= Site percolation
{wiki}
In independent site percolation with parameter $p$, each <vertex> of a <graph> is open with <probability> $p$, independently of the others. Retain those <vertices> and every original <edge> whose two endpoints are retained. This gives an <induced subgraph>, whose <connected components of a graph> are the <percolation clusters>. A closed <vertex> is absent from the usable network, whereas <bond percolation> keeps all <vertices> and instead removes individual <edges>.
A specified simple <graph path> of $m$ <edges> has $m+1$ distinct <vertices>. Its <probability> of being entirely open is therefore $p^{m+1}$, including the starting <vertex>. Conditional on that <vertex> being open, it is $p^m$ by <independence>. For a two-<edge> <graph path> at $p=1/2$, the unconditional <probability> is $1/8$, compared with $1/4$ in <bond percolation>. This comparison concerns a fixed route; <percolation critical probabilities> require analysis of all possible routes on the specified infinite <graph>.
Back to article page
