A Gradient-based Topology Design Algorithm Applied to Second-order Network for Pursuing Best Possible Synchronization*

As is well-known, synchronization acts as one of the most significant dynamic properties of interaction-induced networks, which is depicted by the ratio of the maximal and the second smallest eigenvalues of topology Laplacian matrix and thus could be optimized by tuning the topology of network. Second-order interactions involving three units have been prevalent to describe some real organization of social, biological systems, etc. correspondingly, pursuing the best possible synchronization of second-order network is converted into an optimization problem of second-order topology design. Although some traditional searching technologies, such as simulated annealing, have been adopted to find a candidate optimal second-order topology, the inherent defection on those algorithms, such as the well-known ease of falling into the slow convergence, limiting their application. In this paper, the gradient of eigenvalues of Laplacian matrix is described by a function of the associated eigenvector and the objective hyperlinks. By utilizing Taylor’s formula, a gradient-based topology optimization algorithm is designed to search for the minimum eigen-ratio to improve second-order network synchronization, which can also be extended to higher-order networks. Simulations demonstrate the effectiveness of the developed gradient-based method. It is observed that our algorithm could achieve an optimized second-order topology within less time on an order of magnitude compared to simulated annealing and thus can be applied to large-scale networks.