Linear iterative strategies for transmitting streams of values through sensor networks

For any given fixed network of interconnected nodes, with some nodes designated as sources and some nodes designated as sinks, we investigate the problem of transmitting a stream of values from every source node to all of the sink nodes (possibly after some delay). We study linear iterative strategies for transmitting this information through the network, whereby at each time-step, each node in the network transmits a value that is a linear combination of the most recent transmissions of its neighbors. We show that this linear iterative strategy can be conveniently modeled as a linear dynamical system in state-space form. We then use techniques from control theory pertaining to dynamic system inversion and structured linear systems to show that each sink node can reconstruct the data streams if and only if there are node disjoint paths in the network from the set of all source nodes to each sink node. Furthermore, this reconstruction can be accomplished after a delay of at most N - |S| + 1 time-steps (where N is the number of nodes in the network, and |S| is the number of sources). This holds true for almost any choice of weights in the linear iteration.