Shortest path and Schramm-Loewner Evolution

Posé, Nicolas; Schrenk, K. J.; Araújo, Nuno A. M.; Herrmann, Hans J.

doi:10.1038/srep05495

Cited by 22 publications

(16 citation statements)

References 80 publications

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“…To identify and analyse the CP structure we use the t -core decomposition, as proposed in (ref. 4 ). Similar to the k -core decomposition 17 , this method progressively prunes a network by recursively removing nodes that are part of the least number of triangles.…”

Section: Resultsmentioning

confidence: 99%

“…The network undergoes a transition from a state where the size of the core is comparable to the system size but is of small coreness to a state with a small core and relatively large coreness. Since the empirical WAN is known to have a small core size of ∼2.3% but high inter-connectivity within the core 4 , it should be found in regime B with l min < θ ≤ l max , where the value of λ is largest (see Fig. 4 ).…”

Section: Resultsmentioning

confidence: 99%

“…As presented in (ref. 4 ), the empirical network is very sensitive to the removal of high degree nodes and the size of the largest component drops very quickly ( Supplementary Note 8 and Supplementary Fig. 14 ).…”

Section: Resultsmentioning

confidence: 99%

“…One of the remarkable features of the world airline network (WAN) is its small core (consisting of ∼2.5% of the airports) that is almost fully connected and surrounded by a vast periphery that is nearly tree-like and connected to the core through many regional and national hubs 4 . This block arrangement is prominently known as the core–periphery (CP) structure 5 6 7 8 , which was also reported for other infrastructure networks, such as the world trade network 9 10 , the autonomous Internet network 10 and the financial interbank lending markets 11 , where the fraction of peripheral nodes varies from 45 to 85%.…”

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confidence: 99%

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Emergence of core–peripheries in networks

et al. 2016

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A number of important transport networks, such as the airline and trade networks of the world, exhibit a characteristic core–periphery structure, wherein a few nodes are highly interconnected and the rest of the network frays into a tree. Mechanisms underlying the emergence of core–peripheries, however, remain elusive. Here, we demonstrate that a simple pruning process based on removal of underutilized links and redistribution of loads can lead to the emergence of core–peripheries. Links are assumed beneficial if they either carry a sufficiently large load or are essential for global connectivity. This incentivized redistribution process is controlled by a single parameter, which balances connectivity and profit. The obtained networks exhibit a highly resilient and connected core with a frayed periphery. The balanced network shows a higher resilience than the world airline network or the world trade network, revealing a pathway towards robust structural features through pruning.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

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confidence: 99%

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Emergence of core–peripheries in networks

et al. 2016

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“…the shortest paths on critical percolation clusters with κ = 1.04±0.02 [100], are examples of unusual universality classes. Watersheds can also be defined in higher dimensions [117].…”

Section: Introductionmentioning

confidence: 99%

Recent advances in percolation theory and its applications

2015

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Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied to describe a large variety of natural, technological and social systems. Percolation models serve as important universality classes in critical phenomena characterized by a set of critical exponents which correspond to a rich fractal and scaling structure of their geometric features. We will first outline the basic features of the ordinary model.Over the years a variety of percolation models has been introduced some of which with completely different scaling and universal properties from the original model with either continuous or discontinuous transitions depending on the control parameter, dimensionality and the type of the underlying rules and networks. We will try to take a glimpse at a number of selective variations including Achlioptas process, half-restricted process and spanning cluster-avoiding process as examples of the so-called explosive percolation. We will also introduce non-self-averaging percolation and discuss correlated percolation and bootstrap percolation with special emphasis on their recent progress. Directed percolation process will be also discussed as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state.In the past decade, after the invention of stochastic Löwner evolution (SLE) by Oded Schramm, two-dimensional (2D) percolation has become a central problem in probability theory leading to the two recent Fields medals. After a short review on SLE, we will provide an overview on existence of the scaling limit and conformal invariance of the critical percolation. We will also establish a connection with the magnetic models based on the percolation properties of the Fortuin-Kasteleyn and geometric spin clusters. As an application we will discuss how percolation theory leads to the reduction of the 3D criticality in a 3D Ising model to a 2D critical behavior.Another recent application is to apply percolation theory to study the properties of natural and artificial landscapes. We will review the statistical properties of the coastlines and watersheds and their relations with percolation. Their fractal structure and compatibility with the theory of SLE will also be discussed. The present mean sea level on Earth will be shown to coincide with the critical threshold in a percolation description of the global topography.

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Application of Percolation Theory to Statistical Topographies

Saberi¹

2021

Complex Media and Percolation Theory

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Shortest path and Schramm-Loewner Evolution

Cited by 22 publications

References 80 publications

Emergence of core–peripheries in networks

Emergence of core–peripheries in networks

Recent advances in percolation theory and its applications

Application of Percolation Theory to Statistical Topographies

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