Download Introduction to Flocking {Stochastic Matrices}

Supelec
EECI Graduate School in Control
Introduction to Flocking
{Stochastic Matrices}
A. S. Morse
Yale University
Gif – sur - Yvette
May 21, 2012
CRAIG REYNOLDS - 1987
BOIDS
The Lion King
CRAIG REYNOLDS - 1987
BOID
BOIDS
neighborhood
Flocking Rules
separation
alignment
cohesion
Demetri
Terzopoulos
Flocking Rules
separation
alignment
cohesion
Motivated by simulation results
reported in
V i c s e k e t a l . s i m u l a t e d a ° o c k o f n ag en t s f p ar t i c l esg al l m o v i n g i n t h e
p l an e a t t h e sa m e sp eed s, b u t w i t h d i ®er en t h ea d i n g s µ1 ; µ2 ; : : : ; µn .
s = sp eed
s
µi
µi = h ea d i n g
Each agent’s heading is updated at the same time as the rest using a local rule
based 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
neighbors of
agent i
agent i
Each agent is a neighbor of itself
Each agent has its own sensing radius ri
So neighbor relations are not symmetric
HEADING UPDATE EQUATIONS
s = sp eed
s
µi
µi = h ea d i n g
N i(t) = set of indices of agent i0s neighbors at time t
ni(t) = number of indices in N i(t)
Average at time t of headings of neighbors of agent i.
Another rule:
Vicsek Flocking Problem: Under what conditions do all n headings converge to a
common value?
Convex combination {Requires collaboration!}
Neighbor Graph N of Index Sets N 1, N 2 ,…., N n
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 N i, i 2 {1,2,…,n}
i
j i s a n ei g h b o r o f i
j
A self-arced graph = any graph G with self-arcs at all vertices
1
(1,2)
2
3
4
5
7
6
N ei g h b or g r ap h s = sel f - ar c ed g r a p h s
State Space Model
Adjacency Matrix AG of a graph G 2 G: An n£ n matrix of 0‘s and 1’s with aij = 1
whenever there is an arc in G from i to j.
_
4
1
(1,2)
In-degree = 4, out-degree = 1
2
3
In-degree of vertex i = number of
arcs entering vertex i
4
5
7
6
Out-degree of vertex i = number of
arcs leaving vertex i
State Space Model
Adjacency Matrix AG of a graph G 2 G: An n£ n matrix of 0‘s and 1’s with aij = 1
whenever there is an arc in G from i to j.
Flocking Matrix FN of a neighbor graph N 2 G:
bijection
where
D N = d i a g o n a l f d1 ; d2 ; : : : ; dn g
an d
ni = di = i n - d eg r ee o f v er t ex i =
Xn
j= 1
Update Eqns:
State Model:
aj i
Vicsek flocking problem: Under what conditions do all n headings converge to a
common value?
A switched linear system
No common quadratic Lyapunov function exists
B u t t h e n o n - n eg at i v e f u n c t i o n
V ( µ) = m a x f µi g ¡ m i n f µi g
i
is at least non-increasing along trajectories
But it takes much more to conclude that V ! 0
µ( t + 1 ) = F N( t ) µ( t )
i
Verify this!
Vicsek flocking problem: Under what conditions do all n headings converge to a
common value?
µ( t + 1 ) = F N( t ) µ( t )
Problem reduces to determining conditions on the sequence N(0), N(1), ...
under which
where
For if this is so, then
where
and so
{Right} Stochastic Matrices
Sn£ n= stochastic if
1 . i t h as o n l y n o n - n eg a t i v e en t r i es
2. its row sums all equal 1
Stochastic matrices closed under multiplication – flocking matrices are not
Flocking matrices are stochastic
Therefore it is sufficient to determine conditions on an infinite sequence of n£ n
stochastic matrices S1, S2, .... so that
This is a well studied problem in the theory of non-homogeneous Markov chains
If S is a compact set of n£ n stochastic matrices whose members each have
at 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?
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 .
