Refined decay bounds on the entries of spectral projectors associated with sparse Hermitian matrices

Sensitivity of matrix function based network communicability measures: Computational methods and a priori bounds

“…We consider a simple unweighted, undirected example network taken from [19]; see also [10]. The network consists of 15 tine families in the 15th century, and 20 edges, representing marriages between the families. Assume we want to add five edges to the network with the goal to increase its communicability as much as possible.…”

“…It is well-known that the entries of matrix functions f (A) often exhibit an exponential or even super-exponential decay away from the sparsity pattern of A: The larger the geodesic distance d(i, j) of node i and j in the graph of A, the smaller the entry [f (A)] ij can be expected to be. This was first studied for the inverse of banded matrices in [27] and later extended to other functions and matrices with more general sparsity pattern in numerous works; see, e.g., [14,9,15,16,32,33,46,48,49] and the references therein; Specifically, in [47] such an approach was applied in the context of network modifications.…”