Approximation of diagonal entries of matrix functions Sample Clauses

Approximation of diagonal entries of matrix functions. To compute the exponential (resolvent) subgraph centralities of the nodes in a network, only the diagonal entries of the matrix exponential (resolvent) are needed. Efficient, accurate methods for estimating (or, in some cases, bounding) arbitrary entries in a matrix function f (A) have been developed by Xxxxx, Meurant and collaborators (see [60] and references therein) and first applied to problems of network analysis by Benzi and Boito in [7]; see also [18]. They have been implemented in the Matlab toolbox mmq [77]. Here we provide a brief description of these methods. Additional details can be found in [7] and [60]. Let A be a real, symmetric, n × n matrix and let f be a function defined on the spectrum of A. Consider the eigendecomposition A = QΛQT and f (A) = Qf (Λ)QT , where Q = [q1, . . . , qn] and X = xxxx (x0, . . . , xx). For notational simplicity, here we assume that the eigenvalues of A are ordered as λ1 ≤ . . . ≤ λn. For given vectors u and v we have uT f (A)v = uT Qf (Λ)QT v = wT f (Λ)z = Σn k=1 f (λk)wkzk, (1.9) where w = QT u = (wk) and z = QT v = (zk). In particular, for f (A) = eA we obtain Σn uT eAv = k=1 eλk wkzk. (1.10) Choosing u = v = ei (the vector with the ith entry equal to 1 and all the remaining ones equal to 0) we obtain an expression for the subgraph centrality of node i: Σn SC(i) :=