Ali Jadbabaie

1. Workshop on computational worldview and sciences Caltech 03/15/07 Distributed Motion Coordination inNetworked Dynamic Systems: From Flocking and Synchronization to Coverage Verification Ali Jadbabaie Department of Electrical and Systems Engineering and GRASP Laboratory University of Pennsylvania

2. Nonlinear/uncertain hybrid/stochastic etc. Complex networked systems Single Agent Flocking Synchronization Consensus Multi-agent systems Networked dynamical systems Complexity of dynamics Complexity of interconnection

3. Nonlinear/uncertain hybrid/stochastic etc. Complex networked systems Single Agent Flocking/synchronization Consensus/ Coverage Multi-agent systems Networked dynamical systems Complexity of dynamics Complexity of interconnection

4. Flocking/synchronization consensus Multi-agent systems Nonlinear/uncertain hybrid/stochastic etc. Complex networked systems Complexity of dynamics Single Agent Complexity of interconnection Jadbabaie et al

5. Statistical Physics and emergence of collective behavior Simulations and conjectures but few “proofs’

6. Working systems but few “proofs’

7. r neighbors of agent i agent i Multi-agent setting: Vicsek’s kinematic model • How can a group of moving agents collectively decide on direction, based on nearest neighbor interaction? How does global behavior emerge from local interactions?

8. Distributed consensus algorithm for kinematic agents = speed MAIN QUESTION :Under what conditions do all headings converge to the same value and agents reach a consensus on where to go? = heading For small angles For angles in (-/2,/2), the nonlinear problem becomes linear with a coordinate change!

9. switching signal , adjacency matrix finite set of indices corresponding to allgraphs overnvertices. Valence matrix The underlying proximity graph • We use graphs to represent neighboring relations • vertices: • edges: 4 3 5 2 6 1

10. Conditions for reaching consensus Theorem (Jadbabaie et al. 2003): If there is a sequence of bounded, non-overlapping time intervals Tk, such that over any interval of length Tk, the network of agentsis “jointly connected ”, then all agents will reach consensus on their velocity vectors. This happens to be both necessary and sufficient for exponential coordination, boundedness of intervals not required for asymptotic coordination. (see Moreau ’04)

11. History of consensus Algorithms • Consensus based on repeated “local” averaging • Estimation-modification process among a group of experts in a fixed network: average your “opinion” with that of your neighbors” Degroot’74 • Update opinions via a nonhomogeneous Markov chain Chaterjee and Seneta’74. Based on work of Hajnal’58, Sarymsakov’61, Paz’71. • Consensus among multi processors in concurrency theory Lynch, Herlihy, Shostak ,… • Consensus algorithms as examples of asynchronous distributed gradient methods Tsitsiklis’84, Tsitsiklis et al.’86 • In control theory literature, in the context of velocity alignment among kinematic agents Jadbabaie et al.’03,

12. is a randomswitching signal Theorem: necessary and sufficient condition for almost sure convergence is ,i.e., the average system needs to reach deterministic agreement. Consensus in random networks Theorem : Assuming graphs are randomly chosen and independent, reaching consensus is a trivial event, i.e., either it happens almost surely or almost never, i.e., it satisfies the Kolmogorov 0-1 law.

13. The Laplacian of the graph 1 B is the (n x e) incidence matrix of graph G. • The graph Laplacian (n x n) encodes structural properties of the graph • Some properties of the Laplacian: • It is positive semi-definite • The multiplicity of the zero eigenvalue is the number of connected components • The kernel (for connected graph) is the span of vector of ones, • First nonzero eigenvalue is called algebraic connectivity. • Its corresponding eigenvector, called the Fiedler vector. Its sign paper encodes a lot of information about “bottlenecks” and “cutpoints” 2 3 4 W is diagonal

14. Consensus in Continuous time As before, (t) is a piecewise constant switching signal The model is now a hybrid or switching dynamical system Need to assume a dwell time on each graph to avoid complications The result is virtually the same, as exponentials of Laplacians are stochastic matrices

15. Vicsek’s Model with Periodic Boundary Condition • Autonomous agents with constant speed and adjustable headings • Local interaction rule: A consensus algorithm if the graphs are jointly connected over time • Periodic boundary conditions (Motion over a flattened unit torus) to avoid boundary effects. • Location on a torus can be seen as the fractional part of the location of an agent moving in an infinite plane. • Reasonable assumption for statistical physics simulations, for large populations • Vicsek’s simulations: velocity alignment without any assumption on connectivity.

