introduction to flocking {stochastic matrices} a. s. morse yale university gif – sur - yvette may...

Introduction to Flocking {Stochastic Matrices}

A. S. Morse

Yale University

Gif – sur - Yvette May 21, 2012

Supelec

EECI Graduate School in Control

CRAIG REYNOLDS - 1987

BOIDS

The Lion King

separation cohesionalignment

CRAIG REYNOLDS - 1987

BOID

neighborhood

BOIDS

Flocking Rules

separation cohesionalignment

Flocking Rules

DemetriTerzopoulos

Motivated by simulation results reported in

Vicsek et al. simulated a °ock of n agents fparticlesg all moving in theplane at the same speed s, but with di®erent headings µ1;µ2; : : : ;µn.

sµi

s = speed

µi = heading

Each agent’s heading is updated at the same time as the rest using a local rulebased on the average of its own current heading plus the headings of its“neighbors.”

Vicsek’s simulations demonstrated that these nearest neighbor rules can cause all agents to eventually move in the same direction despite

1. the absence of a leader and/or centralized coordination

2. the fact that each agent’s set of neighbors changes with time.

Vicsek Model

ri

agent i

neighbors ofagent i

Each agent is a neighbor of itself

Each agent has its own sensing radius ri

So neighbor relations are not symmetric

sµi

s = speed

µi = heading

HEADING UPDATE EQUATIONS

Average at time t of headings of neighbors of agent i.

Vicsek Flocking Problem: Under what conditions do all n headings converge to acommon value?

Ni(t) = set of indices of agent i0s neighbors at time t

ni(t) = number of indices in Ni(t)

Another rule:

Convex combination {Requires collaboration!}

7

4

1

3

5

2

6

(1,2)

ji j is a neighbor of i

Neighbor graphs = self-arced graphs

Neighbor Graph N of Index Sets N1, N2 ,…., Nn

G = all directed graphs with vertex set V = {1,2,…,n}

N = graph in G with an arc from j to i whenever j 2 Ni, i 2 {1,2,…,n}

A self-arced graph = any graph G with self-arcs at all vertices

7

4

1

3

5

2

6

(1,2)

State Space ModelAdjacency Matrix AG of a graph G 2 G: An n£n matrix of 0‘s and 1’s with aij = 1whenever there is an arc in G from i to j.

In-degree of vertex i = number of arcs entering vertex i

Out-degree of vertex i = number of arcs leaving vertex i

In-degree = 4, out-degree = 1

_4

State Model:

State Space Model

Update Eqns:

Adjacency Matrix AG of a graph G 2 G: An n£n matrix of 0‘s and 1’s with aij = 1whenever there is an arc in G from i to j.

Flocking Matrix FN of a neighbor graph N 2 G:

bijection

j

DN = diagonalf d1;d2; : : : ;dng

and

di = in-degree of vertex i =nX

=1aj i

where

ni =

Vicsek flocking problem: Under what conditions do all n headings converge to a common value?

No common quadratic Lyapunov function exists

A switched linear system

is at least non-increasing along trajectories

But the non-negative function

V (µ) = maxi

f µig¡ mini

f µig

µ(t + 1) = FN(t)µ(t)

But it takes much more to conclude that V ! 0

Verify this!

and so

where

For if this is so, then

Vicsek flocking problem: Under what conditions do all n headings converge to a common value?

µ(t + 1) = FN(t)µ(t)

Problem reduces to determining conditions on the sequence N(0), N(1), ...under which

where

{Right} Stochastic Matrices

1. it has only non-negative entries

2. its row sums all equal 1

Flocking matrices are stochastic

Stochastic matrices closed under multiplication

Sn£n= stochastic if

– flocking matrices are not

Therefore it is sufficient to determine conditions on an infinite sequence of n£n stochastic matrices S1, S2, .... so that

If S is a compact set of n£n stochastic matrices whose members each haveat least one positive column, then for each sequence of matrices S1, S2, …from S,

and this limit is approached exponentially fast.

Why is this true?

This is a well studied problem in the theory of non-homogeneous Markov chains

Induced Norms and Semi-Norms

For M 2 Rn£n and p > 0, let ||M||p denote the induced matrix p norm on Rn£n .

