Control Using Higher Order Laplacians in Network Topologies

This paper establishes the proper notation and precise inte rpre ation for Laplacian flows on simplicial complexes. In particular, we have shown how to interpret these flows as ti me-varying discrete differential forms that converge to harmonic forms. The stability properties of the correspo nding dynamical system are shown to be related to the topological structure of the underlying simplicial comple x. Finally, we discuss the relevance of these results in the context of networked control and sensing. I. I NTRODUCTION AND MOTIVATION In this paper, we introduce the concept of higher order combi natorial Laplacians of simplicial complexes within the context of networked sensing and control. During the last fe w y ars, the graph Laplacian has been used as a powerful tool that lets the network topology be incorporated directl y into the dynamics of a networked dynamical system, e.g. [14], [16], [17], [19], [20]. Using tools from algebraic toplogy, this paper presents a generalization of these ideas through a generalization of the graph Laplacian. The higher order Laplacians do not exist on graphs, but on higher order combinatorial structures called as simplicial complexes. In particular we will study the family of dynamical systems ∂ω ∂t = −∆kω, k ≥ 0, where∆k is thek-th order Laplacian operator, and w in an element of the exterior algebra of certain vector space s on a simplicial complex. The motivation for studying this dynamical system comes fro m some recent work [4], [10], [12] in which researchers have characterized various properties of a networked syste m uch as coverage and routing by certain topological proper ties of the network. These research efforts have a strong resonan ce with programs in computational algebraic topology, disc rete differential geometry and discrete geometric mechanics [5 ], [6], [11]. The null spaces of the higher order Laplacians have some very deep connections with the computation of topo logical invariants. As these null-spaces correspond to the equilibrium points of the above-mentioned family of dyn amical systems, the computation of topological invariants is related to the convergence properties of these dynamical systems. The approach taken in this paper is desirable for related research questions in networked sensing and contro l as it indicates towards devising more quantitative method s and implementation using distributed computation. Since we use certain concepts from algebraic topology that a re not familiar in the systems and networks community, we elaborate on the precise mathematical interpretation of th is dynamical system. Specifically, we summarize the propert ies of simplicial complexes, their homology & cohomology group s and their relation to harmonic analysis via combinatorial Hodge theory. Using this language we interpret the higher or der Laplacians as operators on discrete differential forms. We also make the connection between the theory of differenti al forms for differential manifolds and the calculus for discrete forms. We will see that this connection provides a p owerful interpretation that lets transport the more famili ar ideas in the continuous setting of differential manifolds t o the discrete setting of simplicial complexes. We also insi st that the theory of combinatorial Laplacians exist independentl y of the continuous setting and should not always be thought as anapproximation. This paper is organized as follows. We outline some basic con cepts of simplicial complexes and their topological invariants in Section II. In Section III we introduce the rea d r to basic Hodge theory. Next, we work out a detailed This work was sponsored by the US Army Research Office through the grant #99838. Abubakr Muhammad is with the Department of Electrical & Syst ems Engineering, University of Pennsylvania, Philadelphi a, PA 19104, USA abubakr@seas.upenn. edu Magnus Egerstedt is with the Department of Electrical & Comp uter Engineering, Georgia Institute of Technology, Atlant a, GA 30332, USA magnus@ece.gatech. edu example to elaborate on these concepts. We conclude this sec t on by giving an interpretation of these results in light of the more familiar theory of differential forms on Riemannia n manifolds. This mathematical machinery is used in Section IV to study Laplacian flows on simplicial complexes. We concl ude by summarizing our observations in Section V. II. SIMPLICIAL COMPLEXES AND THEIRTOPOLOGICAL INVARIANTS We first need to introduce some basic tools from algebraic top ology for generalizing the concepts of algebraic graph theory that will subsequently be used for designing control networked algorithms. (For a thorough treatment of this subject, see for example [2], [13], [18].) A. Simplicial Complexes Graphs can be generalized to more expressive combinatorial objects known as a simplicial complexes. Simplicial complexes are a class of topological spaces that are made of s implices of various dimensions. Given a set of points V , a k-simplex is an unordered subset {v0, v1, . , vk} wherevi ∈ V and vi 6= vj for all i 6= j. The faces of this k-simplex consist of all (k− 1)-simplices of the form{v0, . , vi−1, vi+1, . , vk} for 0 ≤ i ≤ k. A simplicial complex is a collection of simplices which is closed with respect to t he inclusion of faces. Graphs are a concrete example, where the vertices of the graph correspond to V and edges correspond to 1-simplices. The orderings of the ve rtices correspond to anorientation. A k-simplex{v0, . , vk} together with an order type is an oriented k-simplex denoted by [v0, . , vk] (see Figure 1 for typical examples of simplices in dimension zero through three), where a change in orientation corresponds to a change in the sign of the coeffic ient: [v0, . , vj , . , vi, . , vk] = −[v0, . , vi, . , vj , . , vk]. 0Simplex [v0] 1Simplex [v0, v1] 2Simplex [v0, v1, v2] 3Simplex [v0, v1, v2, v3] v0 v0 v1