Curvature dimension inequalities on directed graphs

0.09 MB·

by  Taiki Yamada

Curvature dimension inequalities on directed graphs
Taiki Yamada
January 9, 2017
Abstract
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Inthispaper,wedefinethecurvaturedimensioninequalitiesCD(m,K)onfinitedirectedgraphs
1
0 modifying the caseofundirectedgraphs. As a mainresult, we evaluatem andK onfinite directed
2 graphs.

n
a
1 Introduction
J

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In Riemannian geometry, we usually use the curvature dimension inequalities CD(m,K) to consider

]
G analytic properties of Ricci curvature. The curvature dimension inequalities are defined by the Γ-
D calculation. To define Γ-calculation on graphs, we need the symmetric graph laplacian. In general,
the graph laplacian is defined using an adjacent matrix. Jost-Liu [2] defined the curvature dimension
.
h
inequalities on finite undirected graphs, and found relations between Ricci curvature and CD(m,K)
t
a
. However, an adjacent matrix of directed graphs is usually asymmetry, so the graph laplacian is also
m
asymmetry. On the other hand, there is an another definition of the graph laplacian by Chung [1].
[
He defined the graph laplacian by using the transition probability matrix, and symmetrized it.

1
We have defined Ricci curvature on directed graphs [5]. So, we define the curvature dimension in-
v
equalities on finite directed graphs, and find specific m and K in this paper.
0
1
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1 2 Curvature dimension inequalities on directed graphs
0
.
1 Throughout the paper, let G = (V,E) be a directed graph. For x,y ∈ V, we write (x,y) as an edge
0
from x to y if any. We denote the set of vertices of G by V(G) and the set of edges by E(G).
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Definition 2.1. (1) A path from vertex x to vertex y is a sequence of edges
:
v
i (x,a1),(a1,a2),··· ,(an−2,an−1),(an−1,y), where n ≥ 1. We call n the length of the path.
X (2) The distance d(x,y) between two vertices x,y ∈ V is the length of a shortest path connecting
r them.
a
Remark 2.2. The distance function has positivity and triangle inequality, but symmetry is not
necessarily satisfied.
Definition 2.3. (1) For any x ∈ V, the out-neighborhood (in-neighborhood) of x is defined as
Sout(x) := {y ∈ V |(x,y) ∈ E} (Sin(x) := {z ∈ V | (z,x) ∈ E}).
(2) For all x ∈ V, The degree of x, denoted by d , is the number of edges from x. i.e., d = |Sout(x)|.
x x
We sometimes use the following notation.
Nout(x) := {x}∪Sout(x) (Nin(x) := {x}∪Sin(x)).
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Definition 2.4. A probability matrix M is defined by the following.
α
α, v = v ,
j i

1−α
(Mα)i,j = mαvi(vj) :=  dvi , (vi,vj) ∈ E(G),
0, otherwise,



where α ∈ [0,1) and (M ) is an (i,j)-element of M .
α i,j α
Remark 2.5. If G is strongly connected, the matrix M is irreducible, so by Perron-Frobenius’
α
theorem, there exists φ such that all elements of φ is positive and φM = φ (see [1]).
α
Definition 2.6. For all f :V(G) → R, a laplacian on directed graphs is defined by the following.
1  
∆(f)(v ) = w (f(v )−f(v ))+ w (f(v )−f(v )) ,
i 2φ(v ) X ij j i X ki k i 
i vj∈Sout(vi) vk∈Sin(vi)
 
where w = φ(v )(M )
ij i α i,j
Definition 2.7. For all functions f, g :V(G) → R, a Γ-calculation is defined by the following.
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Γ(f,g) = {∆(fg)−f∆(g)−g∆(f)},
2
1
Γ (f,g) = {∆(Γ(f,g))−Γ(f,∆(g))−Γ(∆(f),g)}.
2
2
Definition 2.8. We say a graph G is satisfies the curvature dimension inequalities CD(m,K) if for
all functions f : V(G) → R and for all vertices v ∈ V(G),
i
1
Γ (f,f)(v ) ≥ (∆f(v ))2+K(v )Γ(f,f)(v ),
2 i i i i
m
where m ∈ [1,+∞]
Remark 2.9. We say K(v ) is Bakry-Emery’s curvature. In Riemannian geometry, this curvature is
i
a lower bound of the Ricci curvature.
3 Main theorem
Theorem 3.1. Let G be a finite directed graph satisfying strongly connected. Then G satisfies
CD(2,C −(1−α)). i.e. for all vertices v ∈ V(G)
i
1
Γ (f,f)(v ) ≥ (∆f(v ))2 +{C(v )−(1−α)}Γ(f,f)(v ),
2 i i i i
2
where 1 > C(vi) = minvj∈Sout(vi),vk∈Sin(vi){wij/φ(vj),wki/φ(vk)} > 0.
We prove some lemmas about the property of the Γ-calculation and the laplacian before the main
theorem.
Lemma 3.2. For all functions f : V(G) → R and all vertices v ∈ V, then we have
i
1  
Γ(f,f)(v )= w (f(v )−f(v ))2+ w (f(v )−f(v ))2 .
i 4φ(v ) X ij j i X ki k i 
i vj∈Sout(vi) vk∈Sin(vi)
 