1. Nonnegative: |M|p ¸ 0

2. Homogeneous: |rM|p = r|M|p

3. Triangle inequality: |M1 + M2|p · |M1|p + |M

2|p verify!

These three properties mean that |¢ |p is a semi-norm

{If |M|p = 0 were to imply M = 0, then |¢|p would be a norm.}

We will be interested primarily in the cases p = 1, 2, 1:

For any such p, define

|M|p = 0 ; M = 0

2. Sub-multiplicative: Suppose M is a subset of Rn£n such that M1 = 1 for all M 2 M. Then

1. |M|p · 1 if ||M||p · 1

M is semi - contractive in the p semi-norm if |M|p < 1

Additional Properties of

Let c0 ,c1 and c2 denote values of c which minimize ||M2M1 - 1c||p, ||M1-1c||p, and ||M2-1c||p respectively.

Because |M|p · ||M||p

Proof:

1 = M21

Suppose M is a subset of Rn£n such that M1 = 1 for all M 2 M. Let p be fixed and let C be a compact set of semi - contractive matrices in M. Let

Then for each infinite sequence of matrices M1 , M2, ... in C, the matrix product converges as i ! 1 as fast as ¸i converges to zero, to a rank one matrix of the form 1c.

A stochastic matrix S is semi-contractive in the semi-norm | ¢ |1 if S has a positive column.

To do this , it is enough to show that:

We want to use this fact to prove that:



Proof: See board

Any stochastic matrix S can be written as S = 1c + T where c is the largest row vector for which S - 1c is nonnegative and T = S – 1c

jSj1 = mind

jjS¡ 1d0jj1 · jjS¡ 1cjj1 = jjT jj1 = (1¡ c1)

T1 = S1 – 1c1 = (1 - c1)1 so all row sums of T = (1 - c1)

Moreover c 0 if and only if S has a positive column. verify!

¸ 0 because T ¸ 0

Therefore (1 – c1) < 1 if and only if S has a positive column

A stochastic matrix S is semi-contractive in the semi-norm | ¢ |1 if S has a positive column.

Transitioning from Matrices to Graphs

For a nonnegative matrix Mn£n, °(M) is that graph whose adjacency matrix is the transpose of the matrix which results when each non-zero entries in M isreplaced by a 1.

In other words, for a nonnegative matrix M, °(M) is that graph which hasan arc (i, j) from i to j whenever mj,i 0.

strongly rooted graph


For a nonnegative matrix Mn£n, °(M) is that graph whose adjacency matrix is the transpose of the matrix which results when each non-zero entries in M isreplaced by a 1.

°( FN ) = °( A0N ) = N

A graph is strongly rooted if at least one vertex is adjacent to every vertex inthe graph

°(M) is strongly rooted , M has a positive column

For any nonnegative matrix M, °(M) has an arc (i, j) whenever mj,i 0.

Motivation for strongly rooted:




If S is a compact set of n£n stochastic matrices whose members each havea strongly rooted graph, then for each sequence of matrices S1, S2, …from S,


. . .

T Tq T2 T1

Thus establishing convergence to 1c of an infinite product of stochastic matrices boils down to determining when the graph of a product of stochastic matrices is strongly rooted.


When does

If

then

°(BA) = °(B) ± °(A)


By the composition of graph G2 2 G with graph G1 2 G , written G2 ± G1, is that directed graph in G which has an arc (i, j) from i to j whenever there is an integer k such that (i, k) is an arc in G1 and (k, j) is an arc in G2.

If A and B are nonnegative n£n matrices and C = BA , then

Thus cji 0 if and only if for some k, bjk 0 and aki 0.

Therefore (i , j) is an arc in °(C) if and only if for some k, (i, k) is an arc in °(A) and (k, j) is an arc in °(B).

As before G = set of all directed graphs with vertex set {1,2,....,n}.

What motivates this definition?

Thus deciding when a finite product of stochastic matrices has a strongly rootedgraph is the same problem as deciding when a finite composition of graphs isstrongly rooted. So............


