Download - Graph partitioning and characteristic polynomials of Laplacian matrics of Roach-type graphs
Graph partitioning and eigen polynomials ofLaplacian matrices of Roach-type graphs
Yoshihiro Mizoguchi
Institute of Mathematics for Industry,Kyushu University
Algebraic Graph Theory,Spectral Graph Theory and Related Topics
5th Jan. 2013 at Nagoya University
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 1 / 32
Table of contents
...1 Introduction
...2 Chebyshev polynomials
...3 Tridiagonal matrices
...4 Laplacian Matrix
...5 Mcut, Lcut and spectral clustering
...6 Conclusion
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 2 / 32
Biogeography-Based Optimization (1)
Let Ps be the probability that the habitat contains exactly S species. Wecan arrange Ps equations into the single matrix equation
P0
P1
P2...Pn−1
Pn
=
−(λ0 + µ0) µ1 0 · · · 0
λ0 −(λ1 + µ1) µ2. . .
......
. . .. . .
. . ....
.... . . λn−2 −(λn−1 + µn−1) µn
0 . . . 0 λn−1 −(λn + µn)
P0P1P2...Pn−1Pn
where λs and µs are the immigration and emigration rates when there are
S species in the habitat.Generally λ0 > λ1 > · · · > λn and µ0 < µ1 < · · · < µn hold and weassume λs =
n−sn and µs =
sn in this talk.
[Sim08] D.Simon, Biogeography-Based Optimization,IEEE Trans. on evolutionary computation, 2008.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 3 / 32
Biogeography-Based Optimization (2)
.Theorem..
......
The (n + 1) eigenvalues of the biogeography matrix
A =
−1 1/n 0 · · · 0
n/n −1 2/n . . ....
.... . .
. . .. . .
......
. . . 2/n −1 n/n0 . . . 0 1/n −1
are {0,−2/n,−4/n, . . . ,−2}.
[IS11] B.Igelnik, D. Simon, The eigenvalues of a tridiagonal matrix inbiogeography, Appl. Mathematics and Computation, 2011.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 4 / 32
Heat equation (Crank-Nicolson method) (1)
ut = uxx x ∈ [0, 1] and t > 0
with initial and Dirichlet boundary condition given by:
u(x, 0) = f (x), u(0, t) = g(t) and u(1, t) = h(t)
The finite difference discretization can be expressed as:
Aun+1 = Bun + c
where
A =
1 + α −r/2 0−r/2 1 + r −r/2
. . .. . .
. . .
−r/2 1 + r −r/20 −r/2 1 + α
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 5 / 32
Heat equation (Crank-Nicolson method) (2)
and
B =
1 − β r/2 0r/2 1 − r r/2
. . .. . .
. . .
r/2 1 − r r/20 r/2 1 − β
.
We note un = (un1, un
2, . . . , un
m)T. The parapmeters α and β are given by:
α = β = 3r/2 for the implicit boundary conditions;
α = r and β = 2r for the explicit boundary conditions
The iteration matrix M(r) = A−1B controls the stability of the numericalmethod to compute Aun+1 = Bun + c.
[CM10] J.A. Cuminato, S. McKee, A note on the eigenvalues of a specialclass of matrices, J. of Computational and Applied Mathematics, 2010.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 6 / 32
Tridiagonal Matrix (1)
An =
−α + b c 0 0 · · · 0 0a b c 0 · · · 0 00 a b c · · · 0 0· · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · b c0 0 0 0 · · · a −β + b
n×n
.Theorem..
......
Suppose α = β =√
ac , 0. Then the eigenvalues λk of An are given by
λk = b + 2√
ac coskπn
and the corresponding eigenvectors u(k) = (u(k)j
)
are given by u(k)j= ρ j−1 sin
k(2 j − 1)π2n
for k = 1, 2, · · · , n − 1 and
u(n)j= (−ρ) j−1 where ρ =
√a/c.
[Yue05] W-C. Yueh, Eigenvalues of several tridiagonal matrices, AppliedMathematics E-Notes, 2005.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 7 / 32
Tridiagonal Matrix (2)
Consider the n × n matrix C = (min{ai − b, a j − b})i, j=1,...,n..Proposition..
......
For a > 0 and a , b, the tridiagonal matrix of order n
Tn =
1 + aa−b −1−1 2 −1
......
...−1 2 −1
−1 1
is the inverse of (1/a)C.
[dF07] C.M. da Fonseca, On the eigenvalues of some tridiagonal matrices,J. of Computational and Applied Mathematics, 2007.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 8 / 32
Chebyshev polynomials
For n ∈ N and x ∈ R, we define functions Tn(x) and Un(x) as follows.
T0(x) = 1, T1(x) = x,U0(x) = 1, U1(x) = 2x,
Tn+1(x) = 2xTn(x) − Tn−1(x), and
Un+1(x) = 2xUn(x) − Un−1(x).
