# Consensus on nonlinear spaces and Graph coloring

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1 Consensus on nonlinear spaces and Graph coloring Alain Sarlette Ghent University, Belgium

2 What temperature should we have in this room? Synchronization = consensus in a vector space move towards your neighbors

3 What temperature should we have in this room? Synchronization = consensus in a vector space move towards your neighbors

4 What temperature should we have in this room? Synchronization = consensus in a vector space move towards your neighbors

5 What temperature should we have in this room? Synchronization = consensus in a vector space move towards your neighbors

6 What temperature should we have in this room? Synchronization = consensus in a vector space move towards your neighbors

7 Where is the center of the world? Synchronization on a sphere

8 Where is the center of the world? Synchronization on a sphere Where is the mean position? How do agents move?

9 Consensus on nonlinear spaces & Graph coloring 1. Some examples to motivate nonlinear consensus 2. Formalizing consensus on nonlinear spaces 3. Link with graph coloring : (just) a complexity result

10 I. Orientation synchronization e.g. in formations of spacecraft State space of orientations = manifold of rotation matrices SO(3) DARWIN interferometer (NASA / ESA concept study)

11 II. Coordination on the circle appears in problems involving oscillator networks

12 III. Distributed sensor networks e.g. to collect ocean data (Naomi Leonard et al.) Autonomous underwater vehicles, sparse communication Buoyancy-driven at constant speed ~ 40 cm/s Goal : collective trajectory planning on a simplified AUV model

13 Agreement on collective motion involves nonlinear spaces

14 NB: In nonlinear spaces, coordinated motion differs (more difficult) from consensus

15 NB: In nonlinear spaces, coordinated motion differs (more difficult) from consensus

16 NB: In nonlinear spaces, coordinated motion differs (more difficult) from consensus Algorithms for coordinated motion on Lie groups, see: Coordinated motion design on Lie groups A. Sarlette, S. Bonnabel and R. Sepulchre, IEEE Trans. Automatic Control, vol. 55 nr. 5, pp (2010)

17 IV. Coordination on nonlinear spaces is linked to algorithmic applications

18 Setting

19 Consensus on nonlinear spaces & Graph coloring 1. Some examples to motivate nonlinear consensus 2. Formalizing consensus on nonlinear spaces Synchronization: from vector spaces to the circle Formalization on compact homogeneous manifolds Global synchronization properties 3. Link with graph coloring : (just) a complexity result

20 A linear algorithm achieves global exponential synchronization on vector spaces

21 A linear algorithm achieves global exponential synchronization on vector spaces

22 This result has two fundamental limitations The convergence result involves a condition on G. But often interconnections depend on the states of the agents. What about state-dependent graphs? under investigation see Bullo et al., Aeyels/De Smet, Blondel/Hendrickx The global convergence argument does not extend to nonconvex spaces like the circle, sphere,... How do synchronization algorithms behave globally on manifolds? topic of this talk

23 An algorithm with the same local behavior can be designed on the circle

24 In the following we will extend this to other perfectly symmetric nonlinear spaces = compact homogeneous manifolds (CCH) Formally : quotient manifold of a Lie group by a subgroup Intuitively: all points are identical Examples: sphere S n rotation matrices SO(n) (and all other compact groups) Grassmann manifolds (see last part) In this talk:

25 An alternative distance measure yields convenient properties

26 An alternative distance measure yields convenient properties

27 An alternative distance measure yields convenient properties

28 The induced arithmetic mean of the chordal distance is easily computable

29 The induced arithmetic mean of the chordal distance is easily computable

30 The induced arithmetic mean allows to define several specific configuration types

31 The gradient of VG yields consensus algorithms

32 These developments can be adapted to more complex agent dynamics

33 Consensus on nonlinear spaces & Graph coloring 1. Some examples to motivate nonlinear consensus 2. Formalizing consensus on nonlinear spaces Synchronization: from vector spaces to the circle Formalization on compact homogeneous manifolds Global synchronization properties 3. Link with graph coloring : (just) a complexity result

34 Synchronization is ensured locally. The global behavior is a priori unclear.

35 Fixed but directed graphs can lead to limit cycles, quasi-periodic behavior,...

36 Undirected graphs ensure convergence to an equilibrium set, but which one? What about repulsive agents? - on R n - on circle Agents drive away to infinity Stable equilibria are not trivial

37 The existence of local equilibria is sensitive to the attraction profile between connected agents

38 Alternative algorithms can overcome spurious local equilibria of standard consensus motion Gossip algorithm = forced asynchrony At each time, select a single link, and only its 2 agents move towards each other Thm: If G is uniformly connected, synchronizes with probability 1 also on the circle, sphere,...

39 Alternative algorithms can overcome spurious local equilibria of standard consensus motion Auxiliary variables (can be written with agent-based coordinates)

40 Consensus on nonlinear spaces & Graph coloring 1. Some examples to motivate nonlinear consensus 2. Formalizing consensus on nonlinear spaces Synchronization: from vector spaces to the circle Formalization on compact homogeneous manifolds Global synchronization properties 3. Link with graph coloring : (just) a complexity result

41 Consensus algorithms seem much harder to analyze on nonlinear spaces Attractive agents, fixed undirected interaction graph seems difficult to say if synchronization is the only stable equilibrium How hard can equilibrium characterization be? Consensus on nonlinear spaces and graph coloring A. Sarlette, CDC Orlando, pp (2011)

42 Idea: interacting agents setting graph coloring Graph theory many complexity results Graph parametrizes interacting agents in continuous dynamics Analog computation: continuous dynamics solve computational problem

43 Result: equilibrium characterization on projective space is NP-hard graph k-coloring NP-hard for k>2 (robust) repulsion on projective space P k-1 R

44 Graph coloring is a classical computational problem Given graph G(V,E) and integer k, find ϱ : V {1,2,...,k} ( colors) s.t. ϱ(a) ϱ(b) for all (a,b) E Ex. country maps, Sudoku,...