When is the composition of a finite number of graphs strongly rooted?

°(S2S1) = °(S2) ± °(S1)

Graph composition is defined so that for any two n£n stochastic matrices S1 and S2

3 roots

rooted graph

A rooted graph is any graph in G which has has at least one vertex v which, foreach vertex i 2 V there is a directed path from v to i.

Every composition of (n – 1)2 or more self-arced, rooted graphs in G is strongly rooted.

The set of self-arced, rooted graphs in G is the largest set of set of self-arced graphs in G for which every sufficiently long composition is strongly rooted.

When is the composition of a finite number of graphs strongly rooted?

Proof: See notes.

Proof: See notes.


If G1 has a self-arc at i and (i, j) 2 A(G2) for some j, then (i, j) 2 A(G2 ± G1)

If G2 has a self-arc at j and (i, j) 2 A(G1) for some i, then (i, j) 2 A(G2 ± G1)

If G1 and G2 both have self-arcs at all vertices, then A(G2)[A(G1) ½ A(G2±G1).

For given graphs G1, G2 2 G , G2 ± G1, is that graph in G for which (i, j) 2 A (G2 ± G1) whenever there is an integer k such that (i, k) 2 A(G1) and (k, j) 2 A(G2 ).

If G1 has a self-arc at i , then (i, i) 2 A(G1).

If G2 has a self-arc at j , then (j, j) 2 A(G2).

In general for A(G2)[A(G1) A(G2±G1) even if both graphs are self-arced.

However for self-arced graphs, if there is a directed path between i and j in G2±G1 then there is a directed bath between i and j in G2[G1

Define A(G) = set of arcs in G

If S is a compact set of n£n stochastic matrices whose members each havea strongly rooted graph, then for each sequence of matrices S1, S2, …from S,


If S is a compact set of n£n stochastic matrices whose members each havea self-arced, rooted graph, then for each sequence of matrices S1, S2, …from S,


We can generalize further still...........

. . .

strongly rooted

strongly rooted

strongly rootedrooted

An infinite sequence of graphs G1, G2, ... in G is repeatedly jointly rooted if there is a finite positive integer m for which each of the sequences Gm(k -1)+1, ....... Gk -1, k¸ 1, is jointly rooted.

An finite sequence of graphs G1, G2, ..., G p in G is jointly rooted if the composed graph Gp ± Gp-1 ± ± G1 is rooted.

Repeatedly Jointly Rooted Sequences

. . .

rooted repeatedlyjointly rooted

rooted rooted

If S is a compact set of n£n stochastic matrices whose members each havea self-arced, rooted graph, then for each sequence of matrices S1, S2, …from S,


compact set


Suppose S is a compact set of n£n stochastic matrices whose members each havea self-arced graph. Suppose that S1, S2, ..... is an infinite sequence of matrices from S whose corresponding sequence of graphs °(S1), °(S2), .... is repeatedly jointly rooted by sub-sequences of length m. Suppose in addition that the set of all products of m matrices from S with rooted graphs, written C(m), is closed. Then

Compactness of S does not in general imply compactness of C(m).

Exception: If S is finite and thus compact {as in flocking applications} so is C(m)

Construct an example for 2£2 matrices with m = 2.

Exception: If S is the set of stochastic matrices modeling the convex combo flocking rule, then S and C(m) are both compact. verify this!

Flocking Theorem: For each trajectory of the Vicsek flocking system µ(t +1) = FN(t) µ(t) along which the sequence of neighbor graphs N(0), N(1), .... is repeatedly jointly rooted, there is a constant steady state heading µss which µ(t) approaches exponentially fast, as t ! 1.

Collectively Rooted Sequences

The flocking theorem relies on the notion of jointly rooted sequences:

An finite sequence of graphs G1, G2, ..., G p in G is jointly rooted if the composed graph Gp ± Gp-1 ± ± G1 is rooted.

By the union of G1 , G 2 2 G is meant that graph G1 [ G 2 in G with arc set A (G1) [ A (G2).

An finite sequence of graphs G1, G2, ..., G p in G is collectively rooted if the union graph Gp [Gp-1 [ [ G1 is rooted.

