Executive Summary : | Typical study of complex networks includes modeling and dynamical analysis, but is often restricted to single isolated networks. However, in real situations networks are interdependent: they usually interact with each other to form network of networks. Power grids, communication and computer networks, for example, are known to be interdependent. Malfunction in one of them may lead to the failure of others, that in turn can result in collapse of the whole system. In biology, different cell modules (communities), or neural networks of various kinds work together to perform different functions including metabolism and cognition. Objectives: How do such networks having very different intrinsic properties maintain co-ordination for enabling smooth functioning of the entire system? Further, how are failure/defects in selected nodes responsible for the breakdown of the entire system? From a dynamical systems’ point of view, and in general, these are very interesting questions. In this proposed work, starting from the study of paradigmatic models forming network of networks, we wish to study its collective properties of the such networks. We plan to explore how the collective response of such inter-dependent networks depends on network/system parameters, conditions required to maintain its properties, and finally would like to generalize our analysis. Plan: Interdependent networks with time-delay: time-delay is inherent in couplings due to finite signal-speed, and it is crucial to determine network-dynamics. Hence to start with, we plan to consider modular networks composed of delay-coupled oscillating units such as phase, limit cycle or chaotic oscillators, such as Kuramoto and Landau–Stuart, or Lorenz models. Using analytical (such as network theory analysis) and numerical methods, we wish to understand its collective dynamics. We hope to extend this study in various directions using, for example, multiple delays, various coupling topologies and heterogeneities. Impact: Various empirical, numerical and analytical methods are applied to understand the organization, evolution, robustness and synchronizability of variety of networks. With inclusion of real-life factors, such as delay, coupling strategies and heterogeneities, we hope to understand the resulting dynamics of network of networks. A systematic study will help us get insight into interesting spatio-temporal behaviors observed in such complex systems. The two approaches to explore various properties of complex networks are, to examine connection-properties of the network to understand how they affect its characteristics, and to focus on the dynamics of the individual nodes to determine collective behavior of the ensemble. The interplay between the node dynamics and the network connections, however, is still poorly understood. Learning more about the role of connection-properties and node-dynamics interplay in the context of network of networks would be the long term goal of this project. |