Percolation-induced explosive synchronization in pinning control

Percolation and synchronization are two paradigmatic examples of phase transitions that can take place in network dynamical systems. In this paper, we show how percolation can induce an explosive, first-order transition to synchronization in pinning control, a classical feedback strategy that exploits the network topology to drive the network dynamics to a desired synchronous solution. When the number of control signals is limited by physical or economical constraints, only a fraction of the network nodes can be effectively synchronized onto the desired trajectory. Determining the sensitivity of this fraction to the number of pinning control signals is then key to decide if it is worth adding additional control signals. We find that when the network graph percolates, that is, it becomes endowed with a giant strongly connected component (GSCC), an explosive transition to synchronization occurs as we increase the fraction of nodes where we can inject the pinning signals. Motivated by this numerical observation, we exploit the probabilistic conditions that ensure the presence of a GSCC to predict the number of pinning signals such that all of its nodes will converge to the desired synchronous trajectory. We test the validity and robustness of our analytical derivations through numerical simulations on both synthetic and real networks, proving the benefit of such analysis in supporting decision-making for control design.