In general, for self-arced graphs A( Gp [Gp-1 [ [ G1) is a strictly proper subset of A(Gp ± Gp-1 ± ± G1)

However, for each arc (i, j) 2 A(Gp ± Gp-1 ± ± G1) there must be a directed path between (i, j) in Gp [Gp-1 [ [ G1

Therefore for self-arced graphs, the sequence G1, G2, ..., G p in G is jointlyrooted if and only if it is collectively rooted.

Leader Following

Suppose that one of the agents in the group, namely agent k, ignores Vicsek’s update rule and decides instead to move with some arbitrary but fixed heading 0.

Suppose that the remaining agents are unaware of this non-conformist’s decision and continue to follow Vicsek’s rule just as before.

Note that under these conditions, agent k must have no neighbors to follow which means that vertex k of any neighbor graph N for the group cannot have any incident arcs.

Because of this, the only possible way such a graph N could be rooted or strongly rooted would be if vertex k were the root of N and the only root of N.

All of the preceding results are applicable to this case without change.

Thus for example, all agents in the group will eventually move in the same direction as agent k if the sequence of neighbor graphs is repeatedly jointly rooted.

However more can be said in this special case

Leader Following

For example, suppose that the neighbor graphs N(1), N(2), ..... are allrooted.

Then each N(t) must be rooted at k.

It was noted before that the composition of any (n -1)2 self-arced rooted graphs in G must be strongly rooted.

However in the special case of self-arced, rooted graphs in G which all have a root at the same vertex v, it takes the composition of only (n -1) of them to produce a strongly rooted graph. See notes for a proof

Because of this, one would expect faster convergence than in the leaderless case,all other things being equal.

FOLLOWING RED LEADER

FOLLOWING RED LEADER Leader’s Neighbors Yellow

FOLLOWING RED LEADER Rectangle Pattern

Leader’s Neighbors Yellow

Symmetric Neighbor Relations

The original version of the flocking problem considered the case when all neighborrelations were symmetric – that is if agent i is a neighbor of agent j then agent j isa neighbor of agent i.

Mathematically, a symmetric neighbor relation means that i 2 Nj , j 2 Ni

The corresponding neighbor graph N would thus be “symmetric” as well.

A directed graph G 2 G is symmetric if (i, j) 2 A (G) , (j, i) 2 A (G )

A rooted symmetric graph is the same thing as a “strongly connected’’ symmetric graph

A graph G 2 G is strongly connected if there is a directed path betweenany two distinct vertices i and j.

Another Way to Write the Vicsek Flocking System

L(t) is a symmetric matrix if N(t) is symmetric.

Simplified Rule for Symmetric Neighbor Relations

1. Symmetric

Simplified flocking matrix:

2. Nonnegative if g > max di

L1 = D1 – A1 = 03. Fs1 = 1

Fs is stochastic if g > max di

Comparing Flocking Matrices

Let N be a given self-arcd directed, symmetric, neighbor graph.

°(Fs) = °(F) = N

Therefore all convergence results hold without change for the simplified flocking ruleassuming symmetric neighbor relations.

A = A0

Can extend the symmetric case to continuous time.

Convergence Rates

First we will consider this matter in relation to the semi-norm |¢ |1

Let C be a compact set of n£n stochastic matrices which are semi - contractive in the infinity norm. Then for each infinite sequence of matrices S1 S2, ... in C, the matrix product converges as i ! 1 to a rank one matrix as fast as

converges to zero.

Note that the ith entry ci in c must be the smallest entry in the ith column of S.

jSj1 · (1 ¡ ci) 8i

Since (1 – c1) · 1 - ci for any i,

Any stochastic matrix S can be written as S = 1c + T where c is the largest row vector for which S - 1c is nonnegative and T = S – 1c

jSj1 = mind

jjS¡ 1d0jj1 · jjS¡ 1cjj1 = jjT jj1 = (1¡ c1)

So what we’d like is a uniform upper bound on |S|1 over C. = a convergence rate bound

Suppose that F is a flocking matrix whose graph is strongly rooted at vertex k

Then °(F) must have an arc from vertex k to each other vertex in the graphwhich means that the kth row of adjacency matrix A of °(F) must be [1 1 1 1]

Since F = D-1A0 where D = diagonal {n1,n2, ..., nn}, the kth column of F must be

The smallest entry in this column is bounded below by

Therefore for any flocking matrix F with a strongly rooted graph

Convergence Rate Bound for Flocking Matrices with Strongly Rooted Graphs

= a convergence rate bound

If ci is the smallest element in the ith column of S, then |S| 1 · 1 – ci

An Explicit Formula for the Infinity Semi-Norm

If M is a stochastic matrix, the quantity on the right is known as the coefficient ofergodicity.