16. Kronecker’s Theorem, Weyl’s Theorem and Number Theory • Kronecker’s Theorem: The numbers are dense in the unit interval if is an irrational number. • Weyl’s Theorem: If the sequence grows fast enough, then for almost every the numbers are equidistributed in the unit interval . • Equidistributedness implies denseness and more! • Weyl’s theorem can be generalized to higher dimensions.

17. Flocking, artificial life, and computer graphics • Reynolds [Reynolds 87] named the autonomous systems that behave like members of animal groups boids (bird + oids) • He developed a descriptive model for flocking behavior based on the combined action of alignment and cohesion-separation forces • alignment: steer towards the average heading of flockmates • separation: steer to avoid crowding flockmates • cohesion: steer towards the average position of flockmates

18. Distributed coordination with dynamic models: Flocking with collision avoidance • Double integrator model • Neighbors of i distance dependent: • Cohesion/Separation • Alignment For dynamic models, Proximity graph Connectivity implies emergence of Collective motion (Tanner, Jadbabaie, Pappas)

19. Flock artificial potential energy • Each Vij is required to be: • increasing as • unbounded as • unique minimum • The potential energy Vi of boid i depends on its neighbors Example (Ogren et al ’04)

20. Dynamic Topology Local sensing/communication Graph changes with time Unreliable communication, no dwell time Control is discontinuous Non smooth Lyapunov theory Fixed Topology Fixed (logical) network Graph is constant Control is smooth Classic Lyapunov theory Topology dictates analysis

21. Enforcing connectivity Treat loss of connectivity as an obstacle Zavlanos, Jadbabaie, Pappas ’07 Everything else is the same!

22. Synchronization Fireflies Flashing

23. KuramotoModel All-to-all interaction Model for pacemaker cells in the heart and nervous system, collective synchronization of pancreatic beta cells, synchronously flashing fire flies, rhythmic applause, gait generation for bipedal robots, … Benchmark problem in physics Not very well understood over arbitrary networks Introduced by Kuramoto in 1975 as a toy model of synchronization

24. Kuramoto model & graph topology 2 1 3 6 4 5 B is the incidence matrix of the graph

25. Frequency and phase locking Frequency entrainment and phase locking as long as graph is connected. Proof same as the continuous-time agreement model Frequency entrainment possible. Phase stability, but no phase locking.

26. Kuramoto model, dual decomposition, and nonlinear utility minimization Minimize the misalignment Kuramoto model is the just a gradient algorithm for minimization of a global utility which measures misalignment between phasors (exactly like TCP!)

27. Phase locking in switching networks with heterogeneous delays • When the natural frequencies are identical, can switch to a rotating frame, and set i to 0. • Theorem (Papachristodoulou and Jadbabaie’06) If f is locally passive (e.g., a sine function), and the switching sequence is such that the union of graphs across uniformly bounded intervals contains a globally reachable node infinitely often, then the synchronized set is asymptotically attracting, so long as the delays are finite.

28. Beyond Graphs in Networked Systems • Main Idea: understanding global properties with local information: algebraic topology • For certain problems, e.g. coverage, makes sense to go beyond graphs and pair-wise interactions • Example: Given a set of sensor nodes in a given domain (possibly bounded by a fence), is every point of the domain under surveillance by at least one node?

29. Coverage Problems • Problem: Given a set of sensor nodes in a domain (possibly bounded by a fence), is every point of the domain under surveillance by at least one node? Figure from De Silva and Ghrist ’06

30. From Graphs to Simplicial Complexes • Simplicial Complex: A finite collection of simplices • Simplex: Given V, an unordered non-repeating subset • k-simplex: The number of points is k+1 • Faces: All (k-1)-simplices in the k-simplex • Orientation

31. From Graphs to Simplicial Complexes • Simplicial complex: made up of simplices of several dimensions • Properties • Whenever a simplex lies in the collection then so does each of its faces • Whenever two simplices intersect, they do so in a common face. • Valid Examples • Graphs • Triangulations Invalid examples

32. Rips-Vietoris Simplicial Complex • 0-simplices : Nodes • 1-simplices : Edges • 2-simplices: A triangle in the connectivity graph ~ 2-simplex (Fill in with a face) • K-simplices: a complete subgraph on k+1 vertices • k-simplex in the Rips complex ~ (k+1) points within communication range of each other Generalization of r-disk graphs

33. Coverage Problems • Intersection of sensing ~ simplicial complexes • Communication graphs ~ simplicial complexes • Holes ~ homology of simplicial complexes • A sensor network has coverage hole if there is a “robust” hole in the simplicial complexes induced by the communication graphs [Ghrist et al.]

34. Rips and Čech Complexes: Topological vs. Geometric information A set of points • (Rips complex of radius ):k-simplex, if the pairwise distance between k points are less than . - Easy to compute in a dsitributed manner. - However, does not preserve the topological properties. • (Čech complex of radius ): k-simplex, if k coverage disks of radius overlap - Hard to compute. - Preserves the topological properties.

