Distributed balancing in digraphs under interval constraints

We consider networks the nodes of which are interconnected via directed edges, each able to admit a flow within a certain interval, with nonnegative end points that correspond to lower and upper flow limits. The paper proposes and analyzes a distributed algorithm for obtaining admissable and balanced flows, i.e., flows that are within the given intervals at each edge and are balanced (the total in-flow equals the total out-flow) at each node. The algorithm can also be viewed as a distributed method for obtaining a set of weights that balance a digraph for the case when there are upper and lower limit constraints on the edge weights. The proposed iterative algorithm assumes that communication among pairs of nodes that are interconnected is bidirectional (i.e., the communication topology is captured by the undirected graph that corresponds to the network digraph), and allows the nodes to asymptotically (with geometric rate) reach a set of balanced feasible flows, as long as the circulation conditions on the given digraph, with the given flow/weight interval constraints on each edge, are satisfied.