Cartesian Products of Z-Matrix Networks:
Factorization and Interval Analysis
Airlie Chapman and Mehran Mesbahi
(work also in part with Marzieh Nabi-Abdolyouse)
Distributed Space Systems Lab (DSSL)
University of Washington
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
andofInterval
Washington
Space
Analysis
Systems1Lab
/ 27(D
The Network in the Dynamics
General Dynamics
ẋ (t ) = f (G, x (t ), u (t ))
y (t ) = g (G, x (t ), u (t ))
Network
Graph Spectrum
Random Graphs
Automorphisms
Graph Factorization
System Dynamics
Rate of convergence
Random Matrices
Homogeneity
Decomposition
&
1
2
Airlie Chapman and Mehran Mesbahi
State Dynamics
Controllability
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
andofInterval
Washington
Space
Analysis
Systems2Lab
/ 27(D
The Network in the Dynamics
First Order, Linear Time Invariant model
ẋi (t ) = −wii xi (t ) + ∑ wij xj (t )+ui (t )
yi (t ) = xi (t )
i ∼j
Dynamics
ẋ (t ) = −A(G)x (t ) + B (S )u (t )
y (t ) = C (R )x (t )
A(G) =
w11
−w21
.
.
.
−wn1
−w12
w22
···
···
.
.
.
..
.
−wn,n−1
A(G): Z-matrix, e.g. Laplacian (wii = ∑ wij ) and Advection matrices
(wii = ∑ wji )
Input node set S = {vi , vj , . . . }, B (S ) = [ei , ej , . . . ]
Output node set R = {vp , vq , . . . }, C (R ) = [ep , eq , . . . ]T
Airlie Chapman and Mehran Mesbahi
−w1n
−wn−1,n
wnn
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
andofInterval
Washington
Space
Analysis
Systems3Lab
/ 27(D
Graph Cartesian Product
Cartesian product GH
Vertex set: V (GH) = V (G) × V (H)
Edge set: (x1 , x2 ) ∼ (y1 , y2 ) is in GH
if
x1 ∼ y1 and x2 = y2 or x1 = y1 and x2 ∼ y2
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
andofInterval
Washington
Space
Analysis
Systems4Lab
/ 27(D
Graph Cartesian Product
Cartesian product GH
Vertex set: V (GH) = V (G) × V (H)
Edge set: (x1 , x2 ) ∼ (y1 , y2 ) is in GH
if
x1 ∼ y1 and x2 = y2 or x1 = y1 and x2 ∼ y2
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
andofInterval
Washington
Space
Analysis
Systems5Lab
/ 27(D
Graph Cartesian Product
Cartesian product GH
Vertex set: V (GH) = V (G) × V (H)
Edge set: (x1 , x2 ) ∼ (y1 , y2 ) is in GH
if
x1 ∼ y1 and x2 = y2 or x1 = y1 and x2 ∼ y2
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
andofInterval
Washington
Space
Analysis
Systems6Lab
/ 27(D
Part 1:
Factoring the State Dynamics
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
andofInterval
Washington
Space
Analysis
Systems7Lab
/ 27(D
State Dynamics Factorization
Lemma 1: Uncontrolled Dynamics
Consider the dynamics
ẋ (t ) = −A(G1 G2 . . . Gn )x (t ).
Then, when properly initialized, one has
x (t ) = x1 (t ) ⊗ x2 (t ) ⊗ · · · ⊗ xn (t )
where ẋi (t ) = −A(Gi )xi (t ).
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
andofInterval
Washington
Space
Analysis
Systems8Lab
/ 27(D
State Dynamics Factorization
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
andofInterval
Washington
Space
Analysis
Systems9Lab
/ 27(D
Graph Factorization
A graph can be factored as well as composed...
Theorem (Sabidussi 1960)
Every connected graph can be factored as a Cartesian product of prime graphs.
Moreover, such a factorization is unique up to reordering of the factors.
