Supporting Information ( SI ) for “ Toward link predictability of complex networks ”

If the adjacency matrix contains degenerate eigenvalues, we must modify the approach using non-degenerate eigenvalues. We denote the eigenvalues as λki, where the index k runs over different eigenvalues and the index i runs over M associated eigenvectors of the same eigenvalue. Note that there is no unique way of choosing a basis for the eigenvectors of the unperturbed network since any linear combination of the eigenvectors belonging to the same eigenvalue is still an eigenvector. Repeated eigenvalues have been shown to be related to the symmetric motifs and graph automorphisms in networks [1]. After a perturbation is added to the network, the symmetry of the nodes will be lifted either partly or completely, so the degenerate eigenvalues must be chosen such that they can be transformed continuously into the perturbed non-degenerate eigenvalues. If we define the chosen eigenvectors to be x̄ki = ∑M j=1 βkjxkj , the eigenfunction becomes ( A +∆A ) x̄ki = ( λki +∆λ̄ki ) x̄ki, (1)