2
1
Proof. By the definition 2.7, Γ(f,f)= ∆(f2)−2f∆(f) . So,
2(cid:8) (cid:9)
1  
2Γ(f,f)(v ) = w (f2(v )−f2(v ))+ w (f2(v )−f2(v ))
i 2φ(v ) X ij j i X ki k i 
i vj∈Sout(vi) vk∈Sin(vi)
 
f(vi) 
− w (f(v )−f(v ))+ w (f(v )−f(v ))
φ(v ) X ij j i X ki k i 
i vj∈Sout(vi) vk∈Sin(vi)
 
1  
= w (f(v )−f(v ))2+ w (f(v )−f(v ))2 .
2φ(v ) X ij j i X ki k i 
i vj∈Sout(vi) vk∈Sin(vi)
 
Lemma 3.3. For all functions f : V(G) → R and all vertices v ∈ V, then we have
i
∆Γ(f,f)(v ) = 1  wij (Win(v )+Wout(v ))+ wki (Win(v )+Wout(v )),
i 8φ(v ) X φ(v ) j j X φ(v ) k k 
i vj∈Nout(vi) j vk∈Nin(vi) k
 
where
Wout(v ) = w ((f(v )−f(v ))2−(f(v )−f(v ))2
j ja a j j i
X
va∈Nout(vj)
= w (f(v )−2f(v )+f(v ))2+2(f(v −f(v ))(f(v )−2f(v )+f(v )) ,
ja a j i j i a j i
X (cid:8) (cid:9)
va∈Nout(vj)
Win(v ) = w ((f(v )−f(v ))2 −(f(v )−f(v ))2
j X bj b j j i
vb∈Nin(vj)
= w (f(v )−2f(v )+f(v ))2+2(f(v −f(v ))(f(v )−2f(v )+f(v )) .
X bj(cid:8) b j i j i b j i (cid:9)
vb∈Nin(vj)
Proof. By the definition 2.6 and lemma 3.2,
2φ(v )∆Γ(f,f)(v ) = w (Γ(f,f)(v )−Γ(f,f)(v )) (3.1)
i i ij j i
X
vj∈Sout(vi)
+ w (Γ(f,f)(v )−Γ(f,f)(v )). (3.2)
ki k i
X
vk∈Sin(vi)
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So, we calculate (3.1). Then we obtain
w (Γ(f,f)(v )−Γ(f,f)(v ))
ij j i
X
vj∈Sout(vi)
= w Γ(f,f)(v )−φ(v )Γ(f,f)(v )
ij j i i
X
vj∈Nout(vi)
= wij  w (f(v )−f(v ))2 + w (f(v )−f(v ))2
X 4φ(v ) X ja a j X bj b j 
vj∈Nout(vi) j va∈Nout(vj) vb∈Sin(vj)
 
1 
− w (f(v )−f(v ))2+ w (f(v )−f(v ))2
4 X ij j i X ki k i 
vj∈Nout(vi) vk∈Sin(vi)
 