See notes of a proof of this fact.

For any real numbers x and y

So for a stochastic matrix S

For any nonnegative n£n matrix M,

1. |S|1 · 1 Because |S|1 · ||S||1 = 1

2. |S|1 = 0 if and only if all rows are equal = iff S = 1c

Therefore |S|1 < 1 iff for each distinct i and j, sik > 0 and sjk > 0 for at leastone value of k.

A stochastic matrix with this property is called a scrambling matrix

*

For fixed i and j, the kth term in the sum in will be positive iff sik > 0 and sjk > 0 *Therefore the sum in will be positive iff sik > 0 and sjk > 0 for at least one value of k *

Summary

A stochastic matrix S is a scrambling matrix for each distinct i and j, sik > 0 and sjk > 0 for at least one value of k.

Equivalently, a stochastic matrix is a scrambling matrix if no two rows are orthogonal.

A stochastic matrix is a semi-contraction in the infinity norm iff it is a scrambling matrix.

An explicit formula for the infinity semi-norm of any stochastic matrix S is

The Graph of a Scrambling Matrix.

A graph G 2 G is neighbor shared if each two distinct vertices i and j have a common neighbor k

A stochastic matrix S is a scrambling matrix if and only if its graph is neighbor shared.

A stochastic matrix S is a scrambling matrix for each distinct i and j, sik > 0 and sjk > 0 for at least one value of k.

In a strongly rooted graph there must be a root which is the neighbor of each vertexin the graph. So...........

Every strongly rooted graph is neighbor shared.

The converse is clearly false.

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2 and 4 share 4

1 and 2 share 2

1 and 3 share 2

1 and 4 share 4

3 and 4 share 1

2 and 3 share 2

Neighbor-Shared Directed Graph

A neighbor shared graph is a directed graph in which each pair of distinct vertices share a common neighbor

Suppose G 2 G is neighbor shared.

Then any pair of vertices (i, j) must be reachable from a {common neighbor} vertex k.

Suppose for some integer p 2 {2, 3, ..., n -1}, each subset of p vertices is reachablefrom a single vertex.

Let {v1, v2, ..., vp} be any any such set and let v be a vertex from which allof the vi can be reached.

Let w be any vertex not in the set {v1, v2, ..., vp}.

Since G is neighbor shared, w and v can be reached from a common vertex y

Therefore every vertex in the set {v1, v2, ..., vp , w} can be reached from y.

So every subset of p + 1 vertices in the graph is reachable from a single vertex.

Every neighbor-shared graph in G is rooted.

Converse is false. Verify by constructing an example.

Strongly rooted graphs ½ neighbor shared graphs ½ rooted graphs

So by induction all n vertices are reachable from a single vertex.

Strongly rooted graphs ½ neighbor shared graphs ½ rooted graphs

The composition on any n – 1 or more self-arced, rooted graphs in G is neighbor shared.

Proof: See notes.

The composition on any n – 1 or more neighbor-shared graphs in G is strongly rooted.

We will use this fact a little later to get a convergence rate bound for products offlocking matrices whose sequence of graphs is repeatedly jointly rooted.

Compositions of rooted and neighbor shared graphs

Let’s outline a proof of this:

A graph G 2 G is k neighbor shared if each set of k distinct vertices in G share a common neighbor.

A 2 neighbor shared graph is thus a neighbor shared graph and an n neighbor shared graph is obviously strongly rooted.

Suppose G is neighbor shared and H is k neighbor shared for some k < n

Let {v1, v2, ..., vk+1} be distinct vertices.

