Influence of cumulative damage on synchronization of Kuramoto oscillators on networks

Abstract

In this paper, we study the synchronization of identical Kuramoto phase oscillators under cumulative stochastic damage to the edges of networks. We analyze the capacity of coupled oscillators to reach a coherent state from initial random phases. The process of synchronization is a global function performed by a system that gradually changes when the damage weakens individual connections of the network. We explore diverse structures characterized by different topologies. Among these are deterministic networks as a wheel or the lattice formed by the movements of the knight on a chess board, and random networks generated with the Erd\H{o}s-R\'enyi and Barab\'asi-Albert algorithms. In addition, we study the synchronization times of 109 non-isomorphic graphs with six nodes. The synchronization times and other introduced quantities are sensitive to the impact of damage, allowing us to measure the reduction of the capacity of synchronization and classify the effect of damage in the systems under study. This approach is general and paves the way for the exploration of the effect of damage accumulation in diverse dynamical processes in complex systems.