w
= ij Wout(v )
j
X 4φ(v )
vj∈Nout(vi) j
w
+ ij w (f(v )−f(v ))2 (3.3)
bj b j
X 4φ(v ) X
vj∈Nout(vi) j vb∈Sin(vj)
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− w (f(v )−f(v ))2. (3.4)
ki k i
4 X
vk∈Sin(vi)
We calculate (3.2) similarly. Then we obtain
w (Γ(f,f)(v )−Γ(f,f)(v ))
ki k i
X
vk∈Sin(vi)
w
= ki Wout(v )
k
X 4φ(v )
vk∈Nin(vi) k
w
+ ki w (f(v )−f(v ))2 (3.5)
dk d k
X 4φ(v ) X
vk∈Nin(vi) k vd∈Sin(vk)
1
− w (f(v )−f(v ))2 (3.6)
ij j i
4 X
vj∈Sin(vi)
We combine (3.3) and (3.6). Thus,
w 1
ij w (f(v )−f(v ))2− w (f(v )−f(v ))2
bj b j ij j i
X 4φ(v ) X 4 X
vj∈Nout(vi) j vb∈Sin(vj) vj∈Sin(vi)
w
= ij Win(v )
j
X 4φ(v )
vj∈Nout(vi) j
w
We combine (3.4) and (3.5) similarly, and we obtain ki Win(v ).
Pvk∈Nin(vi) 4φ(vk) k
Lemma 3.4. For all functions f : V(G) → R and all vertices v ∈ V, then we have
i
1
2Γ(∆f,f)(v ) = (∆f)2(v )+  w W (v )+ w W (v ),
i i 2φ(v ) X ij ∆ j X ki ∆ k
i vj∈Sout(vi) vk∈Sin(vi) 
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where W (v ) = ∆f(v )(f(v )−f(v )).
∆ j j j i
Proof. By the definition of 2.7, 2Γ(∆f,f)= ∆(∆f,f)−(∆f)2−f∆(∆f). So,
2φ(v )∆(∆f,f)(v ) = w (f(v )∆f(v )−f(v )∆f(v ))
i i ij j j i i
X
vj∈Sout(vi)
+ w (f(v )∆f(v )−f(v )∆f(v ))
ki k k i i
X
vk∈Sin(vi)
= w (∆f(v )−∆f(v ))f(v )+ w (f(v )−f(v ))∆f(v )
ij j i j ij j i i
X X
vj∈Sout(vi) vj∈Sout(vi)
+ w (∆f(v )−∆f(v ))f(v )+ w (f(v )−f(v ))∆f(v )
ki k i k ki k i i
X X
vk∈Sin(vi) vk∈Sin(vi)
= w (∆f(v )−∆f(v ))(f(v )−f(v ))
ij j i j i
X
vj∈Sout(vi)
+ w (∆f(v )−∆f(v ))(f(v )−f(v ))
X ki k i k i
vk∈Sin(vi)
 
+ f(v ) w (∆f(v )−∆f(v ))+ w (∆f(v )−∆f(v ))
i  X ij j i X ki k i 
vj∈Sout(vi) vk∈Sin(vi)
 
 
+ ∆f(v ) w (f(v )−f(v ))+ w (f(v )−f(v ))
i  X ij j i X ki k i 
vj∈Sout(vi) vk∈Sin(vi)
 