We note cos nθ = Tn(cos θ), and sin(n + 1)θ = Un(cos θ) sin θ for θ ∈ R..Proposition..
......
Let x = cos θ. Then
Tn(x) = 0 ⇔ x = cos((2k + 1)π
2n) (k = 0, · · · , n − 1).
Un(x) = 0 ⇔ x = cos(kπ
n + 1) (k = 1, · · · , n).
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 9 / 32
Tridiagonal matrix An(a, b)We define a n × n matrix An(a, b) as follows:
An(a, b) =
a b 0 · · · · · · · · · 0
b a b 0...
0 b a b 0...
.... . .
. . .. . .
. . .. . .
...... 0 b a b 0... 0 b a b0 · · · · · · · · · 0 b a
.
We put |A0(a, 1)| = 1, then |An(a, 1)| = a|An−1(a, 1)| − |An−2(a, 1)|,|A1(a, 1)| = a and An(a, b) = bn · An (a/b, 1)..Proposition..
......|An(a, b)| = bn ·
sin(n + 1)θsin θ
where cos θ =a
2b.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 10 / 32
Tridiagonal matrix Bn and Cn (1)Let n ≥ 3.
Bn(a0, b0, a, b) =
a0 b0 0 · · · 0b00 An−1(a, b)...0
Cn(a, b, a0, b0) =
0
An−1(a, b)...0b0
0 · · · 0 b0 a0
We note that
|Bn(a0, b0, a, b)| = a0|An−1(a, b)| − b20|An−2(a, b)|, and
||Cn(a, b, a0, b0)|| = |a0|An−1(a, b)| − b20|An−2(a, b)||.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 11 / 32
Tridiagonal matrix Bn and Cn (2)We define functions
gn(β) = 2 sin((n + 1)β) + sin nβ − sin((n − 1)β), and
hn(β) = 2 sin((n + 1)β) − sin nβ − sin((n − 1)β)
before introducing the next Lemma..Proposition..
......
Let n ≥ 3.∣∣∣∣∣∣∣Bn(λ − 1,1√
2, λ − 1,
12
)
∣∣∣∣∣∣∣ = 12n−1
cos nα, (λ = 1 + cos α),∣∣∣∣∣∣∣Cn(η − 23,
13, η − 1
2,
1√
6)
∣∣∣∣∣∣∣ = 12 · 3n · sin β
gn(β), (η =23
(1 + cos β)),∣∣∣∣∣∣∣Cn(µ − 43,
13, µ − 3
2,
1√
6)
∣∣∣∣∣∣∣ = 12 · 3n · sin β
hn(β), (µ =23
(2 + cos β)).
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 12 / 32
Tridiagonal matrix Qn(a0, b0, a, b)
Let n ≥ 4. We define n × n matrix Qn(a0, b0, a, b) as follows:
Qn(a0, b0, a, b) =
a0 b0 0 · · · 0
b0...
0 An−2(a, b) 0... b00 · · · 0 b0 a0
We note that
|Qn(a0, b0, a, b)| = a0|Cn−1(a, b, a0, b0)| − b20|Cn−2(a, b, a0, b0)|.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 13 / 32
Laplacian matrix of a graph.Definition (Weighted normalized Laplacian)..
......
The weighted normalized Laplacian L(G) = (ℓi j) is defined as
ℓi j =
1 − w j j
d jif i = j,
− wi j√di d j
if vi and v j are adjacent and i , j,
0 otherwise.
The adjacency matrix A(P5) and the normalized Laplacian matrix L(P5) ofa path graph P5.
A(P5) =
0 1 0 0 01 0 1 0 00 1 0 1 00 0 1 0 10 0 0 1 0
L(P5) =
1 − 1√2
0 0 0− 1√
21 − 1
2 0 00 − 1
2 1 − 12 0
0 0 − 12 1 − 1√
20 0 0 − 1√
21
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 14 / 32
Characteristic polynomial of L(Pn).Proposition..
......
Let n ≥ 4.
|λIn − L(Pn)| = −(
12
)n−2
(sin α sin((n − 1)α))
where λ = 1 + cos α. That is λ = 1 − cos(kπ
n − 1) (k = 0, . . . , n − 1).
We note
L(Pn) = Qn
1,− 1√
2, 1,−1
2
, and
λIn − L(Pn) = Qn
λ − 1,1√
2, λ − 1,
12
.Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 15 / 32
Characteristic polynomial of L(Pn,k) (1)
Let n ≥ 3 and k ≥ 3. Then
L(Pn,k) =
Bn(1,− 1√2, 1,− 1
2 ) Xn,k
X tn,k
Ck( 23 ,−
13 ,
12 ,−
1√6)
where Xn,k is the n × k matrix defined by
Xn,k =
0 · · · · · · 0...
