Asymptotic Stability of Switched Higher Order Laplacians

Recently, several properties in networked sensing and distributed systems have been modeled by various researchers [1,2,3,5,7,9,11,12] using topological spaces and their topological invariants. The unifying theme in these approaches has been that the local properties of a network, as dictated by local interactions among agents, can be captured by certain topological spaces. These spaces are mostly combinatorial in nature and are a generalization of the more familiar graphical models. Moreover, the global properties of the network characteristics correspond to certain topological invariants of these spaces such as genus, homology, homotopy, and the existence of simplicial maps. Examples of such modeling attempts include coverage problems for sensor networks [1,2,3,7]; consensus & concurrency modeling in asynchronous distributed systems [9]; and routing in networks without geographical information [5]. One notable characteristic of these studies has been that the topological abstractions preserve many global geometrical properties of the network while abstracting away the redundant geometrical details at small scales. This promises great simplification of algorithms as well as hardware, which is an important requirement for realizing large-scale robust networks.