function P = expectedA(A)
% EXPECTEDA computes the expected number of connections between each vertex
% P = EXPECTEDA(A) computes the matrix P of the expected number of
% connections between each vertex, given the adjacency matrix A.
% (1) the expected number of connection is computed based on the
% assumption of the "null model": i.e. random connections between nodes
% of the same degree sequence as A. Thus, for undirected graphs, Pij =
% (ki*kj) / 2m. For directed graphs, we have to make the additional
% assumption that the in-degree and out-degree distributions are
% themselves not correlated - i.e. that we can place an edge between any
% randomly selected pair of available out-nodes and in-nodes.
% (2) In addition, the underlying "null model" allows multiple and
% self-edges, and hence has values along the diagonal.
% References: Newman, M. E. J. (2006) "Finding community structure in
% networks using the eigenvectors of matrices". Phys Rev E, 74, 036104.
% Mark Humphries 24/8/2006
[n c] = size(A);
inks = sum(A);
outks = sum(A');
m = sum(inks); % number of edges
P = (outks' * inks) ./ m;