We will be interested primarily in the cases p = 1, 2, 1 :
For any such p, define
1. Nonnegative:
|M|p ¸ 0
2. Homogeneous:
|rM|p = r|M|p
3. Triangle inequality:
|M1 + M2|p · |M1|p + |M2|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.}
|M|p = 0 ;
M=0
Additional Properties of
1. |M|p · 1 if ||M||p · 1
Because |M|p ·
||M||p
M is semi - contractive in the p semi-norm if |M|p < 1
2. Sub-multiplicative: Suppose M is a subset of Rn£ n such that M1 = 1
for all M 2 M . Then
Proof: Let c0 ,c1 and c2 denote values of c which minimize ||M2M1 - 1c||p,
||M1-1c||p, and ||M2-1c||p respectively.
1 = M2 1
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.
Proof: See board
We want to use this fact to prove that:
If S is a compact set of n£ n stochastic matrices whose members each have
at least one positive column, then for each sequence of matrices S1, S2, …
from S,
and this limit is approached exponentially fast.
To do this , it is enough to show that:
A stochastic matrix S is semi-contractive in the semi-norm | ¢|1 if S has a positive column.
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
T1 = S1 – 1c1 = (1 - c1)1
so
all row sums of T = (1 - c1)
¸ 0
because T ¸ 0
Moreover c ≠ 0 if and only if S has a positive column.
verify!
Therefore (1 – c1) < 1 if and only if S has a positive column
jSj 1 = m in jjS¡ 1d0jj 1 · jjS¡ 1cjj 1 = jjT jj 1 = ( 1¡ c1)
d
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 is
replaced by a 1.
In other words, for a nonnegative matrix M, ° (M) is that graph which has
an arc (i, j) from i to j whenever mj,i ≠ 0.
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 is
replaced by a 1.
° ( FN ) = ° ( A0N ) = N
A graph is strongly rooted if at least one vertex is adjacent to every vertex in
the graph
strongly rooted graph
Motivation for strongly rooted:
For any nonnegative matrix M, ° (M) has an arc (i, j) whenever mj,i ≠ 0.
° (M) is strongly rooted
,
M has a positive column
Transitioning from Matrices to Graphs
If S is a compact set of n£ n stochastic matrices whose members each have
at least one positive column, then for each sequence of matrices S1, S2, …
from S ,
and this limit is approached exponentially fast.
If S is a compact set of n£ n stochastic matrices whose members each have
a strongly rooted graph, then for each sequence of matrices S1, S2, …
from S ,
and this limit is approached exponentially fast.
Transitioning from Matrices to Graphs
When does
...
T
Tq
T2
T1
If
then
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.
Transitioning from Matrices to Graphs
As before G = set of all directed graphs with vertex set {1,2,....,n}.
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.
What motivates this definition?
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).
° (BA) = ° (B) ± ° (A)
Transitioning from Matrices to Graphs
Graph composition is defined so that for any two n£ n stochastic matrices S1 and S2
° (S2S1) = ° (S2) ± ° (S1)
Thus deciding when a finite product of stochastic matrices has a strongly rooted
graph is the same problem as deciding when a finite composition of graphs is
strongly rooted. So............
When is the composition of a finite number of graphs strongly rooted?
A rooted graph is any graph in G which has has at least one vertex v which, for
each vertex i 2 V there is a directed path from v to i.
3 roots
When is the composition of a finite number of graphs strongly rooted?
rooted graph
Every composition of (n – 1)2 or more self-arced, rooted graphs in G is strongly rooted.
Proof: See notes.
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.
Proof: See notes.
Every composition of (n – 1)2 or more self-arced, rooted graphs in G is strongly rooted.
Every composition of (n – 1)2 or more self-arced, rooted graphs in G is strongly rooted.
Define A (G ) = set of arcs in G
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 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 , then (j, j) 2 A (G2).
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).
In general for A (G2)[ A (G1) ≠ A (G 2±G 1) even if both graphs are self-arced.
