Synchronized interval in random networks: The role of number of edges

We study the synchronized interval in undirected and unweighted random networks of coupled oscillators as a function of the number of edges. In many coupled oscillator systems, synchronization is stable in a finite interval of coupling parameter, which we define as the synchronized interval. We find in random networks, the width of the synchronized interval is maximum for an optimal number of edges. We derive analytically an estimation for this optimal value of the number of edges. In small networks, the analytical estimation deviates from the numerical results. However, in large networks the analytical and numerical results are in excellent agreement.