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