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.

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The Structure and Function of Complex Networks

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.

https://iopscience.iop.org/article/10.1088/0305-4470/9/8/029/pdf

Topology of cosmic domains and strings

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.

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Stochastic Geometry for Wireless Networks

Topology of Cosmic Domains and Strings

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.

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Stochastic geometry and random graphs for the analysis and design of wireless networks

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.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/C00CC4943F48228F8AC8031092FE84EC/S0305004100032680a.pdf/div-class-title-percolation-processes-div.pdf

S.R. Broadbent

Papers

Percolation processes. I. Crystals and Mazes