Network Analysis via Partial Spectral Factorization and Gauss Quadrature

Abstract

Abstract. Large-scale networks arise in many applications. It is often of interest to be able to identify the most important nodes of a network or to ascertain the ease of traveling between nodes. These and related quantities can be determined by evaluating expressions of the form u T f (A)w, where A is the adjacency matrix that represents the graph of the network, f is a nonlinear function, such as the exponential function, and u and w are vectors, for instance, axis vectors. This paper describes a novel technique for determining upper and lower bounds for expressions u T f (A)w when A is symmetric and bounds for many vectors u and w are desired. The bounds are computed by first evaluating a low-rank approximation of A, which is used to determine rough bounds for the desired quantities for all nodes. These rough bounds indicate for which vectors u and w more accurate bounds should be computed with the aid of Gauss-type quadrature rules. This hybrid approach is cheaper than only using Gauss-type rules to determine accurate upper and lower bounds in the common situation when it is not known a priori for which vectors u and w accurate bounds for u T f (A)w should be computed. Several computed examples, including an application to software engineering, illustrate the performance of the hybrid method.

New block quadrature rules for the approximation of matrix functions

“…The matrix Vfem is not an adjacency matrix for a network, but stems from the discretization of a partial differential problem in electromagnetics. [4,16] for undirected networks and in [2,5] for directed ones. In [17] block algorithms were developed for both cases.…”

Estimating the diagonal of matrix functions

“…Therefore, the most important node in a network can be determined by estimating the diagonal of f ( A ) and then computing the index of the largest element in the vector estimate of the diagonal. A hybrid approach that employs Gauss‐type quadrature rules is also developed in for the efficient determination of the most important nodes in a network. In Table , we determine the most important node of the test networks email , autobahn , internet , and collaborations that represent large real‐world networks , using the matrix resolvent with parameter a = 0.85/ λ 1 .…”

“…A hybrid approach that employs Gauss‐type quadrature rules is also developed in for the efficient determination of the most important nodes in a network. In Table , we determine the most important node of the test networks email , autobahn , internet , and collaborations that represent large real‐world networks , using the matrix resolvent with parameter a = 0.85/ λ 1 . In the second column of Table , we see the total number of nodes, that is, the dimension p of the corresponding adjacency matrix.…”

Block Gauss and Anti-Gauss Quadrature with Application to Networks

“…This method is applied to seven real-world undirected unweighted networks, some of which have been investigated numerically in [25]. The networks have the following properties:…”

“…Thus, also in the situation when only subgraph centralities are needed, the block method is competitive. In particular, it may be attractive to apply block Lanczos methods in the hybrid scheme for identifying the k nodes of a network with the largest subgraph centrality proposed in [25]. This scheme first computes a low-rank approximation of the adjacency matrix to determine a short list of candidate nodes.…”