I will talk about some of the statistical properties of the G(n,p) graph model, due to Erdos and Renyi.
This model has two parameters: the number of vertices (n), and the edge probability (p). For each pair of vertices, u and v, p is the probability that the edge (u,v) is present, independent of all other edges.

We will show that such a random graph has many interesting global properties. Namely:

(1) When p=d/n for some constant d, the number of triangles in G(n,p) is independent of n.
(2) The property that G(n,p) has diameter two has a sharp threshold at p = \sqrt(\dfrac{2\ln n}{n}).
(3) The disappearance of isolated vertices in G(n,p) has a sharp threshold at p = \dfrac{\ln n}{n}. 
(4) When p=d/n, d > 1, there is a giant component consisting of a constant fraction of the vertices.

I will be covering material from the following book: http://www.cs.cornell.edu/jeh/book11April2014.pdf