  Consensus strategies find a variety of applications in distributed
coordination and decision making in multi-agent systems. In particular, average
consensus plays a key role in a number of applications and is closely
associated with two classes of digraphs, weight-balanced (for continuous-time
systems) and bistochastic (for discrete-time systems). A weighted digraph is
called balanced if, for each node, the sum of the weights of the edges outgoing
from that node is equal to the sum of the weights of the edges incoming to that
node. In addition, a weight-balanced digraph is bistochastic if all weights are
nonnegative and, for each node, the sum of weights of edges incoming to that
node and the sum of the weights of edges out-going from that node is unity;
this implies that the corresponding weight matrix is column and row stochastic
(i.e., doubly stochastic). We propose two distributed algorithms: one solves
the weight-balance problem and the other solves the bistochastic matrix
formation problem for a distributed system whose components (nodes) can
exchange information via interconnection links (edges) that form an arbitrary,
possibly directed, strongly connected communication topology (digraph). Both
distributed algorithms achieve their goals asymptotically and operate
iteratively by having each node adapt the (nonnegative) weights on its outgoing
edges based on the weights of its incoming links (i.e., based on purely local
information). We also provide examples to illustrate the operation,
performance, and potential advantages of the proposed algorithms.
