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function [K, M, epsilon, edgelist] = computeEdge2EdgeMeasures(A)
%COMPUTEEDGE2EDGEMEASURES Given a adjacency matrix (weighted
% adjacency matrix) A compute the flow-redistribution matrix, the
% edge-to-edge transfer function matrix and the embeddedness vector
% (c) M. Schaub, Imperial College London, mts09@ic.ac.uk
%
% If you make use of this code please cite the associated article:
% "Structure of complex networks: Quantifying edge-to-edge relations by
% failure-induced flow redistribution", Michael T. Schaub, Jörg Lehmann,
% Sophia N. Yaliraki, Mauricio Barahona, Network Science 2(1), 2014,
% pp. 66--89; see also arXiv:0707.0609
%
% remove diagonal from matrix
A = A-diag(diag(A));
% compute number of links
links= nnz(A)/2;
% find all indices of nodes
[i,j,v] = find(triu(A));
edgelist =[(1:links)' i j v];
%combinatorial Laplacian
L = diag(sum(A,2))-A;
E = createIncidenceMatrix(A);
%M = diag(v)*E'*pinv(full(L))*E;
% faster than above: compute pseudoinverse directly
%Lp = (L- ones(size(L))/length(L))^-1 + ones(size(L))/length(L);
%M = diag(v)*E'*Lp*E;
% even faster: use Gaussian elimination, (note that constant matrix is in
% null space of incidence matrix, so can be neglected)
M = diag(v)*E'*( (L- ones(size(L))/length(L))\ E );
epsilon = 1-diag(M);
if any(epsilon < 1e-12)
warning('Some edges have an embeddedness essentially zero! Associated columns in K might be wrong.')
end
K = M/(diag( epsilon ) );
end
function [E, edgelist] = createIncidenceMatrix(A)
% For a given (undirected) graph with adj matrix A
% create node-edge incidence matrix E
% Note that self loops are not taken into account here
% compute number of links
links= nnz(A)/2;
nodes = length(A);
% find all indices of nodes
[i,j, v] = find(triu(A));
% allocate incidence matrix
% insert +1/-1 in both node rows and column given by edge id
E = sparse(i,1:links,-1,nodes,links);
E = E + sparse(j,1:links,1,nodes,links);
% edgelist in format edge_id, from, to, strength
edgelist =[(1:links)' i j v];
end