Analysing region of attraction of load balancing on complex network

Many complex engineering systems network together functional elements to balance demand spikes but suffer from stability issues due to cascades. The research challenge is to prove the stability conditions for any arbitrarily large and dynamic network topology with any complex balancing function. Most current analyses linearize the system around fixed equilibrium solutions. This approach is insufficient for dynamic networks with multiple equilibria, for example, with different initial conditions or perturbations. Region of attraction (ROA) estimation is needed in order to ensure that the desirable equilibria are reached. This is challenging because a networked system of non-linear dynamics requires compression to obtain a tractable ROA analysis. Here, we employ master stability-inspired method to reveal that the extreme eigenvalues of the Laplacian are explicitly linked to the ROA. This novel relationship between the ROA and the largest eigenvalue in turn provides a pathway to augmenting the network structure to improve stability. We demonstrate using a case study on how the network with multiple equilibria can be optimized to ensure stability.