Control contribution identifies top driver nodes in complex networks

Abstract: We propose a new measure to quantify the impact of a node $i$ in controlling a directed network. This measure, called `control contribution' $\mathcal{C}{i}$, combines the probability for node $i$ to appear in a set of driver nodes and the probability for other nodes to be controlled by $i$. To calculate $\mathcal{C}{i}$, we propose an optimization method based on random samples of minimum sets of drivers. Using real-world and synthetic networks, we find very broad distributions of $C_{i}$. Ranking nodes according to their $C_{i}$ values allows us to identify the top driver nodes that control most of the network. We show that this ranking is superior to rankings based on control capacity or control range. We find that control contribution indeed contains new information that cannot be traced back to degree, control capacity or control range of a node.