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 ; and routing in networks without geographical information . 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.
KeywordsSensor Network Asymptotic Stability Simplicial Complex Homology Group Topological Invariant
Unable to display preview. Download preview PDF.
- 1.de Silva, V., Ghrist, R.: Coordinate-free coverage in sensor networks with controlled boundaries via homology. Intl. J. Robotics Research, to appearGoogle Scholar
- 2.de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Algebraic and Geometric Topology (To appear)Google Scholar
- 3.de Silva, V., Ghrist, R., Muhammad, A.: Blind Swarms for Coverage in 2-D. In: Robotics: Science and Systems, MIT, Cambridge, MA, June 8-11 (2005)Google Scholar
- 5.Fang, Q., et al.: Glider: gradient landmark-based distributed routing for sensor networks. In: Proc. IEEE Infocom, IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
- 6.Gage, D.: Command control for many-robot systems. In: Nineteenth Annual AUVS Technical Symposium, Huntsville, Alabama, USA, pp. 22–24 (1992)Google Scholar
- 7.Ghrist, R., Muhammad, A.: Coverage and Hole-Detection in Sensor Networks via Homology. In: The Fourth International Conference on Information Processing in Sensor Networks (IPSN’05), UCLA, Los Angeles, CA, April 25-27 (2005)Google Scholar
- 11.Muhammad, A., Egerstedt, M.: Control Using Higher Order Laplacians in Network Topologies. In: Proc. of 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan (2006)Google Scholar
- 12.Muhammad, A., Jadbabaie, A.: Decentralized Computation of Homology Groups in Networks by Gossip. American Controls Conference (Submitted) (2007)Google Scholar
- 13.Muhammad, A., Jadbabaie, A.: From Consensus in Switching Graphs to Coverage in Switching Simplicial Complexes. University of Pennsylvania Technical Report (2006)Google Scholar