From Complex Networks to Time Series Analysis and Viceversa: Application to Metabolic Networks

In this work we present a simple and fast approach to generate network structures based on time series recurrence plots and viceversa. In addition, we discuss the application of the different analysis techniques developed in both fields, i.e. complex networks and time series analysis. Concerning the transformation from time series to networks, we propose a deterministic growth procedure which produces a new types of complex network structures that have some interesting features. This simple and fast approach is able to generate deterministic network structures based on time series recurrence plots. The generated networks contain several properties of the original time series. In this case, networks generated from chaotic attractors display interesting features from the point of view of robustness which could help in designing systems with high tolerance against errors and transfer of information. Chaotic networks based on the Lorenz attractor show that they are highly tolerant against attacks and they have a high ability for the transfer of information or on the contrary they are able to transmit infections faster. It is still necessary to investigate if such chaotic networks exist already in natural or man-made systems or, if possible, to construct such networks and test their properties. On the other hand, the transformation from networks to time series presents some problems concerning the selection of the initial time or in our case the initial node and the way in which the nodes are visited. If a network has been generated following a certain growth law it seems logical to choose the first node as the origin and then proceed following the network growth pattern. However, the situation is not so clear for example with metabolic networks, where it is difficult to select which is the first metabolite. Similar concerns would apply to other types of biological networks. In this case several alternatives could be considered, e.g. ordering using the number of connections. However, we have still to find if there are some invariant/preserved properties in the generated time series from the same network. We have found that rescaled range analysis does not preserve the fractal structure in the time series. In any case, if time series parameters would be invariant against the initial node selection, then they could be used to analyze the networks that have generated said time series. Our future work will continue along these lines. .