Stationary waves in cyclic swarms

The authors discuss the design of private swarms, i.e., swarms which can carry out the external task without knowing it. They have developed a methodology for designing cyclic swarms by solving asynchronously systems of linear equations describing a difference equation with periodic boundary conditions. The solutions form a large class of functions which can be used, with appropriate normalizations and rescalings, as orthonormal basis functions for producing arbitrary distributions. The convergence of the protocol devised to solve asynchronously the difference equation corresponding to the stationary wave equation with periodic boundary conditions has been proved and verified, starting from arbitrarily highly ordered or highly disordered initial distributions. The main advantages of this self-organization is in the privacy of the process. No unit is ever aware of the distribution it is working to realize. For the same reason, nobody could, by simply observing the swarm externally, predict to what distribution it is tending. Applications to the realization of sensing swarms are presented