ∞) such that P(W x > w) = w −(τ −1) L(w) for some τ > 1 and with L a slowly varying function.…”

Section: Proof Of Tightnessmentioning

confidence: 99%

Branching process and homogeneization for epidemics on spatial random graphs

Bansaye¹,

Salvi²

2023

Preprint

View full text Add to dashboard Cite

Consider a graph where the sites are distributed in space according to a Poisson point process on R n . We study a population evolving on this network, with individuals jumping between sites with a rate which decreases exponentially in the distance. Individuals give also birth (infection) and die (recovery) at constant rate on each site. First, we construct the process, showing that it is well-posed even when starting from non-bounded initial conditions. Secondly, we prove hydrodynamic limits in a diffusive scaling. The limiting process follows a deterministic reaction diffusion equation. We use stochastic homogenization to characterize its diffusion coefficient as the solution of a variational principle. The proof involves in particular the extension of a classic Kipnis-Varadhan estimate to cope with the non-reversiblity of the process, due to births and deaths. This work is motivated by the approximation of epidemics on large networks and results are extended to more complex graphs including percolation of edges.

show abstract

“…Recently, precise results about convergence of clustering coefficients, and scaling of the clustering function as k grows to infinity, for Hyperbolic Random Graphs has been obtained in [15]. Also, it was shown in [11] that under suitable conditions, the CSFP model has non-zero clustering in the limit.…”

Section: Consequences Of Local Convergence: Clusteringmentioning

confidence: 99%

Local limits of spatial inhomogeneous random graphs

Hofstad¹,

Hoorn²,

Maitra³

2021

Preprint

View full text Add to dashboard Cite

Consider a set of n vertices, where each vertex has a location in R d that is sampled uniformly from the unit cube in R d , and a weight associated to it. Construct a random graph by placing edges independently for each vertex pair with a probability that is a function of the distance between the locations, and the vertex weights.Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on R d with vertex locations given by a homogeneous Poisson point process, having weights which are i.i.d. copies of limiting vertex weights. Our setup covers many sparse geometric random graph models from the literature, including Geometric Inhomogeneous Random Graphs (GIRGs), Hyperbolic Random Graphs, Continuum Scale-Free Percolation and Weight-dependent Random Connection Models.We prove that the limiting degree distribution is mixed Poisson, and the typical degree sequence is uniformly integrable, and obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a by-product of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.

show abstract

Scale-free percolation in continuous space: quenched degree and clustering coefficient

Cited by 6 publications

References 37 publications

Scale-free percolation mixing time

Scale-free percolation mixing time

Branching process and homogeneization for epidemics on spatial random graphs

Local limits of spatial inhomogeneous random graphs

Contact Info

Product

Resources

About