# Threshold Function and Phase Transitions¶

• t(n) is a threshold function for a monotone property A(N) if
• Pr[A(N) | p(n)] -> 1 if p(n)/t(n) -> infinity, and
• Pr[A(N) | p(n)] -> 0 if p(n)/t(n) -> 0
• $1/n^2$ - the network has some links (avg deg 1/n)
• $1/n^{3/2}$ - the network has a component with at least three links (avg deg $1/n^{1/2}$)
• $1/n$ - the network has a cycle, the network has a unique giant component: a component with at least n^a nodes some fixed a<1; (avg deg 1)
• $log(n)/n$ - the network is connected; (avg deg log(n))