Ali Jadbabaie
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Transcript of Ali Jadbabaie
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Distributed Motion Coordination inNetworked Dynamic Systems: From Flocking and Synchronization to Coverage Verification
Ali JadbabaieDepartment of Electrical and Systems Engineering
and GRASP LaboratoryUniversity of Pennsylvania
Workshop on computational worldview and sciences Caltech 03/15/07
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Networked dynamical systemsNetworked dynamical systems
Complexity of
dynamics
Complexityof interconnection
Single Agent
Single Agent
Nonlinear/uncertainhybrid/stochastic etc.
Multi-agent systems
Multi-agent systems
FlockingSynchronizationConsensus
Complex networked
systems
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Networked dynamical systemsNetworked dynamical systems
Complexity of
dynamics
Complexityof interconnection
Single Agent
Single Agent
Nonlinear/uncertainhybrid/stochastic etc.
Multi-agent systems
Multi-agent systems
Flocking/synchronizationConsensus/Coverage
Complex networked
systems
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Complexity of
dynamics
Complexityof interconnection
Single Agent
Single Agent
Nonlinear/uncertainhybrid/stochastic etc.
Multi-agent systems
Multi-agent systems
Flocking/synchronizationconsensus
Complex networked
systems
Jadbabaie et al
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Statistical Physics and Statistical Physics and emergence of collective behavioremergence of collective behavior
Simulations and conjectures but few “proofs’
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Working systems but few “proofs’
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
r
agent i
neighbors ofagent i
Multi-agent setting: Vicsek’s kinematic modelMulti-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?
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
= heading
= speed
MAIN QUESTION MAIN QUESTION :: Under what conditions do all headings converge to the same value and agents reach a consensus on where to go?
Distributed consensus algorithm for Distributed consensus algorithm for kinematic agentskinematic agents
For small angles
For angles in (-/2,/2), the nonlinear problem becomes
linear with a coordinate change!
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
The underlying proximity graph The underlying proximity graph
We use graphs to represent neighboring relations
vertices:
edges:
1
2
34
5
6
switching signal ,
adjacency matrix
Valence matrix
finite set of indices corresponding to all graphs over n vertices.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Conditions for reaching consensusConditions 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 agents is “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)
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
History of consensus AlgorithmsHistory 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 chainChaterjee 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,
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Theorem: necessary and sufficient condition for almost sure convergence is ,i.e., the average system needs to reach deterministic agreement.
Consensus in random networksConsensus 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.
is a random switching signal
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
The Laplacian of the graphThe Laplacian of the graph
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”
W is diagonal
B is the (n x e) incidence matrix of graph G.
1
2
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A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Consensus in Continuous timeConsensus 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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Vicsek’s Model withVicsek’s Model with Periodic Boundary Condition 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.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Kronecker’s Theorem, Weyl’s TheoremKronecker’s Theorem, Weyl’s Theorem and Number Theory 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.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Flocking, artificial life, and Flocking, artificial life, and computer graphicscomputer 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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Distributed coordination with dynamic Distributed coordination with dynamic models: Flocking with collision avoidancemodels: 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)
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Flock artificial potential energyFlock 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)
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Dynamic Topology
Local sensing/communication Graph changes with timeUnreliable communication, no dwell time Control is discontinuous Non smooth Lyapunov theory
Topology dictates analysisTopology dictates analysis
Fixed Topology
Fixed (logical) networkGraph is constant
Control is smooth
Classic Lyapunov theory
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Enforcing connectivityEnforcing connectivity
Treat loss of connectivity as an obstacle Zavlanos, Jadbabaie, Pappas ’07
Everything else is the same!
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Fireflies Flashing
Synchronization
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
1
sin( )N
ii j i
j
d K
dt N
: Number of oscillators
: Natural frequency of oscillator , 1, , .
: Phase of oscillator , 1, , .
: Coupling strength
i
i
N
i i N
i i N
K
All-to-all interaction
Kuramoto Model
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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
1
sin( )N
ii ij j i
j
d KA
dt N
1
2
6
3
5
4
0 1 1 0 0 1
1 0 1 0 0 0
1 1 0 1 0 0
0 0 1 0 1 1
0 0 0 1 0 1
1 0 0 1 1 0
A
Kuramoto model & graph topology
B is the incidence matrix of the graph
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
1
sin( )N
iij j i
j
d KA
dt N
Frequency entrainment and
phase locking as long asgraph is connected.
Proof same as the continuous-time agreement model
sin( )x
x
1
sin( )N
ii ij j i
j
d KA
dt N
Frequency entrainment possible. Phase stability, but no phase
locking.
