Extracting Connectivity of Bidirectional Networks from Dynamics

The study of networks has become increasingly important in many disciplines ranging from physics to biology. A network consists of many elementary components which interact with one another. The elementary components are known as the nodes while the interactions among the nodes are the links or connections of the network. In order to understand the function and properties of a network, a crucial first step is to determine how the nodes are linked or connected to each other. It is often difficult to directly measure or observe the connectivity between nodes. Thus a method that infers the connectivity of an unknown network from measurements would be highly valuable. Many methods have been proposed to address this difficult inverse problem of network reconstruction. One approach is to model networks by dynamical systems with the time evolution of the nodes given by ordinary differential equations. We have recently developed a method that reconstructs the connectivity of a network using only the measured time series of the nodes. Earlier studies required additional information such as the coupling strength or the response of the network to external perturbations, which are difficult to obtain in realistic situations. Our method is based on a noise-induced relation between the network connectivity and the dynamical covariance of the nodes, and is applicable for networks in which the coupling between nodes is bidirectional. In realistic networks, the strength of interaction among nodes often varies from node to node but most network reconstruction studies focus on the simpler case of uniform coupling strength. For weighted networks with non-uniform coupling strength, it is especially hard to predict the relative strength or weight of the links. Our method can also infer the normalized coupling strength of the links for weighted networks. In this thesis, we shall present our method and demonstrate that it gives satisfactory reconstruction of the connectivity and the normalized coupling strength for different types of networks with both linear and nonlinear dynamics.