Design of inner coupling matrix for robustly self-synchronizing networks

A self-synchronizing network may undergo change of scale and topology during its functioning, thus adjustment of parameters is necessary to enable the synchronization. The adjustment cost and runtime-break demand a method to maintain continuous operation of the network. To address these issues, this paper presents an analytical method for the design of the inner coupling matrix. The proposed method renders the synchronization robust to change of network scale and topology. It is usual in network models that scale and topology are represented by outer coupling matrix. In this paper we only consider diffusively coupled networks. For these networks, the eigenvalues of the outer coupling matrix are all non-positive. By utilizing this property, the designed inner coupling matrix can cover the entire left half of complex plane within the synchronized region to underlie robustness of synchronization. After elaborating the applicability of several types of synchronization state for a robustly self-synchronizing network, the analytical design method is given for the stable equilibrium point case. Sometimes the Jacobian matrix of the node dynamical equation may lead to an unrealizable complex inner coupling matrix in the method. We then introduce a lemma of matrix transformation to prevent this possibility. Additionally, we investigated the choice of inner coupling matrix to get a desirable self-synchronization speed. The corresponding condition in the design procedure is given to drive the network synchronization faster than convergence of each node. Finally, the article includes examples that show effectiveness and soundness of the method.