Measuring the importance of individual units in producing the collective behavior of a complex network

A quantitative evaluation of the contribution of individual units in producing the collective behavior of a complex network can allow us to understand the potential damage to the structure integrity due to the failure of local nodes. Given a time series for each unit, a natural way to do this is to find the information flowing from the unit of concern to the rest of the network. In this study, we show that this flow can be rigorously derived in the setting of a continuous-time dynamical system. With a linear assumption, a maximum likelihood estimator can be obtained, allowing us to estimate it in an easy way. As expected, this "cumulative information flow" does not equal the sum of the information flows to other individual units, reflecting the collective phenomenon that a group is not the addition of individual members. For the purpose of demonstration and validation, we have examined a network made of Stuart-Landau oscillators. Depending on the topology, the computed information flow may differ. In some situations, the most crucial nodes for the network are not the hubs, i.e., nodes with high degrees; they may have low degrees and, if depressed or attacked, will cause the failure of the entire network. This study can help diagnose neural network problems, control epidemic diseases, trace city traffic bottlenecks, identify the potential cause of power grid failure (e.g., the 2003 great power outage that darkened much of North America), build robust computer networks, and so forth.