High-order Strictly Local Swarms

Abstract

In physical realizations of Swarms, the interactions are restricted to short range in order to avoid time-delays and complexity of connections. However, short-range interactions restrict the Swarm to low-order equations. In order to take full advantage of the possibility of producing patterns generated by higher order differential equations while maintaining physical locality of interaction, and hence guarantee no communication time delays, we have recast the model so that only nearest neighbor interactions are considered. The tradeoff is that each Swarm unit updates n variables (a multi-channel process) where n is the order of the differential equation. We show by simulation and experiments on systems of microprocessors that the multi-channel algorithm converges and that the rate of convergence is not slower than for the standard Swarm algorithm. Hence, strictly local high-order Swarms appear to be practically realizable.