35. A Čech complex can be bounded by two Rips complexes: Ghrist and Muhammad’05 Use two communication power levels , Check for holes in by Finding generators for homology groups. - A sufficient condition for coverage holes If a hole exists in , then check for holes in - A necessary condition. There is a gapbetween the necessary and the sufficient conditions. Topological information from Rips complex

36. Boundary Maps: Generalization of Incidence matrices • : the vector space whose basis is the set of oriented k-simplices of X • The boundary map is the linear transformation • k-cycles: • k-boundaries: • Note: Homology groups : Hk(X) = Zk (X) / Bk (X)

37. Relevance of Homology • dim H0(X) ~ no of connected components of X • dim H1(X) ~ types of loops in X that surround ‘punctures’ • dim Hk(X) ~ no of k+1-dimensional ‘voids’ in X • Available software • Plex (Stanford) • CHomP (Georgia Tech)

38. Combinatorial k-Laplacians • Since X is finite we can represent the boundary maps in matrix form • Moreover, we can get the adjoint • [Eckmann 1945] The Combinatorial k-Laplacian is given by • Note: incidence matrix

39. k-Laplacian at the Simplex Level • Adjacency of a simplex to other simplices • Upper adjacency if they share a higher simplex (e.g. 2 nodes connected by an edge) • Lower adjacency if they share a common lower simplex (e.g. two edges share a node) • ‘Local’ formula with orientations

40. k-Laplacian at the Simplex Level • Adjacency of a simplex to other simplices • Upper adjacency if they share a higher simplex (e.g. 2 nodes connected by an edge) • Lower adjacency if they share a common lower simplex (e.g. two edges share a node) • ‘Local’ formula with orientations Hodge theory, 1940’s: Kernel of the Laplacian ~ cohomologies

41. Laplacian Flows • Laplacian flows : a semi-stable dynamical system (Recall heat equation for k = 0) • [Muhammad-Egerstedt MTNS’06] System is asymptotically stable if and only if rank(Hk(X)) = 0. • A method to detect ‘no holes’ locally

42. Laplacian Flows (contd.) • System converges to the unique harmonic cycle if rank(Hk(X)) = 1. • A method to detect ‘proximity to hole’ locally when single hole • When rank(Hk(X)) > 1 : System converges to the span of harmonic homology cycles

43. Decentralized Computation of Homology • Compute eigenvector decomposition of the k-Laplacian • All eigenvalues are non-negative • Eigenvectors corresponding to zero eigenvalues ~ harmonic homology classes • ‘Small’ positive eigenvalues ~ near-harmonic • There is a recent decentralized algorithm for eigenvectors computation. Each entry is computed only based on information from neighboring nodes. • Use the distributed Algorithm for spectral analysis with a complexity of , where mix is the mixing rate of the random walk on the network. Kempe and McSherry, 2004

44. Decentralized computation • Step 1 : Build the simplicial (Rips) complex • Computations take place at node level • Need protocols at the node level for • Simplex membership : What simplices are a node part of ?

45. Decentralized computation • Step 1 : Build the simplicial (Rips) complex • Computations take place at node level • Need protocols at the node level for • Simplex membership : What simplices are a node part of ?

46. Decentralized computation • Step 1 : Build the simplicial (Rips) complex • Computations take place at node level • Need protocols at the node level for • Simplex membership : What simplices are a node part of ?

47. Decentralized computation • Step 1 : Build the simplicial (Rips) complex • Computations take place at node level • Need protocols at the node level for • Simplex membership : What simplices are a node part of ? • Simplex owner-ship : What simplices (and therefore a subcomplex) are owned by a node? • e.g. the node with smallest label gets the simplex Owned sub-complexes have trivial homology in all dimension

48. Example, eigenvectors of 1 Network 1st homology class 2nd homology class ‘Fiedler-like’- eigenvector

49. Consensus in Switching Graphs • Mobility, switching graphs and consensus : switched linear system • Joint connectedness (Jadbabaie’ 2003) • Theorem : Consensus if and only if there is a sequence of bounded, non-overlapping time intervals, such that over any interval, the network of agents is jointly connected.

50. Coverage in Switching Simplicial Complexes • Can we repeat similar analysis for switching simplicial complexes? YES! • Jointly `hole-free’ simplicial complexes • Joint hole-free implies trivial homology in union complex • Theorem (Muhammad, Jadbabaie ’06): Switched linear system is globally asymptotically stable if and only if there exists an infinite sequence of bounded intervals, across each of which the simplicial complexes encountered are jointly hole-free.