Function that spectrally reorders a similarity matrix and outputs the spectrum of eigenvalues and eigenvectors

Function supplemental to the text of: "The Spectral Transformation in Neuroimaging: A review and MATLAB tool" Bajada et al (in prep)

Matlab implementation partially based on an algorithm published in supplemetary text of: Johansen-Berg et al. (2004) PNAS 10.1073/pnas.0403743101. For in depth mathematical background, see: Barnard, S.T., Pothen, A., & Simon, H.D. (1995) Numer. Linear Algebra Appl. 2,317-334 For a tutorial on the different approaches please see: von Luxburg, U. (2007) Statistics and Computing 17 (4), 2007

Here we implement 4 different methods: method = 'geig' Based on the generalised spectral decomposition of the un-normalised Laplacian (default approach) method = 'sym' Based on the spectral decomposition of the normalised symmetric Laplacian - results should match geig method = 'rw' Based on the spectral decomposition of the normalised non-symmetric (random-walk) Laplacian method = 'unnorm' Based on the spectral decomposition of the un-normalised Laplacian

Function takes the form: [sortedB, p, v2, v3D, sortedEigenValues , sortedEigenVectors] = spectral_reorder(B,method);

B = similarity matrix (accepts values between -1 and 1) sortedB = spectrally reordered matrix p = the reordering vector that allows you to reorder your subjects in the same way as your matrix (sortedB) v2 = Fiedler vector (eigen vector corresponding to the second smallest eigenvalue) v3D = First 3 smaller eigenvectors (the first one is the Fiedler vector) sortedEigenValues = the complete set of sorted eigenvalues sortedEigenVectors = the complete set of sorted eigenvectors

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

If you only assign one output (i.e. of the form sortedB = extract_spectral_reorder(B), only the spectrally reordered matrix not the reordering vector or the other outputs - will be produced)

Created by: Claude J. Bajada - University of Manchester Nelson Trujillo-Barreto - University of Manchester

Edited by: Owen Falzon - University of Malta

Neuroscience and Aphasia Research Unit (NARU) University of Manchester 2014