Primes: G = G1 G2 implies that either G1 or G2 is
K1
Number of prime factors is at most log |G|
Algorithms
O |V |4.5
O |V |4 from isometrically
Feigenbaum (1985) Winkler (1987) -
embedding graphs by Graham
and Winkler (1985)
Feder (1992) -
O (|V | |E |)
Imrich and Peterin (2007) -
O (|E |)
C++ implementation by Hellmuth and Staude
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
10Lab
/ 27(D
Controlled Factorization Lemma
Lemma 2: Controlled Dynamics
Consider the dynamics
ẋ (t ) = −A(G1 G2 . . . Gn )x (t ) + B (S1 × S2 × · · · × Sn )u (t ).
y (t ) = C (R1 × R2 × · · · × Rn )x (t )
Then, when properly initialized and u (t ) = u1 (t ) ⊗ u2 (t ) ⊗ · · · ⊗ un (t ), one has
y (t ) = yu (t ) + yf (t ),
yu (t ) = y1u (t ) ⊗ y2u (t ) ⊗ · · · ⊗ ynu (t )
yf (t ) =
Z t
0
ẏ1f (τ) ⊗ ẏ2f (τ) ⊗ · · · ⊗ ẏnf (τ)d τ
where ẋi (t ) = −A(Gi )xi (t ) + B (Si )ui (t ) and yi (t ) = C (Ri )xi (t ).
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
11Lab
/ 27(D
Controlled Factorization Lemma
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
12Lab
/ 27(D
Controlled Factorization Lemma
Lemma 2: Controlled Dynamics
Consider the dynamics
ẋ (t ) = −A(∏ Gi )x (t ) + B (∏ Si )u (t )
×
y (t ) = C (∏ Ri )x (t )
×
Then, when properly initialized and u (t ) = ∏⊗ ui (t ), one has
y (t ) = ∏ yiu (t ) +
⊗
Z t
0
∏ ẏif (τ)d τ
⊗
where ẋi (t ) = −A(Gi )xi (t ) + B (Si )ui (t ) and yi (t ) = C (Ri )xi (t ).
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
13Lab
/ 27(D
Controlled Factorization - Idea of the Proof
Firstly
A(G1 G2 ) = A(G1 ) ⊗ I + I ⊗ A(G2 ) = A(G1 ) ⊕ A(G2 )
Also
Using
e A(G1 )⊕A(G2 ) = e A(G1 ) ⊗ e A(G2 )
B (S1 × S2 ) = B (S1 ) ⊗ B (S2 )
As well as
e −A(G1 G2 )t Bu (t ) = (e −A(G1 )t ⊗ e −A(G2 )t )(B (S1 ) ⊗ B (S2 ))(u1 (t ) ⊗ u2 (t ))
= e −A(G1 )t B (S1 )u1 (t ) ⊗ e −A(G2 )t B (S2 )u2 (t )
The proof follows from these observations.
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
14Lab
/ 27(D
Approximate Graph Products
A(G) ∈ A(G 1 G 2 ), A(G 1 G 2 )
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
15Lab
/ 27(D
Approximate Factorization Lemma
Lemma 3: Approximation
Consider the dynamics
ẋ (t ) = −A(G)x (t ) + B (S )u (t ),
y (t ) = C (R )x (t )
A(G) ∈ A(∏ G i ), A(∏ G i ) , S ∈ ∏× S i , ∏× S i , R = ∏× R i , ∏× R i and
positive u (t ) ∈ [∏⊗ u i (t ), ∏⊗ u i (t )]. Then, when properly initialized, one has
∏ y iu (t ) +
⊗
Z t
0
∏ ẏ if (τ)d τ ≤ y (t ) ≤ ∏ y iu (t ) +
⊗
⊗
Z t
0
∏ ẏ if (τ)d τ
⊗
where ẋ i (t ) = −A(G i )x i (t ) + B (S i )u i (t ), y i (t ) = C (R i )x i (t ) and similarily for
y i (t ).
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
16Lab
/ 27(D
Approximate Graph Products
Can a graph be approximately factored as well as approximately composed?
Distance between graphs d (G, H) is the smallest integer k such that
V (G 0 )4V (H0 ) + E (G 0 )4E (H0 ) ≤ k .
A graph G is a k -approximate graph product if there is a product H such that
d (G, H) ≤ k .
Lemma (Hellmuth 2009)
For xed k all Cartesian k -approximate graph products can be recognized in
polynomial time.
