Laplacian-Based Consensus on Spatial Computers

Abstract

Robotic swarms, like all spatial computers, are a challenging environment for the execution of distributed consensus algorithms due to their scale, diameter, and frequent failures. Exact consensus is generally impractical on spatial computers, so we consider approximate consensus algorithms. In this paper, we show that the family of self-organizing protocols based on the graph Laplacian of a network[19] are impractical as well. With respect to the structure of a finiteneighborhood spatial computer, we find that these protocols have an expected convergence time of O(diameter 2) when the inputs are strongly correlated with location. Verifying this result in simulation, we further determine that the constant factor on the convergence time is high, rendering Laplacian-based approximate consensus unsuitable for general use on spatial computers.