Abstract
Various kinds of fluctuations can occur in different natural or artificial networks and it can easily endanger the stability of the system. In this thesis, we investigate ways of enhancing network stability against fluctuations in the network flow model and the Kuramoto model. For the network flow model, we introduce the message passing algorithm and discrete Green's function to elucidate how resource fluctuations determine flow fluctuations in a network optimizing a global cost function. To strengthen the robustness of the network against fluctuations, we develop the schemes of optimal bandwidth allocation in links and optimal resource adjustment in nodes. The changes of loads and currents prescribed by the optimal bandwidth allocation and resource adjustment schemes are correlated with each other, except for some nodes that exhibit relay effects. To examine the synchronization stability in the network, we study the linear stability of the Kuramoto model using state dependent algebraic connectivity. Similarly, we propose an optimal resource adjustment scheme to improve the stability against probabilistic uncertainties of the natural frequencies. In addition, we introduce a vulnerability measure of individual nodes to discuss how the topology of the network affects the synchronization stability.