...
0 0...
− 1√6
0 · · · 0
.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 16 / 32
Characteristic polynomial of L(Pn,k) (2)
.Proposition..
......
Let
pn,k(λ) =1
2n3k sin β(gk(β) cos(nα)) − gk−1(β) cos((n − 1)α)).
Then ∣∣∣λIn+k − L(Pn,k)∣∣∣ = pn,k(λ),
where λ = 1 + cos α and λ =23
(1 + cos β).
∣∣∣λIn+k − L(Pn,k)∣∣∣ =
∣∣∣∣∣∣∣∣Bn(λ − 1, 1√
2, λ − 1, 1
2 ) Xn,k
X tn,k
Ck(λ − 23 ,
13 , λ −
12 ,
1√6)
∣∣∣∣∣∣∣∣Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 17 / 32
Characteristic polynomial of L(Rn,k) (1)
1 2 3 4 5 6
7 8 9 10 11 12
Let n ≥ 3 and k ≥ 3. Then L(Rn,k) =Bn(1,− 1√
2, 1,− 1
2 ) Xn,k O OX t
n,kCk(1,− 1
3 , 1,−1√6) O Ck(− 1
3 , 0,−12 , 0)
O O Bn(1,− 1√2, 1,− 1
2 ) Xn,k
O Ck(− 13 , 0,−
12 , 0) X t
n,kCk(1,− 1
3 , 1,−1√6)
.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 18 / 32
Characteristic polynomial of L(Rn,k) (2).Proposition..
......
Let n ≥ 3, k ≥ 3 and
pn,k(λ) =1
2n3k sin β(gk(β) cos(nα)) − gk−1(β) cos((n − 1)α)), and
qn,k(λ) =1
2n3k sin γ(hk(γ) cos(nα) − hk−1(γ) cos((n − 1)α)). Then
|λIn+k − L(Rn,k)| = pn,k(λ) · qn,k(λ).
where λ = 1 + cos α =23
(1 + cos β) =23
(2 + cos γ).
∣∣∣λIn+k − L(Rn,k)∣∣∣ =
∣∣∣∣∣∣∣∣Bn(λ − 1, 1√
2, λ − 1, 1
2 ) Xn,k
X tn,k
Ck(λ − 23 ,
13 , λ −
12 ,
1√6)
∣∣∣∣∣∣∣∣×
∣∣∣∣∣∣∣∣Bn(λ − 1, 1√
2, λ − 1, 1
2 ) Xn,k
X tn,k
Ck(λ − 43 ,
13 , λ −
32 ,
1√6)
∣∣∣∣∣∣∣∣= pn,k(λ) × qn,k(λ).
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 19 / 32
∣∣∣∣Bn(λ − 1, 1√2, λ − 1, 1
2 )∣∣∣∣ (Calculation)
Let λ = 1 + cos α. Then we have∣∣∣∣∣∣∣Bn(λ − 1,1√
2, λ − 1,
12
)
∣∣∣∣∣∣∣ = (λ − 1)∣∣∣∣∣An−1(λ − 1,
12
)∣∣∣∣∣ − 1
2
∣∣∣∣∣An−2(λ − 1,12
)∣∣∣∣∣
= (λ − 1)(
12
)n−1 sin nαsin α
− 12
(12
)n−2 sin(n − 1)αsin α
=
(12
)n−1
· 1sin α
((λ − 1) sin nα − sin(n − 1)α)
=
(12
)n−1
· 1sin α
(cos α sin nα − sin(nα − α))
=
(12
)n−1
· 1sin α
(cos nα sin α)
=
(12
)n−1
cos nα.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 20 / 32
∣∣∣∣Bn(λ − 1, 1√2, λ − 1, 1
2 )∣∣∣∣ (Mathematica)
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 21 / 32
∣∣∣∣Cn(η − 23 ,
13 , η −
12 ,
1√6)∣∣∣∣ (Calculation)
Let η =23
(1 + cos β) and gn(β) = 2 sin(n + 1)β + sin nβ − sin(n − 1)β. Then we have∣∣∣∣∣∣∣Cn(η − 23,
13, η − 1
2,
1√
6)
∣∣∣∣∣∣∣ =
(η − 1
2
) ∣∣∣∣∣∣An−1
(η − 2
3,
13
)∣∣∣∣∣∣ − 16
∣∣∣∣∣∣An−2
(η − 2
3,
13
)∣∣∣∣∣∣=
(13
)n−1 ((η − 1
2
) sin nβsin β
− 12
sin(n − 1)βsin β
)=
(13
)n−1 ((16+
23
cos β) sin nβ
sin β− 1
2sin(n − 1)β
sin β
)=
(13
)n−1 16 sin β
(sin nβ + 4 cos β sin nβ − 3 sin(nβ − β))
=
(13
)n−1 16 sin β
(2 sin(n + 1)β + sin nβ − sin(n − 1)β)
=
(13
)n 12 sin β
gn(β).
