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


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


Networked dynamical systemsNetworked dynamical systems

Complexity of

dynamics


Single Agent

Single Agent


Multi-agent systems

Multi-agent systems

Flocking/synchronizationConsensus/Coverage

Complex networked

systems


Complexity of

dynamics


Single Agent

Single Agent


Multi-agent systems

Multi-agent systems

Flocking/synchronizationconsensus

Complex networked

systems

Jadbabaie et al


Statistical Physics and Statistical Physics and emergence of collective behavioremergence of collective behavior

Simulations and conjectures but few “proofs’


Working systems but few “proofs’


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?


= 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!


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.


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)


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,


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


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

34


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


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.


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.


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


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)


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)


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


Enforcing connectivityEnforcing connectivity

Treat loss of connectivity as an obstacle Zavlanos, Jadbabaie, Pappas ’07

Everything else is the same!


Fireflies Flashing

Synchronization


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


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


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


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


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.


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?


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


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


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


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


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.]


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 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


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)


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)


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


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


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


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


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


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


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


Example, eigenvectors of Example, eigenvectors of 11

Network 1st homology class

2nd homology class ‘Fiedler-like’- eigenvector


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.


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.


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


Jointly hole free simplicial complex Amalgamated complexes

Union vs. amalgamated complex


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


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


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

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