Generally, percolation refers to qualitative changes in connectivity in systems (specially large ones) as its components are added or removed. In particular, percolation most often refers to the case where a system goes from being "mostly disconnected" to "mostly connected", in some sense. A more general mathematical model inspired by percolation and the Potts model is the Random-cluster model.

Percolation theory, from the perspective of Network theory describes the behavior of connected clusters in a network (often modelled as a random graph), as some substructures in the network are added or removed. The most common types are random site and bond percolation, where one removes either nodes or edges with a uniform probability, known as the occupation probability. However, there are other types (see below).

Again, from the perspective of networks, the transition from the system being "disconnected" to "connected", is most often made precise by the appearance of a giant connected component. See below.

_{Often, the theory of percolation is concerned with the clustering properties of identical objects which are randomly and uniformly distributed through space with a given occupation probability. However, these uniformity assumptions may be relaxed in other types of percolation.}

Keywords: Network science, Complex systems.

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References

Newman's book, and Mason and Gleeson tutorial have good reviews. See more at References for percolation

Percolation theory

Mathematical theory of percolation, with several important results, and discoveries.

Percolation phase transition

Percolation phase transition

A phase transition occurs between a phase without a giant connected component and a phase with one. A giant connected component, or GCC, is a connected component that contains a finite fraction of the nodes as the network size $N \rightarrow \infty$, i.e. it has an "extensive" scaling. The transition occurs at a critical value of the occupation probability, known as the percolation threshold.

Critical phenomena in percolation

Types of percolation models

Main types:

• Site percolation
• Bond percolation
• K-core percolation
• Explosive percolation
• Bootstrap percolation
• Directed percolation
• Limited path percolation
• K-clique percolation
• Percolation in Multilayer networks
• etc.

Applications of percolation models

Applications to porous materials

Applications to the study of landscapes

Applications in topography (study of landscapes) has been found, in particular relating to:

• Statistical and fractal properties of watersheds.
• Percolation of water bodies as water level rises in a landscape.