Efficiently reaching consensus on the largest entries of a vector

We consider a problem where agents gossip on a d-dimensional state vector. The goal is to achieve a consensus on the average. However, instead of computing the average of the entire d-dimensional state, the goal is to have all agents reach a consensus on the largest k entries of the average initial state vector. For example, the value in each entry could correspond to the agents' opinions about a different item, in which case the goal is to determine which are the k most popular items, on average. A primary challenge is that the indices of the k largest entries are not known a priori, and so the agents must adaptively identify which entries are the largest while also computing their values. We consider an asynchronous gossip-style algorithm where pairs of agents interact, communicate, and update only those state entries which either agent currently believes to be one of the largest k. We show that, as long as the underlying communication graph is connected, the algorithm converges to a state where all agents reach a consensus on the indices and values of the largest k entries of the initial average. We study, via numerical simulation, the convergence rate of the algorithm in terms of the total number of scalar values transmitted to reach a desired level of accuracy.