Computes the maximum-degree heuristic edge weights

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function [w,rho] = max_deg(A)

% [W,RHO] = MAX_DEG(A) gives a vector of maximum-degree edge weights for a
% graph described by the incidence matrix A (NxM). N is the number of
% nodes, and M is the number of edges. Each column of A has exactly one +1
% and one -1. RHO is computed from the weights W as follows:
%    RHO = max(abs(eig( eye(n,n) - (1/n)*ones(n,n) - A*W*A' ))).
%
% Maximum-degree edge weights are all equal to one over the maximum
% degree of the nodes in the graph.
%
% For more details, see the references:
% "Fast linear iterations for distributed averaging" by L. Xiao and S. Boyd
% "Fastest mixing Markov chain on a graph" by S. Boyd, P. Diaconis, and L. Xiao
% "Convex Optimization of Graph Laplacian Eigenvalues" by S. Boyd
%
% Almir Mutapcic 08/29/06

% maximum degree solution
[n,m] = size(A);

% max degrees of the nodes
Lunw = A*A';        % unweighted Laplacian matrix
degs = diag(Lunw);

% max degree weight allocation
max_deg = max(degs);
w = (1/max_deg)*ones(m,1);

% compute the norm
if nargout > 1,
    rho = norm( eye(n) - A*diag(w)*A' - (1/n)*ones(n) );
end