6 FIG. 2: (Color online) Results for the three examples of real-world networks G 2 (top row), G 3 (middle row) and G 4 (bottom row) in Table I. The left column [(a), (d), and (g)] shows the eigenvector components sorted in the increasing order. The middle column [(b), (e), and (h)] shows the true SI vs approximated SI for the removal of edges, while the right column [(c), (f), and (i)] is for the removal of nodes. The meaning of the symbols is the same as in Fig. 1. the left eigenvector u , and the same convergence rates apply. Although other iterative schemes for the compu- tation of the dominant eigenvalue and the corresponding eigenvector are available, the power method is used here since its iterations directly highlight the local information of a graph. Assuming that the spectrum of A satisfies | λ | > | λ$_{2}$ | ≥ . . . ≥ | λ$_{n}$ | , one can adopt the power method [14–16] to solve for the dominant eigenvalue λ and its corresponding left and right eigenvectors u and v . The starting point of this method is a normalized vector v $^{(0)}$, that in this case can be taken as Then, for t = 1 , 2 , . . . , until convergence, the following is iterated: The convergence of both the eigenvalue and eigenvector is geometric, with rate O ( | λ$_{2}$/λ | $^{t}$) [39]. This algorithm can be straightforwardly adjusted for the computation of Indeed, the first iteration of the power method gives the degree, the number of connections that each node has, up to a normalization constant. The k th component of u (1) is proportional to the in-degree of node k , d in k = ∑ n i =1 a$_{ik}$ , while the k th component of v (1) scales with the out-degree of the same node, d out k = ∑ n i =1 a$_{ki}$ . If the graph is undirected and unweighted, the k th component of both u (1) and v (1) are proportional to the number of direct neighbors that node k has (counting itself, if a self-loop is in place). v (0) = 1 √$_{n}$ [1 , 1 , · · · , 1] $^{T}$. (29) y ( t ) = Av ( t − $^{1)}$, λ( t ) = || y ( t $^{)}$| | $_{2}$, v ( t ) = y ( t $^{)}$/ || y ( t $^{)}$| | $_{2}$. (30)