# Percolation theory

**Percolation theory** relates to how networks/graphs behave when edges or vertices are removed from the system
- Representative question:
- “given that a liquid is poured onto some porous material, will the liquid be able to make its way from hole to hole and reach the bottom?”
- often modeled as a 3-dimensional Graph of size
`n x n x n`

, where edges (bonds) between vertices (sites) allow the liquid to pass through probabilistically
- edges will independently allow liquid between two given vertices with probability
`p`

, or not with probability `1 - p`

- so the likelihood of percolation depends on the “site vacancy probability”
`p`

- “therefore, for a given
`p`

, what is the probability that an open path exists for the liquid?”
- particularly when
`n`

(the dimension of the porous material) is large

- a system “percolates” if and only if (iff) there is an open path from top to bottom
- reflected in many real-world systems:

model |
system |
vacant site |
occupied site |
percolates |

electricity |
material |
conductor |
insulated |
conducts |

fluid flow |
material |
empty |
blocked |
porous |

social interaction |
population |
person |
empty |
communicates |