Behavioral Intervention and Non-Uniform Bootstrap Percolation
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Bootstrap percolation is an often used model to study the spread of diseases, rumors, and information on sparse random graphs. The percolation process demonstrates a critical value such that the graph is either almost completely affected or almost completely unaffected based on the initial seed being larger or smaller than the critical value. To analyze intervention strategies we provide the first analytic determination of the critical value for basic bootstrap percolation in random graphs when the vertex thresholds are nonuniform and provide an efficient algorithm. This result also helps solve the problem of "Percolation with Coinflips" when the infection process is not deterministic, which has been a criticism about the model. We also extend the results to clustered random graphs thereby extending the classes of graphs considered. In these graphs the vertices are grouped in a small number of clusters, the clusters model a fixed communication network and the edge probability is dependent if the vertices are in close or far clusters. We present simulations for both basic percolation and interventions that support our theoretical results.Keywords:
Percolation (cognitive psychology)
Continuum percolation theory
Percolation Theory
Percolation theory is a powerful framework for studying strongly disordered systems. However, almost all the interesting physics of percolation happens in a very small range in parameter space, and often it may seem difficult to justify why a physical system should happen to be near this special point, the percolation threshold. In this paper we review some aspects of percolation theory from a different viewpoint than the usual one. In this formulation, it becomes natural why the percolation threshold plays an important role in many physical systems.
Percolation (cognitive psychology)
Directed percolation
Percolation Theory
Continuum percolation theory
Physical space
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Porous media are often modeled as systems of overlapping obstacles, which leads to the problem of two percolation thresholds in such systems, one for the porous matrix and the other for the void space. Here we investigate these percolation thresholds in the model of overlapping squares or cubes of linear size k > 1 randomly distributed on a regular lattice. We find that the percolation threshold of obstacles is a nonmonotonic function of k, whereas the percolation threshold of the void space is well approximated by a function linear in 1/k. We propose a generalization of the excluded volume approximation to discrete systems and use it to investigate the transition between continuous and discrete percolation, finding a remarkable agreement between the theory and numerical results. We argue that the continuous percolation threshold of aligned squares on a plane is the same for the solid and void phases and estimate the continuous percolation threshold of the void space around aligned cubes in a 3D space as 0.036(1). We also discuss the connection of the model to the standard site percolation with complex neighborhood.
Continuum percolation theory
Percolation Theory
Percolation (cognitive psychology)
Lattice (music)
Void (composites)
Directed percolation
Unit cube
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We propose a simple percolation criterion for arbitrary percolation problems. The basic idea is to decompose the system of interest into a hierarchy of neighborhoods, such that the percolation problem can be expressed as a branching process. The criterion provides the exact percolation thresholds for a large number of exactly solved percolation problems, including random graphs, small-world networks, bond percolation on two-dimensional lattices with a triangular hypergraph, and site percolation on two-dimensional lattices with a generalized triangular hypergraph, as well as specific continuum percolation problems. The fact that the range of applicability of the criterion is so large bears the remarkable implication that all the listed problems are effectively treelike. With this in mind, we transfer the exact solutions known from duality to random lattices and site-bond percolation problems and introduce a method to generate simple planar lattices with a prescribed percolation threshold.
Continuum percolation theory
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Directed percolation
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We investigate percolation on a randomly directed lattice, an intermediate between standard percolation and directed percolation, focusing on the isotropic case in which bonds on opposite directions occur with the same probability. We derive exact results for the percolation threshold on planar lattices, and present a conjecture for the value the percolation-threshold for in any lattice. We also identify presumably universal critical exponents, including a fractal dimension, associated with the strongly-connected components both for planar and cubic lattices. These critical exponents are different from those associated either with standard percolation or with directed percolation.
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The geometrical features of discontinuous Pb films are analyzed in terms of the scaling theory of percolation. Above the percolation threshold it is shown that the infinite cluster as well as the backbone has an anomalous mass distribution up to a length of the order of the percolation correlation length (${\ensuremath{\xi}}_{p}$), corresponding to that of self-similar objects. Above ${\ensuremath{\xi}}_{p}$, the mass distribution is homogeneous. Below the percolation threshold, the cluster statistics agrees with scaling theory.
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Percolation Theory
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As one of the common theoretical methods for dealing with strong disorder systems and random geometrical structure,percolation theory provides a well-defined,transparent,and intuitively satisfying model for spatially random processes.The paper systematically expounded the center of percolation theory(ie,there will have a sharp chang of the physical nature at the point of percolation threshold),the basic characteristics of percolation theory;summarized the eight characteristics of percolation model with conductive compound as an example;outlined the applications of percolation theory,and overviewed its development prospects.
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Many modern nanostructured materials and doped polymers are morphologically too complex to be interpreted by classical percolation theory. Here, we develop the concept of a hierarchical percolating (percolation-within-percolation) system to describe such complex materials and illustrate how to generalize the conventional percolation to double-level percolation. Based on Monte Carlo simulations, we find that the double-level percolation threshold is close to, but definitely larger than, the product of the local percolation thresholds for the two enclosed single-level systems. The deviation may offer alternative insights into physics concerning infinite clusters and open up new research directions for percolation theory.
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Continuum percolation theory
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Abstract Percolation theory, the theory of the properties of classical particles interacting with a random medium, is of wide applicability and provides a simple picture exhibiting critical behaviour, the features of which are well understood and amenable to detailed calculation. In this review the concepts of percolation theory and the general features associated with the critical region about the onset of percolation are developed in detail. In particular, several dimensional invariants are examined which make it possible to unify much of the available information, and to extend the insights of percolation theory to processes which have not yet received numerical study. The compilation of the results of percolation theory, both exact and numerical, is believed to be complete through 1970. A selective bibliography is given. In a concluding chapter several recent applications of percolation theory to classical and to quantum mechanical problems are discussed.
Percolation Theory
Percolation (cognitive psychology)
Continuum percolation theory
Directed percolation
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