Erdős–Rényi model (G(n, p) vs G(n, M)) distribution

They do not account for the formation of hubs. Formally, the degree distribution of ER graphs converges to a Poisson distribution, rather than a power law observed in most real-world, scale-free networks.

Originally Posted by CRGreathouse

They're both Poisson-distributed.

In the G(n, p) model, a graph is thought to be constructed by connecting nodes randomly. Each edge is included in the graph with probability p, with the presence or absence of any two distinct edges in the graph being independent. Equivalently, all graphs with n nodes and M edges have equal probability of p^M (1-p)^{{n \choose 2}-M}. The parameter p in this model can be thought of as a weighting function; as p increases from 0 to 1, the model becomes more and more likely to include graphs with more edges and less and less likely to include graphs with fewer edges. In particular, the case p = 0.5 corresponds to the case where all 2^\tbinom{n}{2} graphs on n vertices are chosen with equal probability.

The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed. Therefore it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and np ≤ 10.[1]