See Newman's book

Sample calculations

Average number of edges between two nodes is

$\frac{k_i k_j}{2m-1} \approx \frac{k_i k_j}{2m}$

in the limit of large size. This is approximately equal to the probability of an edge between the two nodes in the limit of large size too.

Excess degree distribution

Generating functions for the small components

See derivation in problem sheet or notes or book, using generating functions (in particular it's "power" property where the g.f. of a sum of independent random variables is the product of g.f.s of these rand. vars.)

Giant component

Can find expression for size of giant component.

One can then derive condition for existence of giant component in the configuration model. It is called the Malloy-Reed condition:

$\langle k^2 \rangle -2\langle k \rangle > 0 \Leftrightarrow \exists \text{ GCC}$

GCC = giant connected component.

Random graphs with clustering

Degree-triangle model

Variant, that has tunable clustering coefficient.

See here and here