Sharp asymptotic for the chemical distance in long‐range percolation

Biskup, Marek; Lin, Jeffrey

doi:10.1002/rsa.20849

“…In [33], it is shown that distances grow linearly with the norm of vertices involved, when α > 2d and degrees have finite variance. The α ∈ (d, 2d) case remains open, and they are believed to grow poly-logarithmically with an exponent depending on α, d, τ based on the analogous results about long-range percolation [16,17].…”

Section: 1mentioning

confidence: 98%

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Komjáthy

¹

,

Lodewijks

²

2020

Stochastic Processes and their Applications

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In this paper we study weighted distances in scale-free spatial network models: hyperbolic random graphs, geometric inhomogeneous random graphs and scale-free percolation. In hyperbolic random graphs, n = Θ(e R/2 ) vertices are sampled independently from the hyperbolic disk with radius R and two vertices are connected either when they are within hyperbolic distance R, or independently with a probability depending on the hyperbolic distance. In geometric inhomogeneous random graphs, and in scale-free percolation, each vertex is given an independent weight and location from an underlying measured metric space and Z d , respectively, and two vertices are connected independently with a probability that is a function of their distance and their weights. We assign independent and identically distributed (i.i.d.) weights to the edges of the obtained random graphs, and investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical weighted distance. In scale-free percolation, we study the weighted distance from the origin of vertex-sequences with norm tending to infinity.In particular, we study the case when the parameters are so that the degree distribution in the graph follows a power law with exponent τ ∈ (2, 3) (infinite variance), and the edge-weight distribution is such that it produces an explosive age-dependent branching process with power-law offspring distribution, that is, the branching process produces infinitely many individuals in finite time. We show that in all three models, typical distances within the giant/infinite component converge in distribution, that is, no re-scaling is necessary. This solves an open question in [51].The main tools of our proof are to develop elaborate couplings of the models to infinite versions, to follow the shortest paths to infinity and then to connect these paths by using weight-dependent percolation on the graphs, that is, we delete edges attached to vertices with higher weight with higher probability. We realise the percolation using the edge-weights: only very short edges connected to high weight vertices are allowed to stay, hence establishing arbitrarily short upper bounds for connections.(3) High clustering. Local clustering is the formation of locally concentrated clusters in a network and it is measured by the average clustering coefficient: the proportion of triangles present at a vertex versus the total number of possible triangles present at said vertex, averaged over all vertices [76]. Many real-life networks show high clustering, e.g. biological networks such as food webs, technological networks as the World Wide Web, and social networks [30,64,69,70,72].The study of real-world networks could be further improved by the analysis of network models. In the last three decades, network science has become a significant field of study, both theoretically and experimentally. One of the first models that was proposed for studying real-world networks was the Watts-Strogatz model [76], that has the small-...

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“…The network is a small world only if this expected maximum link length is larger than the space diameter ∼ n 1/d , which implies ξ > 1/d or β < 2d/c, cf. the second row in Table I with l sup = c. If f (x) grows faster than logarithmically, l inf = ∞, then l(x) decays faster than a power law, ξ = 0, and there are no long links at all, so that our networks are necessarily large worlds, the last regime in Table I. This logic is about the necessary conditions for small worldness, but they have been proven to be also sufficient [31][32][33], and we confirm all the results above in simulations in Fig. 1 (small worldness) and in Fig.…”

mentioning

confidence: 90%

“…growing function of β ranging in values from some minimum value at β = 2d that does not appear to be zero, to its theoretical maximum b = 1/d at zero temperature β → ∞ corresponding to sharp RGGs. The nature of the small-to-large world phase transition at β = 2d appears to be an interesting open question [32]. The simulations can hardly reach network sizes that are sufficiently large to provide any hints regarding whether this transition is continuous or discontinuous, yet the results in Fig.…”

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confidence: 95%

Small worlds and clustering in spatial networks

Boguñá

¹

,

Krioukov

²

,

Almagro

³

et al. 2020

Phys. Rev. Research

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Networks with underlying metric spaces attract increasing research attention in network science, statistical physics, applied mathematics, computer science, sociology, and other fields. This attention is further amplified by the current surge of activity in graph embedding. In the vast realm of spatial network models, only a few reproduce even the most basic properties of real-world networks.Here, we focus on three such properties-sparsity, small worldness, and clustering-and identify the general subclass of spatial homogeneous and heterogeneous network models that are sparse small worlds and that have nonzero clustering in the thermodynamic limit. We rely on the maximum entropy approach where network links correspond to noninteracting fermions whose energy dependence on spatial distances determines network small worldness and clustering. arXiv:1909.00226v1 [physics.soc-ph]

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“…[12,Section 1.4] and [14,Section 10.6] for background and many references. Although substantial progress on these questions has been made over the last forty years, with highlights of the literature including [5,12,13,21,24,25,42,72], many further important problems remain open.…”

Section: Introductionmentioning

confidence: 99%

Power-law bounds for critical long-range percolation below the upper-critical dimension

Hutchcroft

¹

2021

Probab. Theory Relat. Fields

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We study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

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Sharp asymptotic for the chemical distance in long‐range percolation

Cited by 21 publications

References 28 publications

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Small worlds and clustering in spatial networks

Power-law bounds for critical long-range percolation below the upper-critical dimension

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