Since H is k neighbor shared, in H {v1, v2, ..., vk} share a common neighbor p and {v2, v3, ..., vk +1} share a common neighbor q

Since G is neighbor shared, in G p and q share a common neighbor w.

In H±G, vertices v1, v2, ..., vk must have w for a neighbor as must vertices v2, v3, ..., vk +1

Therefore in H±G, vertices v1, v2, ..., vk+1 must have w for a neighbor.

Complete the proof using induction.

The composition on any n – 1 or more neighbor-shared graphs in G is strongly rooted.

Therefore H±G must be k + 1 neighbor shared.

Let C be a compact set of n£n stochastic matrices which are semi - contractive in the infinity norm. Then for each infinite sequence of matrices S1 S2, ... in C, the matrix product converges as i ! 1 to a rank one matrix as fast as

converges to zero.

Let C be a compact set of n£n scrambling matrices. Then for each infinite sequence of matrices S1 S2, ... in C, the matrix product converges as i ! 1 to a rank one matrix as fast as

converges to zero.

Scrambling matrices are semi-contractive in the infinity norm.

What can be said about convergence rate for scrambling matrices which are also flocking matrices?

Convergence rate bounds for products of scrambling matrices

Since all di · n, all non-zero fij satisfy

di = in-degree of vertex iD = diagonal {d1, d2, …, dn}n£ n

aij = 1 if i is a neighbor of j aij = 0 otherwise

A = [aij]

°(F) = N = neighbor shared

Worst Case |F|1 for F = D-1A0 = Scrambling

Since all di · n, all non-zero fij satisfy

di = in-degree of vertex iD = diagonal {d1, d2, …, dn}n£ n

aij = 1 if i is a neighbor of j aij = 0 otherwise

A = [aij]

Fix distinct i and j and let k be a shared neighbor. Then fik 0 fjk.



Vertex 1 has only itself as a neighbor

Vertex 2 has every vertex as a neighbor

n ¸ 3

For i > 2, vertex i has only itself and vertex 1 as neighbors

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43

How tight is this bound?



There exist infinite product of n £ n flocking matrices with neighbor-shared graphswhich actually converge to a rank-one matrix product 1c at this rate.

Flocking Matrices with Neighbor-Shared Graphs

Summary

Every infinite product of n £ n flocking matrices with neighbor-shared graphsconverges to a rank-one matrix product 1c at a rate no slower than

Let S be a compact set of n£n rooted matrices and write C for the compact set of all products of n – 1 matrices from S. Then for each infinite sequence of matrices S1 S2, ... in S, the matrix product converges as i ! 1 to a rank one matrix as fast as

converges to zero


Let C be a compact set of n£n scrambling matrices. Then for each infinite sequence of matrices S1 S2, ... in C, the matrix product converges as i ! 1 to a rank one matrix as fast as

converges to zero.

Convergence rates for products of stochastic matrices with rooted graphs

What can be said about the convergence rate for the product of an infinite sequence of flocking matrices whose sequence of graphs is repeatedly jointly rooted?

For any nonzero matrix M ¸ 0, define Á(M) = smallest nonzero element of M.

For S1 and S2 n£n stochastic matrices

We need a few ideas

Note that M can be written as

where

By induction

Recall that

Suppose S is scrambling

Claim that

Since S is scrambling, for any distinct i and j there must be a k such that

F(p) = { Fp Fp-1 F1: Fi 2 F, { ° (F1 ), ° (F2 ) , ... ,° (Fp) } is jointly rooted }

F = set of all n£n flocking matrices

Fk(p) = set of all products of k matrices from F(p)

Each matrix in F k(p) is scrambling if k ¸ n - 1

Each matrix in F (p) is rooted

For any F 2 F,

If S 2 Fk(p) , then S is the product of kp flocking matrices so

If S is scrambling |S|1 · 1 - Á(S)


If k = n – 1, then S is scrambling and

Therefore a convergence rate bound for the infinite product of flocking matrices whose sequence of graphs is repeatedly jointly rooted is

introduction to flocking {stochastic matrices} a. s. morse yale university gif – sur - yvette may...

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