However for self-arced graphs, if there is a directed path between i and j in
G 2±G 1 then there is a directed bath between i and j in G 2[ G 1
If S is a compact set of n£ n stochastic matrices whose members each have
a strongly rooted graph, then for each sequence of matrices S1, S2, …
from S ,
and this limit is approached exponentially fast.
...
rooted
strongly
rooted
strongly
rooted
strongly
rooted
If S is a compact set of n£ n stochastic matrices whose members each have
a self-arced, rooted graph, then for each sequence of matrices S1, S2, …
from S ,
and this limit is approached exponentially fast.
We can generalize further still...........
Repeatedly Jointly Rooted Sequences
An finite sequence of graphs G 1, G 2, ..., G
if the composed graph
G p ± G p-1 ±  ± G 1
is rooted.
p
in G is jointly rooted
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
G m(k -1)+1, ....... G k -1, k¸ 1,
is jointly rooted.
...
repeatedly
jointly rooted
rooted
rooted
rooted
If S is a compact set of n£ n stochastic matrices whose members each have
a self-arced, rooted graph, then for each sequence of matrices S1, S2, …
from S ,
and this limit is approached exponentially fast.
Suppose S is a compact set of n£ n stochastic matrices whose members each have
a 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
and this limit is approached exponentially fast.
Compactness of S does not in general imply compactness of C(m).
Construct an example for
2£ 2 matrices with m = 2.
Exception: If S is finite and thus compact {as in flocking applications} so is C(m)
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 ±  ± G 1 is rooted.
By the union of G 1 , G 2 2 G is meant that graph G 1 [ G
with arc set A (G 1) [ A (G 2).
2
in G
An finite sequence of graphs G 1, G 2, ..., G p in G is collectively rooted
if the union graph G p [ G p-1 [ [ G 1 is rooted.
In general, for self-arced graphs A ( G p [ G p-1 [ [ G 1) is a strictly proper
subset of A (G p ± G p-1 ±  ± G 1)
However, for each arc (i, j) 2 A (G p ± G p-1 ±  ± G 1) there must be a
directed path between (i, j) in G p [ G p-1 [ [ G 1
Therefore for self-arced graphs, the sequence G 1, G 2, ..., G
rooted if and only if it is collectively rooted.
p
in G is jointly
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 θ0.
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 all
rooted.
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 neighbor
relations were symmetric – that is if agent i is a neighbor of agent j then agent j is
a neighbor of agent i.
Mathematically, a symmetric neighbor relation means that i 2 N j ,
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 between
any 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
Simplified flocking matrix:
1. Symmetric
2. Nonnegative if g > max di
3. Fs1 = 1
Fs is stochastic if g > max di
L1 = D1 – A1 = 0
Comparing Flocking Matrices
Let N be a given self-arcd directed, symmetric, neighbor graph.
A = A0
° (Fs) = ° (F) = N
Therefore all convergence results hold without change for the simplified flocking rule
assuming symmetric neighbor relations.
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.
over C. = a convergence rate bound
So what we’d like is a uniform upper bound on |S|1
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
jSj 1 = m in jjS¡ 1d0jj 1 · jjS¡ 1cjj 1 = jjT jj 1 = ( 1¡ c1)
d
Note that the ith entry ci in c must be the smallest entry in the ith column of S.
Since (1 – c1) ·
1 - ci
for any i,
jSj 1 · ( 1 ¡ ci )
8i
Convergence Rate Bound for Flocking Matrices with Strongly Rooted Graphs
If ci is the smallest element in the ith column of S, then |S| 1 ·
1 – ci
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 graph
which 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
= a convergence rate bound
An Explicit Formula for the Infinity Semi-Norm
For any nonnegative n£ n matrix M,
See notes of a proof of this fact.
If M is a stochastic matrix, the quantity on the right is known as the coefficient of
ergodicity.
For any real numbers x and y
So for a stochastic matrix S
*
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
For fixed i and j, the kth term in the sum in
Therefore the sum in
*
*
will be positive iff sik > 0 and sjk > 0
will be positive iff sik > 0 and sjk > 0 for at least one value of k
Therefore |S|1 < 1 iff for each distinct i and j, sik > 0 and sjk > 0 for at least
one value of k.
