As in (Atasoy et al., 2016), we use the graph Laplacian L as the discrete counterpart of the Laplace operator applied to the brain connectivity matrix. The graph Laplacian can be interpreted as a particular case of the discrete approximation of the continuous Laplace-Beltrami operator, a generalization of the Laplace operator to Riemannian manifolds. We use the graph Laplacian (Levy, 2006) defined as:

where A is the adjacency matrix of the combined connectome consisting of white-matter and gray-matter connectivity, and D its degree matrix. Note that other formulae exist for computing the discrete Laplacian. Our choice of this Laplacian formula is mainly driven by its numerical stability as discussed in (Levy, 2006). The combined connectome is the adjacency matrix A obtained by combining the local connectivity adjacency matrix A derived from the cortical surface mesh and the long-range connectivity adjacency matrix Ac constructed by tractography methods:

whereby both A and Ac are m × m matrices with m being the number of vertices of the cortical surface mesh. The long-range connectivity adjacency matrix Ac is derived by removing the weakest weights of the z-scored long-range connectivity matrix Cz (referred in this article as thresholding):

for all i,j=1,,m, with zC being the adjacency weight threshold. Note that we applied z-score normalization to the long-range connectivity matrix C in order to have integer values of zC relating to the standard deviation of connection weights:

where μC is the mean connectivity strength of the long-range connectome C, and σC its standard deviation. A represents the local gray matter connectivity matrix. As in (Atasoy et al., 2016), two nodes are locally connected when they are connected through the mesh as direct neighbors.

We then decompose the graph Laplacian L into a finite number of eigenvalues λk and eigenvectors, or connectome harmonics, ψk: