INTERNATIONAL INSTITUTE OF INFORMATION TECHNOLOGY,

Two classes of networks that have been extensively studied in the dynamical analysis of spatially extended systems are: (i) regular networks, wherein each node interacts with a specified number of neighbouring nodes on geometrical lattices, and (ii) random networks, wherein each pair of nodes has a fixed probability of interacting with each other. Recently much attention is being focused on a class of network models that are neither strictly regular nor completely random, but are somewhere in between and exhibit properties of both. Such networks are called small-world networks. Examples for small-world networks are found to occur widely across biological (e.g., neural connection patterns), social (e.g., friendship network, co-authorship) and technological (e.g., the World Wide Web) domains. It has also been observed that a large number of real complex systems such as protein contact networks, protein-protein interaction networks, metabolic networks, exhibit power-law behaviour in their degree distribution, i.e., a few nodes have a very high degree. Such networks are referred to as scale-free networks. To model real physical and biological systems governed by dynamical processes, on such networks the nodes are defined by dynamical systems and the connection between them may be fixed or changing in response to the local dynamics of the nodes, e.g., neuronal excitations. Such networks are called ̳Dynamical Networks‘. Thus, from the point of view of dynamical systems, we have a coupled system of dynamical equations, and we are interested in controlling or manipulating the resulting global dynamics of the system emerging from the interactions between the local dynamics of the individual elements. We propose to use concepts from Graph theory to analyze the coupling structure and its influence in attaining global control of the resulting dynamics in the event of the system exhibiting complex chaotic or excitable dynamics. In an earlier study by Parekh et al, robust control (both global and local) of spatiotemporal dynamics on coupled map lattice (CML) models by applying a constant external perturbation or “pinning‖ has been shown. The advantage of this control approach is that neither any a priori knowledge of the system dynamics, such as stable or unstable fixed points and periodic orbits nor any modification/tracking of the system parameters/variables explicitly is required to suppress chaos. In this study we use the same control approach and analyse the role of connection topologies in achieving global control of the dynamics on small-world and scalefree networks. The analyses have been carried out for dynamics on the nodes governed by (a) logistic map (a one-dimensional nonlinear map exhibiting a wide variety of dynamical behaviour including chaos), and (b) neuron map (a two-dimensional nonlinear map modelling an excitable system, e.g., neuron). The two dynamical systems have been modelled on four different network topologies, viz., regular, random, small-world and scale-free. Since it is practically infeasible to pin all the nodes of the network, here we exploit the geometrical/topological properties of networks to choose nodes for exerting control. We observe that in the case of small-world networks, wherein the connection topology is predominantly local, with a few long range interactions, regularly-spaced controllers provide better control for low pinning densities ~ 20% compared to hubs and other centrality measures which require pinning ~ 50% nodes at similar pinning strengths. The long-range connections only seem to have the effect of spatial noise on small-world network. However, in case of controlling dynamics on the scale-free network, centrality nodes do provide significant advantage in achieving global control. We observe that high-betweenness and high-closeness nodes at very low pinning densities ~ 10 – 15% suffice in suppressing spatiotemporal chaos. Table of