Data analysis of non-standard time series : the role of graph Laplacians and covariance matrices in data and processes of complex systems

In this report we provide an informal mathematical introduction to the use of some methods from graph theory for the data analysis of non-standard time series. By this we mean information from a channel that is not presented to us in the usual format as a regular sequence of numbers. In particular we are interested in the role of the Laplacian matrix operator in helping us to understand the underlying structure and meaning of information generated from a wide range of sources. Before introducing any mathematical methods, we provide a synopsis as to some of the concepts associated with complex networks that generate such non-standard data, hoping to set the background and motivation for the purpose of our work, namely the practical application of spectral graph analysis. Although we do not purport to present any new mathematical results we believe that we introduce two aspects of some novelty and value: in the first instance we present an alternative algebraic description of the Dirichlet sum over a graph which we have, serendipitously, discovered to be useful for interpreting spectral bounds of the Laplacian; on another theme we present a simple proof concerning the connection between the Laplacian of physical processes and covariance-based multivariate analysis. In order to demonstrate the versatility of the Laplacian matrix as a tool to understand complex networks, we provide some of our own examples based on data from written language, computer science and earthquake engineering.