  Performance of standard processes over large distributed networks typically
scales with the size of the network. For example, in planar topologies where
nodes communicate with their natural neighbors, the scaling factor is $O(n)$,
where $n$ is the number of nodes. As the size of the network increases, this
makes global convergence over the entire network less practical. On the other
hand, for several applications, such as load balancing or detection of local
events, global convergence may not be necessary or even relevant. We introduce
simple distributed iterative processes which limit the scope of the
computational task to a smaller subnetwork of the entire network. This is
achieved using one additional local parameter which controls the size of the
subnetwork. We establish termination and convergence rate of such processes in
theory, in experiment, in comparison to the well understood behavior of Markov
processes, and for a variety of network topologies and initial conditions.
