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'' $_(i)$, 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 $_(i)$, 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 can control most of the network. We show that this ranking is superior to rankings based on other control-based measures. We find that control contribution indeed contains new information that cannot be traced back to degree, control capacity or control range of a node.