= w ∆f(v )(f(v )−f(v )) w ∆f(v )(f(v )−f(v ))
X ij j j i X ki k k i
vj∈Sout(vi) vk∈Sin(vi)
+ 2φ(v ) 2(∆f)2(v )+f(v )∆(∆f)(v ) .
i i i i
(cid:8) (cid:9)
The proof is completed.
We use these lemmas and prove the theorem 3.1.
Proof of theorem 3.1. By the definition of 2.7, 2Γ (f,f)= ∆Γ(f,f)−2Γ(∆f,f). By the lemma 3.4,
2
1
2φ(v )(2Γ(∆f,f)(v )−(∆f)2(v )) =  w W (v )+ w W (v )
i i i 2φ(v ) X ij ∆ j X ki ∆ k
i vj∈Sout(vi) vk∈Sin(vi) 
w
ij
= w (f(v )−f(v ))(f(v )−f(v ))
ja a j j i
X 2φ(v ) X
vj∈Sout(vi) j va∈Sout(vj)
w
ij
+ w (f(v )−f(v ))(f(v )−f(v ))
bj b j j i
X 2φ(v ) X
vj∈Sout(vi) j vb∈Sin(vj)
w
ki
+ w (f(v )−f(v ))(f(v )−f(v ))
kc c k k i
X 2φ(v ) X
vk∈Sout(vi) k vc∈Sout(vk)
w
ki
+ w (f(v )−f(v ))(f(v )−f(v )).
dk d k k i
X 2φ(v ) X
vk∈Sout(vi) k vd∈Sin(vk)
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On the other hand, Wout(v ) in the lemma 3.3 can be transformed into the following.
j
Wout(v ) = (f(v )−2f(v )+f(v ))2+2(f(v −f(v ))(f(v )−2f(v )+f(v ))
j a j i j i a j i
X (cid:8) (cid:9)
va∈Nout(vj)
= (f(v )−2f(v )+f(v ))2+2(f(v )−f(v ))(f(v )−f(v ))−2(f(v )−f(v ))2 .
a j i j i a j j i
X (cid:8) (cid:9)
va∈Nout(vj)
Win(v ), Wout(v ), and Win(v ) can be transformed similarly. So, the second terms in Win and Wout
j k k
are set off the value of 2φ(v )(2Γ(∆f,f)(v )−(∆f)2(v )). Moreover, the third terms in Win and Wout
i i i
combine, then we obtain −2Γ(f,f)(v ). Thus, we have
i
1
Γ (f,f)(v ) = (∆f)2(v )−Γ(f,f)(v )+H(f)(v ),
2 i i i i
2
where
w
16φ(v )H(f)(v ) := ij w (f(v )−2f(v )+f(v ))2
i i ja a j i
X φ(v ) X
vj∈Nout(vi) j va∈Nout(vj)
w
+ ij w (f(v )−2f(v )+f(v ))2
X φ(v ) X bj b j i
vj∈Nout(vi) j vb∈Nin(vj)
w
+ ki w (f(v )−2f(v )+f(v ))2
kc c k i
X φ(v ) X
vk∈Nin(vi) k vc∈Nout(vk)
w
+ ki w (f(v )−2f(v )+f(v ))2.
X φ(v ) X dk d k i
k
vk∈Nin(vi) vd∈Nin(vk)
As a result, it is sufficient to consider only H(f)(v ). We remark that the vertex v belongs to Nin(v )
i i j
for all v , and Nout(v ) for all v . Then, we obtain
j k k
w
16φ(v )H(f)(v ) = ij w (f(v )−2f(v )+f(v ))2
i i ja a j i
X φ(v ) X
vj∈Sout(vi) j va∈Sout(vj)
+ α w (f(v )−f(v ))2+α w (f(v )−f(v ))2
ij j i ij j i
X X
vj∈Sout vj∈Sout
w
+ ij w (f(v )−2f(v )+f(v ))2
X φ(v ) X bj b j i
vj∈Sout(vi) j vb∈Sin(vj)
+ α w (f(v )−f(v ))2+α w (f(v )−f(v ))2
X ki k i X ij j i
vk∈Sin vj∈Sout
w
+ ki w (f(v )−2f(v )+f(v ))2
kc c k i
X φ(v ) X
vk∈Sin(vi) k vc∈Sout(vk)
+ α w (f(v )−f(v ))2+α w (f(v )−f(v ))2
ij j i ki k i
X X
vj∈Sout vk∈Sin
w
+ ki w (f(v )−2f(v )+f(v ))2
dk d k i
X φ(v ) X
vk∈Sin(vi) k vd∈Sin(vk)
+ α w (f(v )−f(v ))2+α w (f(v )−f(v ))2
ki k i ki k i
X X
v ∈Sin v ∈Sin
k k
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w
= ij w (f(v )−2f(v )+f(v ))2
ja a j i
X φ(v ) X
vj∈Sout(vi) j va∈Sout(vj)
w
+ ij w (f(v )−2f(v )+f(v ))2
bj b j i
X φ(v ) X
vj∈Sout(vi) j vb∈Sin(vj)
w
+ ki w (f(v )−2f(v )+f(v ))2
X φ(v ) X kc c k i
vk∈Sin(vi) k vc∈Sout(vk)
w
+ ki w (f(v )−2f(v )+f(v ))2+16αφ(v )Γ(f,f)(v )
X φ(v ) X dk d k i i i
vk∈Sin(vi) k vd∈Sin(vk)
4(w )2 4(w )2
≥ ij (f(v )−f(v ))2 + ki (f(v )−f(v ))2
X φ(v ) j i X φ(v ) k i
vj∈Sout(vi) j vk∈Sin(vi) k
+ 16αφ(v )Γ(f,f)(v )
i i
≥ 16φ(v )(C(v )+α)Γ(f,f)(v ).
i i i
Then, the proof is completed.
References
[1] F. Chung, Laplacians and the Cheeger inequality for directed graphs, Annals of Combinatorics 9.1
(2005), 1-19.
[2] J.JostandS.Liu,Ollivier’s Ricci curvature, local clustering and curvature-dimension inequalities
on graphs, Discrete and Computational Geometry 51.2 (2014): 300-322.
[3] Y. Lin, L. Lu and S. T. Yau, Ricci curvature of graphs, Tohoku Math. J. 63 (2011) 605-627.
[4] Y.Ollivier, Ricci curvature of Markov chains on metric spaces, J.FunctionalAnalysis.256 (2009)
810-864.
[5] T. Yamada, Classification of directed regular Ricci-flat graphs , arXiv:1602.07779 (2016).
T. Yamada: Mathematical Institute,Tohoku University,Aoba, Sendai 980-8578, Japan
E-mail address: [email protected]
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