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
17Lab
/ 27(D
30
10
3
4
0
1
2
3
4
0
6
x7
x6
x5
4
5
2
3
4
0
20
x10
x9
30
10
2
3
4
0
8
8
6
6
4
4
3
1
2
3
1
2
time
3
4
Airlie Chapman and Mehran Mesbahi
0
1
2
time
3
0
4
25
20
15
10
5
0
4
2
3
4
5
0
1
2
3
4
4
5
4
3
2
1
0
1
2
3
4
4
10
8
6
4
2
0
1
2
3
4
1
2
time
3
4
12
10
8
6
4
2
4
2
2
0
2
12
10
8
6
4
x14
x13
1
1
x11
2
x15
1
1
x8
2
1
2
3
x12
1
10
0
10
20
10
15
0
15
x4
40
15
1
2
3
10
8
6
4
2
0
6
x16
0
20
x3
50
40
30
20
10
x2
x1
Approximate Factorization Lemma: An Example
4
2
1
2
time
3
4
0
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
18Lab
/ 27(D
Part 2:
Factoring Controllability
(work with Marzieh Nabi-Abdolyouse)
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
19Lab
/ 27(D
Controllability
Dynamics are controllable if for any x (0), xf and tf there exists an input
u (t ) such that x (tf ) = xf .
Signicant in networked robotic systems, human-swarm interaction, network
security, quantum networks.
Challenging to establish for large networks
Known families of controllable graphs for
selected inputs
Paths
Circulants
Grids
Distance regular graphs
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
20Lab
/ 27(D
Controllability Factorization - Product Control
Theorem 1: Product Controllability
The dynamics
ẋ (t ) = −A(∏ Gi )x (t ) + B (∏ Si )u (t )
×
y (t ) = C (∏ Ri )x (t )
×
where A(∏ Gi ) has simple eigenvalues is controllable/observable if and only if
ẋi (t ) = −A(Gi )xi (t ) + B (Si )ui (t )
yi (t ) = C (Ri )xi (t )
is controllable/observable for all i .
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
21Lab
/ 27(D
Controllability Factorization - Idea of the Proof
Popov-Belevitch-Hautus (PBH) test
(A, B ) is uncontrollable if and only if there exists a left eigenvalue-eigenvector pair
(λ , v ) of A such that v T B = 0.
Eigenvalue and eigenvector relationship:
Eigenvalue
Eigenvector
A(G1 ) A(G2 ) A(G1 G2 )
λi
vi
µj
uj
λi + µj
vi ⊗ uj
Also (vi ⊗ ui )T (B (S1 ) ⊗ B (S2 )) = viT B (S1 ) ⊗ uiT B (S2 )
The proof follows from these observations.
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
22Lab
/ 27(D
Controllability Factorization - Layered Control
Theorem 2: Layered Controllability
The dynamics
ẋ (t ) = −A(∏ Gi )x (t ) + B (∏ Si )u (t )
×
y (t ) = C (∏ Ri )x (t )
×
where Si = Ri = V (Gi ) for i = 2, . . . , n is controllable/observable if and only if
ẋ1 (t ) = −A(G1 )x1 (t ) + B (S1 )u1 (t )
y1 (t ) = C (R1 )x1 (t )
is controllable/observable.
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
23Lab
/ 27(D
Controllability Factorization - Layered Control
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
24Lab
/ 27(D
Uncontrollability through Symmetry
Proposition (Rahmani and Mesbahi 2006)
(A(G), B (S )) is uncontrollable if there exists an automorphism of G which xes all
inputs in the set
S (i.e., S is not a determining set.)
The determining number of a graph G , denoted
so that G has a determining set S of size r .
Det (G), is the smallest integer r
Corollary
(A(G), B (S )) is uncontrollable if |S | < Det (G).
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
25Lab
/ 27(D
Breaking Symmetry
Automorphism group for graph Cartesian products
The automorphisms for a connected G is generated by the automorphisms of its
prime factors.
Proposition: Automorphism group for graph Cartesian products
For controllable pairs (A(G1 ), B (S1 )) and (A(G2 ), B (S2 )) where |S1 | = Det (G1 )
and |S2 | = 1. Then S = S1 × S2 is the smallest input set such that
A(G1 G2 , B (S )) is controllable.
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
26Lab
/ 27(D
Conclusion
Explored decomposition of Z-matrix based network into smaller
factor-networks
Provided exact and approximate factorization of state dynamics
Presented a factorization of controllability - a product and layered approach
Linked the factors symmetry to smallest controllable input set
Future work involves examining other graph products in network dynamics
Airlie Chapman and Mehran Mesbahi
Cartesian
(work also
Products
in part of
with
Z-Matrix
MarziehNetworks:
Nabi-Abdolyouse)
Factorization
University
(Distributed
and
of Washington
Interval
Space
Analysis
Systems
27Lab
/ 27(D
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