A stochastic matrix with this property is called a scrambling matrix
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 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.
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.
In a strongly rooted graph there must be a root which is the neighbor of each vertex
in the graph. So...........
Every strongly rooted graph is neighbor shared.
The converse is clearly false.
Neighbor-Shared Directed Graph
A neighbor shared graph is a directed graph in which each pair of distinct vertices
share a common neighbor
3
2
1
1 and 2 share 2
1 and 3 share 2
1 and 4 share 4
2 and 3 share 2
2 and 4 share 4
3 and 4 share 1
4
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 reachable
from a single vertex.
Let {v1, v2, ..., vp} be any any such set and let v be a vertex from which all
of 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.
So by induction all n vertices are 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
Compositions of rooted and neighbor shared graphs
The composition on any n – 1 or more self-arced, rooted graphs in G is neighbor shared.
Proof: See notes.
We will use this fact a little later to get a convergence rate bound for products of
flocking matrices whose sequence of graphs is repeatedly jointly rooted.
The composition on any n – 1 or more neighbor-shared graphs in G is strongly rooted.
Let’s outline a proof of this:
Strongly rooted graphs ½ neighbor shared graphs ½ rooted graphs
The composition on any n – 1 or more neighbor-shared graphs in G is strongly rooted.
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.
Therefore H ±G must be k + 1 neighbor shared.
Complete the proof using induction.
Convergence rate bounds for products of scrambling matrices
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.
Scrambling matrices are semi-contractive in the infinity norm.
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.
What can be said about convergence rate for scrambling matrices which are also
flocking matrices?
Worst Case |F|1 for F = D-1A0 = Scrambling
° (F) = N = neighbor shared
aij = 1
aij = 0
A = [aij]
D = diagonal {d1, d2, …, dn}n£
n
if i is a neighbor of j
otherwise
di = in-degree of vertex i
Since all di · n, all non-zero fij satisfy
Worst Case |F|1 for F = D-1A0 = Scrambling
° (F) = N = neighbor shared
aij = 1
aij = 0
A = [aij]
D = diagonal {d1, d2, …, dn}n£
n
if i is a neighbor of j
otherwise
di = in-degree of vertex i
Since all di · n, all non-zero fij satisfy
Fix distinct i and j and let k be a shared neighbor. Then fik ≠ 0 ≠ fjk.
Worst Case |F|1 for F = D-1A0 = Scrambling
° (F) = N = neighbor shared
n¸ 3
Vertex 1 has only itself as a neighbor
Vertex 2 has every vertex as a neighbor
For i > 2, vertex i has only itself and vertex 1 as neighbors
3
4
2
1
How tight is this bound?
Summary
Flocking Matrices with Neighbor-Shared Graphs
Every infinite product of n £ n flocking matrices with neighbor-shared graphs
converges to a rank-one matrix product 1c at a rate no slower than
There exist infinite product of n £ n flocking matrices with neighbor-shared graphs
which actually converge to a rank-one matrix product 1c at this rate.
Convergence rates for products of stochastic matrices with rooted graphs
The composition on any n – 1 or more self-arced, rooted graphs in G is neighbor shared.
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.
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
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?
We need a few ideas
For any nonzero matrix M ¸ 0, define Á(M) = smallest nonzero element of M.
Note that M can be written as
where
For S1 and S2 n£ n stochastic matrices
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
If S is scrambling |S|1
·
1 - Á(S)
The composition on any n – 1 or more self-arced, rooted graphs in G is neighbor shared.
F = set of all n£ n flocking matrices
F (p) = { Fp Fp-1  F1:
Fi 2 F ,
{ ° (F1 ), ° (F2 ) , ... ,° (Fp) } is jointly rooted }
Each matrix in F (p) is rooted
F k(p) = set of all products of k matrices from F (p)
Each matrix in F k(p) is scrambling if k ¸ n - 1
For any F 2 F ,
If S 2 F k(p) , then S is the product of kp flocking matrices so
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