Paper Title
Krylov Methods for Complex Network Analysis

Abstract
In the recent years an extended attention is directed upon the properties of networks and more precisely, to identify the ‘central’ nodes, which are obtained from the matrix functions of the respective adjacency matrix and its invariants; the diagonal entries of the exponential of the adjacency matrix for networks are used as a centrality measure, to measure the subgraph centrality, Estrada Index; which play an important role in the study of network analysis at the macroscopic level for computing the natural connectivity, the total graph energy etc.The analysis of complex networks as the communicability between two distinct nodes, subgraph centrality, spectral gap and ranking nodes according to their importance for large real networks can be quite expensive. The matrix functions have been studied well from an algebraic point of view, but when it comes to real network, the matrix dimensions increase extremely, which leads into usage of efficient methods for computing the exponential of large and rare matrices, and Krylov subspace methods are a powerful mechanism in this content, matrix-free methods that use only matrix-vector products. In this work we use a variant of Laczos method combined with Grand-Schmidt orthogonalization to compute them, especially in case the problem is ill-conditioned. In order to compare the efficiency of the most used methods and the advantage of our proposition we have performed several numerical tests with real social and biological networks, with size that vary from hundreds to thousands. Keywords - Krylov Subspace, Matrix Function, Real-Life Networks, Sparse Matrix.