Models of network formation
* G(n, m)
- take n nodes, and place m edges among them at random
- there are ^(^nC_2)C_m ways of doing this
- this random graph model is defined as the ensemble of these graphs,
such that every possible of these members is equally likely to be chosen
* Erdos-Renyi model
- n vertices, connect every pair with probability p
- average number of edges and average degree:
- each edge is a Bernoulli random variable with success p
- the expected number of edges is the expectation of sum of Bernoulli
random variables, and hence is nC2 p
- the average degree is then c = (2/m) * nC2p = (n-1)p
- the degree distribution
- ^(n-1)C_k p^k (1-p)^(n-1)
- in the limit n -> \infty, and c constant, this reduces to the
poisson distribution with mean c: exp(-c) * c^k/k!
- clustering coefficients
- both clustering coefficients tend to 0 in the limit