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 22 / 32
∣∣∣∣Cn(η − 23 ,
13 , η −
12 ,
1√6)∣∣∣∣ (Mathematica)
Some manual computations for gn(θ) (× sin).Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 23 / 32
pn,k(λ) (Mathematica)
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 24 / 32
qn,k(λ) (Mathematica)
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 25 / 32
Minimum Normalized Cut Mcut(G)
.Definition (Normalized cut)..
......
Let G = (V, E) be a connected graph. Let A, B ⊂ V, A , ∅, B , ∅ andA ∩ B = ∅. Then the normalized cut Ncut(A, B) of G is defined by
Ncut(A, B) = cut(A, B)(
1vol(A)
+1
vol(B)
).
.Definition (Mcut(G))..
......
Let G = (V, E) be a connected graph. The Mcut(G) is defined by
Mcut(G) = min{Mcut j(G) | j = 1, 2, . . . }.
Where,
Mcut j(G) = min{Ncut(A,V \ A) | cut(A,V \ A) = j, A ⊂ V}.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 26 / 32
Mcut(R6,3)
G0 (even) G1 (odd)1 2 3 4 5 6
7 8 9 10 11 12
1 2 3 4 5 6
7 8 9 10 11 12
Ncut(A0, B0) = 2 × (116+
110
) =1340= 0.325
Ncut(A1, B1) = 3 × (113+
113
) =6
13≈ 0.462
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 27 / 32
Spectral Clustering.Definition (Lcut(G))..
......
Let G = (V, E) be a connected graph, λ2 the second smallest eigenvalueof L(G), U2 = ((U2)i) (1 ≤ i ≤ |V|) a second eigenvector of L(G) with λ2.We assume that λ2 is simple. Then Lcut(G) is defined asLcut(G) = Ncut(V+(U2) ∪ V0(U2),V−(U2)).
1
2
3
4
-
-
-
-
5
6
7
+
+
+
1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20 21 22
Lcut(G) = Mcut(G) Lcut(R4,7) = Mcut(R4,7)
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
Mcut(R6,4) Lcut(R6,4)
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 28 / 32
Roach Graph Observation
.Proposition..
......
Let Rn,k be a roach-type graph. If Lcut(Rn,k) = Mcut(Rn,k) then a secondeigen vector of L(Rn,k) is an even vector.
.Proposition..
......
Let R2k,k be a roach-type graph, P2k,k a weighted path and P4k a pathgraph.
1. λ2(L(P4k)) = 1 − π4k−1 .
2. λ2(L(R2k,k)) < λ2(L(P4k)).3. λ2(L(P4k)) < λ2(L(P2k,k)).4. A second eigenvector of L(R2k,k) is an odd vector.
5. Mcut(R2k,k) < Lcut(R2k,k).
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 29 / 32
Conclusion
We give followings in this talks:
Tridiagonal matrices, Laplacian of graphs and spectral clusteringmethod.
Concrete formulae of characteristic polynomials of tridiagonalmatrices.
Mathematica computations for characteristic polynomials.
Concrete formulae of eigen-polynomials of (P2k,k) and L(R2k,k).Proof of Lcut does not always give an optimal cut.
We are not able to decide the simpleness of the second eigenvalue forPn,k and Rn,k.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 30 / 32
Reference I
A. Behn, K. R. Driessel, and I. R. Hentzel.The eigen-problem for some special near-toeplitz centro-skewtridiagonal matrices.arXiv:1101.5788v1 [math.SP], Jan 2011.
H-W. Chang, S-E. Liu, and R. Burridge.Exact eigensystems for some matrices arising from discretizations.Linear Algebra and its Applications, 430:999–1006, 2009.
J. A. Cuminato and S. McKee.A note on the eigenvalues of a special class of matrices.Journal of Computational and Applied Mathematics, 234:2724–2731,2010.
C. M. da Fonseca.On the eigenvalues of some tridiagonal matrices.Journal of Computational and Applied Mathematics, 200:283–286,2007.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 31 / 32
Reference II
B. Igelnik and D. Simon.The eigenvalues of a tridiagonal matrix in biogeography.Applied Mathematics and Computation, 218:195–201, 2011.
S. Kouachi.Eigenvalues and eigenvectors of tridiagonal matrices.Electronic Journal of Linear Algebra, 15:115–133, 2006.
D. Simon.Biogeography-based optimization.IEEE Transactions on Evolutionary Computation, 12(6):702–713,2008.
W. Yueh.Eigenvalues of several tridiagonal matrices.Applied Mathematics E-Notes, 5:66–74, 2005.
Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 32 / 32