Frequency and phase locking
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Kuramoto model is the just a gradient algorithm for minimization of a global utility which measures misalignment between phasors (exactly like TCP!)
1 1
1
1min 1 cos( ) ,
2
s.t. sin( )
N N
ij i ji j
Ni
ij i jj
A
NA
K
1 1 1 1 1
11 cos( ) sin( )
2
N N N N Ni i
ij i j ij i i ji j i j i
NL A A
K
sin( ) ( )cos( )i j i j i j
i j
L
1
sin( )N
iij i j
ji
NLA
K
Minimize the misalignment
21
( )
1 ( )
Ni j
i i ijji i j
K L KA
N N
Kuramoto model, dual decomposition,
and nonlinear utility minimization
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Phase locking in switching networks Phase locking in switching networks with heterogeneous delayswith 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.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Beyond Graphs in Networked SystemsBeyond 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?
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Coverage ProblemsCoverage 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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
From Graphs to Simplicial ComplexesFrom Graphs to Simplicial ComplexesSimplicial 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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
From Graphs to Simplicial ComplexesFrom 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 facesWhenever two simplices intersect, they do so in a common face.
Valid ExamplesGraphsTriangulations
Invalid examples
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Rips-Vietoris Simplicial ComplexRips-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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Coverage ProblemsCoverage ProblemsIntersection 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.]
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Rips and Rips and ČČech Complexes: ech Complexes: Topological vs. Geometric informationTopological 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.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
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 gap between the necessary and the sufficient conditions.
Topological information from Rips complexTopological information from Rips complex
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Boundary Maps: Generalization Boundary Maps: Generalization of Incidence matricesof Incidence matrices
: the vector space whose basis is the set of oriented k-simplices of XThe boundary map is the linear transformation
k-cycles:k-boundaries:Note:
Homology groups : Hk(X) = Zk (X) / Bk (X)
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Relevance of Homology 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)
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Combinatorial k-LaplaciansCombinatorial 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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
k-Laplacian at the Simplex Levelk-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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
k-Laplacian at the Simplex Levelk-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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
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
Laplacian FlowsLaplacian Flows
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Laplacian Flows (contd.)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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
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
Decentralized Computation of Decentralized Computation of HomologyHomology
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Decentralized computation 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 ?
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Decentralized computation 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 ?
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Decentralized computation 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 ?
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Decentralized computation 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
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Example, eigenvectors of Example, eigenvectors of 11
Network 1st homology class
2nd homology class ‘Fiedler-like’- eigenvector
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Consensus in Switching Graphs 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.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Coverage in Switching Simplicial Coverage in Switching Simplicial ComplexesComplexes
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.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Computational test for relative Computational test for relative homlogical criterion for coveragehomlogical criterion for coverage
Theorem (De Silva and Ghrist’06): For coverage, H2(R,F) should contain a nontrivial element which does not vanish on the boundary
Computational test: If the above distributed dynamical system converges to a nonzero value which does not vanish on boundary, we have coverage
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Jointly hole free simplicial complex Amalgamated complexes
Union vs. amalgamated complex
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Verification of sweep coverageVerification of sweep coverage
Given a set of sensors with a disk footprint, add an edge when 2 sensors overlap. A face when 3 sensors overlap, …
Construct the 1st Laplacian 1
Rips complex is “jointly persistently hole free over time” intersection
of kernels of Laplacians is zeroSwitched dynamical systems converges to 0
Unfortunately This does NOT imply sweep coverage
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Jointly-hole-free complexes and Jointly-hole-free complexes and sweep coveragesweep coverage
BAD NEWS:If the node is close to the boundary, The union of simplicial complexes is still jointly hole-free and we satisfy the relative homological criterion, BUT, If the node is close to the boundary, we can’t sweep the center
GOOD NEWS If, however, we already have a set possible sensor locations which do verify Coverage when they are static, we can verify sweep coverage when they blink on and off
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Work in Progress ….Work in Progress ….
Distinguish between multiple homology classes by decentralized eigenvector decomposition of k-Laplacian (Kempe’s algorithm)
Combinatorial Laplacian of amalgamated complexes
Quantify ‘proximity to holes’
Quantify fragilities in network : near-harmonic cycles (Fiedler like characterization such as cutpoints for holes? Properties of K-Laplacian Eigenvectors and Eigenvalues)
A “spectral theory” for simplicial complexes?
Is there a percolation result similar to geometric graphs?
Statistical physics of simplicial complexes
Consensus and agreement in CAT-0 topological spaces