Detection of Influential Nodes in Network Dynamical Systems from Time Series

Identifying influential nodes in network dynamical systems requires the manipulation of topological and dynamic characteristics within ideal experiments. However, seldom we have access to experimental settings that could afford targeted interventions or to calibrated mathematical models that could support faithful what/if analyses. Our knowledge of the network dynamical system is often limited to the time series of individual nodes in some real experiments. Using these time series, it is possible to undertake a number of inference tasks, from reconstructing the topology of the network to discovering hidden nodes. Whether time series of real experiments could help pinpoint causal influence within the network is an open question. Here, we address this question in the context of synchronization problems, where the influence of a node is defined as the extent to which adding noise at that particular node affects the overall synchronization of the entire network. For linear time-invariant dynamics and undirected topologies, we demonstrate the possibility of exactly detecting the most influential nodes in the network without a calibrated mathematical model, using only time series of a real experiment, where all nodes are plagued by noise. Beyond illustrating our results on classical and second-order consensus protocols, we consider two real-world datasets: 1) 1) firearm prevalence in the USA and 2) players' movements in a soccer game. Just as our conclusions support the emergence of influential states, which have a less stringent legal environment, they hint at the instrumental role of players, who are critical to the offense strategy of the team.