On the asymptotics of consensus protocols

It is shown that the limiting state vector of the differential consensus seeking model with an arbitrary communication digraph is obtained by multiplying the eigenprojection of the Laplacian matrix of the model by the vector of the initial state. Furthermore, the eigenprojection coincides with the matrix of maximum out-forests of the weighted communication digraph. These statements make the forest consensus theorem. A similar result for DeGroot's iterative pooling model involves the Cesáro limit in the general case. The forest consensus theorem is useful for the analysis of distributed control models.