Adaptation and Percolation in Complex Networks

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Many important dynamical processes take place on networks. Examples include epidemic propagation, genetic regulation, synchronization, information propagation, and many more. Often, these dynamical processes have a modifying effect on the network structure. This project will study the bidirectional interaction of network structure and network processes. As an important and representative case, the synchronization of network-coupled dynamical systems will be studied when network links and oscillator parameters adapt in response to node dynamics. Network and parameter adaptation will be investigated numerically and analytically by developing averaged equations that describe the evolution of the network and oscillator parameters. Possible network fixed points, bifurcations, and attractors in low-dimensional subsets of the space of networks will be studied as a function of various network measures and adaption rules. In a related project, percolation in non-Markovian networks will be studied. The effect of network structure on the percolation threshold has been studied for Markovian networks and for locally tree-like networks. The validity of the Markovian assumption will be tested for various networks arising in applications. Additionally, a way to determine the percolation threshold in non-Markovian networks that are not tree-like will be sought.Network percolation is related to models of epidemic propagation, the propagation of information in a network, or the robustness of networks under attack or random failures. For example, in the epidemic context, the percolation threshold separates networks in which a disease will die out from those in which it will propagate to infect a significant fraction of the population. Our theoretical understanding of how network structure (for example, how people interact with each other during an epidemic) determines this transition is restricted to networks satisfying specific conditions. One of the goals of this project is to directly test whether or not networks found in practice, such as social networks, satisfy them. In addition, existing theoretical tools will be extended to networks that do not satisfy these conditions. Many processes that can be described in terms of networks, such as communication networks of unmanned aerial vehicles, food-chain networks, and neuron networks, do not take place on a static network, but on a network that changes in response to node dynamics. The other part of the project seeks to increase our understanding of how networks change in response to the processes that they mediate, and how they can be described as dynamical objects.