Random Graph: Erdős–Rényi model_random graph model

http://en.wikipedia.org/wiki/Erdos-Renyi_model

One of their most surprising results is the existence of abrupt changes in the characteristics
of random graphs as random edges are added. They showed that for a number of graph
properties there is a threshold value of the probability p below which the property is rare
and above which it is almost certain. This transition is sometimes called a “phase change”
by analogy with physical systems that change state at some critical value of temperature.

One of the properties that displays this kind of transition is connectedness. For a given size n, there is a critical value, p*, such that a random graph G ( n, p ) is unlikely to be connected if p < p* and very likely to be connected if p > p* .