[igraph] Definition of Modularity

[Zhige Xin]

Hi dear all,

I have a question about the definition of Modularity proposed by Mark Newman in 2004. In his formula, for 

any two nodes i and j in the same community in the network N, we sum the following Aij - Ki*Kj/2m and divide

it by 2m, where Aij is the corresponding entry in the adjacency matrix of N and Ki is the degree of node i and m

is the number of edges of N.

So my question is that why Ki*Kj/2m is "the probability of an edge existing between vertices i and j if connections

are made at random but respecting vertex degrees". Of course, I understand Ki*Kj is the all possible events for the 

connection between i and j. Also, I checked the wiki page for Modularity which said this was based on Configuration 

Model but I do not get why Expected [Full edges between i and j] = (Full edges between i and j)/(Total number of rewirings possible) 

and what rewiring means? Besides, the wired thing is that it seems that Mark Newman did not include this  Configuration Model

in his reference. So could someone explain this for me?

Thanks!

Isaiah