Consensus in a heterogeneous influence network.
We consider a dynamical network model in which a number of agents all move on the plane with the same constant absolute velocity. At each time step, each agent's direction is updated as the average of its direction plus the directions of other agents who can influence it. The influencing capability of each agent is represented by its influencing radius, which is randomly chosen according to a power-law distribution with a scaling exponent between 2 and infinity. As the value of the scaling exponent decreases, the radius distribution becomes more heterogeneous and the network becomes much easier to achieve direction consensus among agents due to the leading roles played by a few hub agents. Furthermore, almost all agents will finally move in the same desired direction in a strong heterogeneous influence network, if and only if a small fraction of hub agents can be controlled to move in the desired direction. These results also reflect the "robust yet fragile" feature of a heterogeneous influence network.