Percolation Theory describes the behavior of a network when nodes or links are added.
A percolation is a process wherein given a lattice, we place pebbles with probability at each lattice point. Neighboring pebbles are considered connected.
- An inverse percolation process involves removing nodes instead of adding them from the lattice.
The percolating cluster is an emergent property during a percolation process.
- We may define a threshold function, where, when , we go from a system of tiny clusters to one having a giant cluster. is the breakdown threshold.

Fundamental Quantities

The breakdown threshold is a threshold notated . It is the number of nodes that is needed for a given network to have a percolating cluster.
The clustering coefficient is a quantity, denoted which captures the degree to which the neighbors of link to each other to form a clique. Let be a vertex, with degree . be the number of edges between the neighbors of .

The clustering coefficient is calculated as

It should be noted that

Critical Exponents

In a percolation process the average cluster size is defined as Where is the breakdown threshold is the critical exponent. As this value diverges
The order parameter is a parameter that dictates the probability that a randomly chosen pebble in a percolation process belongs to the largest cluster.

It is notated and follows Where is the critical exponent.
- As , this probability drops to .
In a percolation process, the correlation distance is the mean distance between two pebbles in the same cluster. It is defined as

Where is a critical exponent .
- As , the distance goes from finite to divergent. This implies the size of the largest cluster becomes infinite and percolates the whole lattice.
The critical exponents are a set of three exponents that describe a percolation process
- - average cluster size
- - order parameter
- - correlation length
In a percolation process, the critical exponents are universal properties which are independent of the precise value of or the lattice.