45 Graph coloring is a classical computational problem Given graph G(V,E) and integer k, find ϱ : V {1,2,...,k} ( colors) s.t. ϱ(a) ϱ(b) for all (a,b) E Ex. country maps, Sudoku,... Complexity For k=2: G is 2-colorable G is bipartite (polynomial) For k>2: NP-hard (in #V) to determine if G(V,E) is k-colorable

46 Graph k-coloring & Directions in R k k different equivalent colors {1,2,...,k} k orthogonal lines of R k

47 Lines of R k define the projective space P k-1 R x P k-1 R represents a line of R k v Handy representation: x orthonormal projection Π onto line x Π R k k, rank(π)=1, trace(π)=1 Π = v v T / (v T v) Chordal distance on P k-1 R ϕ x1 x2 v2 v1

48 Repulsive agents try to maximize their mutual distance Cost function graph dependence with g(x) a strictly monotonically increasing function on [0, 2] Gradient dynamics = anti-consensus motion on projective space

49 Goal: relate the stable equilibria to graph coloring solutions Stable equilibria = local maxima of W result about complexity of characterizing stable equilibrium set (as complex as deciding graph coloring) possibility to solve graph coloring by swarm optimization? (continuous evolution of the swarm converges to solution = distributed analog computation)

50 Two particular sets in P k-1 R So = all states belong to a discrete set of k orthogonal lines = colors Sp(G) = every edge is stretched to the maximum distance Properties : Sp(G) can be empty depending on G Sp(G) global maxima of W if (*) So Sp(G) if and only if G is k-colorable (**)

51 The complexity result Question : Given G(V,E) and P k-1 R, is any point in So a stable equilibrium for the repulsive agents? yes/no question (typical decision problem) about specific property of equilibrium set Theorem: This question is as difficult as graph coloring -- that is NP-hard for k>2 -- if g(x) satisfies

52 The condition on coupling function g(x) is not too restrictive... Condition g(x) Kuramoto: g identity x1 ϕ x2 0 2 x = 2 sin 2 ϕ Large class of g(x) coupling functions Allows g(x) identity (canonical consensus) for N/k

53 Proof idea Sp(G) always stable ensure unstable So role of the condition on g(x)? not empty graph is k-colorable stable eq. in So graph k-colorable

54 Simulations with g(x) = atan(x/2) for k=3 Petersen graph, 3-colorable

55 Simulations with g(x) = atan(x/2) for k=3 Grötzsch graph, not 3-colorable

56 Can we use the distributed dynamical system to solve graph coloring? Stable equilibria = local maxima of W result about complexity of characterizing stable equilibrium set (as complex as deciding graph coloring) OK possibility to solve graph coloring by swarm optimization? (continuous evolution of the swarm converges to solution = distributed analog computation)??

57 NO The multi-agent system on P k-1 R does not solve graph-coloring Global maxima of W in So Sp(G) graph k-coloring Sp(G) always stable ensure unstable So Multi-agent system for colorable G converges to Sp(G) So Sp(G) [Kochen-Specker Theorem] There exist non-colorable G for k=3 with Sp(G) A system that converges to a point in Sp(G)\So can correspond to colorable or non-colorable G...

58 The Kochen-Specker theorem discusses fundamentals of quantum measurement element of P k-1 R possible result of projective quantum measurement on R k Kochen-Specker: For k 3, there does not exist a function f from the set of possible measurement projectors Pi P k-1 R to associated measurement results in {0,1} such that for every {Pi} that form a physical observable (i.e. that commute and Pi = I) we have f (Pi) = 1 Use: show a contradiction with classical re-interpretations of quantum laws

59 The Kochen-Specker theorem discusses fundamentals of quantum measurement Proof: Constructs an example of N elements of P k-1 R, where mutually orthogonal lines are connected in a graph. Then assigning f (color 1)=1, f (other colors)=0 would solve the task if colorable They have a counterexample with N=31 agents for k=3 They construct a situation where all pairs of connected agents are orthogonal in R 3, but the graph is NOT 3-colorable

60 Conclusion General geometric interpretation of consensus allows extension to nonlinear spaces Consensus motion yields more complex global behavior than on R n - possible limit cycles, quasi-periodicity,... for directed graphs - multiple equilibria for undirected graphs depending on precise coupling function & interaction graph

61 Conclusion Graph-coloring complexity of consensus on projective space For a class of repulsion functions (robustly difficult) Link not bi-directional: provides no solution for graph coloring Equilibrium stability as key feature to characterize NP-hard for k>2 leaves open the case k=2 correspoding to the circle (which seems not trivial, but further unclear how hard)

62 Consensus on nonlinear spaces and Graph coloring Alain Sarlette Ghent University, Belgium

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