Continuum percolation theory

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In

random geometric graphs
.

Continuum percolation arose from an early mathematical model for

porous material and semiconductors, while becoming a subject of mathematical interest in its own right.[6]

Early history

In the early 1960s

branching processes to show that a threshold value
existed for the infinite or "giant" component.

Definitions and terminology

The exact names, terminology, and definitions of these models may vary slightly depending on the source, which is also reflected in the use of point process notation.

Common models

A number of well-studied models exist in continuum percolation, which are often based on homogeneous Poisson point processes.

Disk model

Consider a collection of points {xi} in the plane ℝ2 that form a homogeneous Poisson process Φ with constant (point) density λ. For each point of the Poisson process (i.e. xi ∈ Φ), place a disk Di with its center located at the point xi. If each disk Di has a random radius Ri (from a common distribution) that is independent of all the other radii and all the underlying points {xi}, then the resulting mathematical structure is known as a random disk model.

Boolean model

Given a random disk model, if the set union of all the disks {Di} is taken, then the resulting structure ⋃i Di is known as a Boolean–Poisson model (also known as simply the Boolean model),[8] which is a commonly studied model in continuum percolation[1] as well as stochastic geometry.[8] If all the radii are set to some common constant, say, r > 0, then the resulting model is sometimes known as the Gilbert disk (Boolean) model.[9]

Percolation in the Boolean–Poisson (constant disk) model.
Simulation of 4 Poisson–Boolean (constant-radius or Gilbert disk) models as the density increases with largest clusters in red.

Germ-grain model

The disk model can be generalized to more arbitrary shapes where, instead of a disk, a random

set union of all the shapes forms a Boolean germ-grain model. Typical choices for the grains include disks, random polygon and segments of random length.[8]

Boolean models are also examples of

stochastic processes known as coverage processes.[10]
The above models can be extended from the plane ℝ2 to general Euclidean space ℝn.

Components and criticality

In the Boolean–Poisson model, disks there can be isolated groups or clumps of disks that do not contact any other clumps of disks. These clumps are known as components. If the area (or volume in higher dimensions) of a component is infinite, one says it is an infinite or "giant" component. A major focus of percolation theory is establishing the conditions when giant components exist in models, which has parallels with the study of random networks. If no big component exists, the model is said to be subcritical. The conditions of giant component criticality naturally depend on parameters of the model such as the density of the underlying point process.

Excluded area theory

The excluded area of a placed object is defined as the minimal area around the object into which an additional object cannot be placed without overlapping with the first object. For example, in a system of randomly oriented homogeneous rectangles of length l, width w and aspect ratio r = l/w, the average excluded area is given by:[11]

In a system of identical ellipses with semi-axes a and b and ratio r = a/b, and perimeter C, the average excluded areas is given by:[12]

The excluded area theory states that the critical number density (percolation threshold) Nc of a system is inversely proportional to the average excluded area Ar:

It has been shown via Monte-Carlo simulations that percolation threshold in both homogeneous and heterogeneous systems of rectangles or ellipses is dominated by the average excluded areas and can be approximated fairly well by the linear relation

with a proportionality constant in the range 3.1–3.5.[11][12]

Applications

Possible coverage model.
A Boolean model as a coverage model in a wireless network.

The applications of percolation theory are various and range from material sciences to

wireless communication systems. Often the work involves showing that a type of phase transition
occurs in the system.

Wireless networks

Wireless networks are sometimes best represented with stochastic models owing to their complexity and unpredictability, hence continuum percolation have been used to develop

sensor networks.[13][14] One of the main limitations of these networks is energy consumption where usually each node has a battery and an embedded form of energy harvesting. To reduce energy consumption in sensor networks, various sleep schemes have been suggested that entail having a subcollection of nodes go into a low energy-consuming sleep mode. These sleep schemes obviously affect the coverage and connectivity of sensor networks. Simple power-saving models have been proposed such as the simple uncoordinated 'blinking' model where (at each time interval) each node independently powers down (or up) with some fixed probability. Using the tools of percolation theory, a blinking Boolean Poisson model has been analyzed to study the latency and connectivity effects of such a simple power scheme.[13]

See also

References

  1. ^ a b Meester, R. (1996). Continuum Percolation. Vol. 119. Cambridge University Press.[ISBN missing]
  2. ^ a b c Franceschetti, M.; Meester, R. (2007). Random Networks for Communication: From Statistical Physics to Information Systems. Vol. 24. Cambridge University Press.[ISBN missing]
  3. ^
    doi:10.1137/0109045
    .
  4. S2CID 1514941
    .
  5. doi:10.1239/jap/1152413741
    .
  6. doi:10.1080/13642818708215336
    .
  7. doi:10.1214/aop/1176992809
    .
  8. ^ a b c Stoyan, D.; Kendall, W. S.; Mecke, J.; Ruschendorf, L. (1995). Stochastic Geometry and Its Applications. Vol. 2. Wiley Chichester.[ISBN missing]
  9. ^ Balister, Paul; Sarkar, Amites; Bollobás, Béla (2008). "Percolation, connectivity, coverage and colouring of random geometric graphs". Handbook of Large-Scale Random Networks. pp. 117–142.[ISBN missing]
  10. ^ Hall, P. (1988). Introduction to the theory of coverage processes. Vol. 1. New York: Wiley.[ISBN missing]
  11. ^
    S2CID 21438506
    .
  12. ^
    ISSN 0378-4371
    .
  13. ^ a b Dousse, O.; Mannersalo, P.; Thiran, P. (2004). "Latency of wireless sensor networks with uncoordinated power saving mechanisms". Proceedings of the 5th ACM International Symposium on Mobile Ad Hoc Networking and Computing. ACM. pp. 109–120.
  14. ^ Gui, C.; Mohapatra, P. (2004). "Power conservation and quality of surveillance in target tracking sensor networks". Proceedings of the 10th Annual International Conference on Mobile Computing and Networking. ACM. pp. 129–143.
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