Percolation processes. I. Crystals and Mazes
01 Jul 1957-Vol. 53, Iss: 3, pp 629-641
TL;DR: In this paper, the authors study how the random properties of a medium influence the percolation of a fluid through it, in a general way, in which the treatment diifers from conventional diffusion theory.
Abstract: The paper studies, in a general way, how the random properties of a ‘medium’ influence the percolation of a ‘fluid’ through it. The treatment diifers from conventional diffusion theory, in which it is the random properties of the fluid that matter. Fluid and medium bear general interpretations: for example, solute diffusing through solvent, electrons migrating over an atomic lattice, molecules penetrating a porous solid, disease infecting a community, etc.
Citations
More filters
TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
17,647 citations
TL;DR: In this article, the possible domain structures which can arise in the universe in a spontaneously broken gauge theory are studied, and it is shown that the formation of domain wall, strings or monopoles depends on the homotopy groups of the manifold of degenerate vacua.
Abstract: The possible domain structures which can arise in the universe in a spontaneously broken gauge theory are studied. It is shown that the formation of domain wall, strings or monopoles depends on the homotopy groups of the manifold of degenerate vacua. The subsequent evolution of these structures is investigated. It is argued that while theories generating domain walls can probably be eliminated (because of their unacceptable gravitational effects), a cosmic network of strings may well have been formed and may have had important cosmological effects.
2,994 citations
Book•
18 Oct 2012TL;DR: This rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects.
Abstract: Covering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects. Practical engineering applications are integrated with mathematical theory, with an understanding of probability the only prerequisite. At the same time, stochastic geometry is connected to percolation theory and the theory of random geometric graphs and accompanied by a brief introduction to the R statistical computing language. Combining theory and hands-on analytical techniques with practical examples and exercises, this is a comprehensive guide to the spatial stochastic models essential for modelling and analysis of wireless network performance.
2,327 citations
01 Jan 1986
TL;DR: In this article, the possible domain structures which can arise in the universe in a spontaneously broken gauge theory are studied, and it is shown that the formation of domain wall, strings or monopoles depends on the homotopy groups of the manifold of degenerate vacua.
Abstract: The possible domain structures which can arise in the universe in a spontaneously broken gauge theory are studied. It is shown that the formation of domain wall, strings or monopoles depends on the homotopy groups of the manifold of degenerate vacua. The subsequent evolution of these structures is investigated. It is argued that while theories generating domain walls can probably be eliminated (because of their unacceptable gravitational effects), a cosmic network of strings may well have been formed and may have had important cosmological effects.
2,274 citations
TL;DR: This tutorial article surveys some of these techniques based on stochastic geometry and the theory of random geometric graphs, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature.
Abstract: Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communication-theoretic results accounting for the networks geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs -including point process theory, percolation theory, and probabilistic combinatorics-have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.
1,893 citations
Cites background or methods from "Percolation processes. I. Crystals ..."
...His investigations were based on prior work on the discrete percolation model of Broadbent and Hammersley [33] and on the theory of branching processes [34]....
[...]
...Theorem 1 (Broadbent and Hammersley [33]) There exists a number 0 < pc < 1 such that θ(p) = 0 for p < pc and θ(p) > 0 for p > pc....
[...]
...The central result of percolation theory is the following: Theorem 1 (Broadbent and Hammersley [33]) There exists a number 0 < pc < 1 such that θ(p) = 0 for p < pc and θ(p) > 0 for p > pc....
[...]
References
More filters
01 Jul 1957
TL;DR: In this paper, the proof of the theorem on the connective constant of a crystal was presented, and a self-contained version of the present paper was presented for the same purpose.
Abstract: The previous paper (1) omitted the proof of the theorem on the connective constant of a crystal. This paper supplies the proof. For ease of reference, we first recall the relevant nomenclature and enunciate the theorem; to this extent the present paper is self-contained.
367 citations
317 citations
167 citations
01 Jul 1949
TL;DR: In this paper, it was shown that the convergence of the series for F(x) is convergent when |x| < |1, and that the series converges with |x | < | 1.
Abstract: A number of important Markoff processes, with a continuous time parameter, can be represented approximately by a discrete process, interesting in its own right, of the following type. A class of individuals gives rise seasonally (in January say) to a number of new individuals (children), the probabilities of an individual having 0, 1, 2, … children being p0, p1, p2, …. These probabilities are the same for all individuals and are independent. The individuals formed each January are regarded as a new generation, and only this generation is capable of reproducing in the next January. Letso that F(x) is the probability generating function (p.g.f.) of the number of children of an individual. Clearly the series for F(x) is absolutely convergent when |x| < |1.
71 citations
Journal Article•
TL;DR: In this paper, the conditions générales d'utilisation (http://www.compositio.org/conditions) of the agreement with the Foundation Compositio Mathematica are described.
Abstract: © Foundation Compositio Mathematica, 1953, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
10 citations