On the Z-eigenvalues of the adjacency tensors for uniform hypergraphs

Linear Algebra and its Applications 439 (2013) 2195–2204

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Linear Algebra and its Applications www.elsevier.com/locate/laa

On the Z-eigenvalues of the adjacency tensors for uniform hypergraphs Jinshan Xie a,b , An Chang b,∗,1 a b

School of Mathematics and Computer Science, Longyan University, Longyan, 364012, PR China Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian, 350003, PR China

a r t i c l e

i n f o

Article history: Received 24 April 2012 Accepted 16 July 2013 Available online 26 July 2013 Submitted by R. Brualdi MSC: 05C65 15A18

a b s t r a c t The adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs, and some fundamental properties for the adjacency tensor and its Z-eigenvalues of a uniform hypergraph are obtained. In particular, some bounds on the smallest and the largest Z-eigenvalues of the adjacency tensors for uniform hypergraphs are presented. © 2013 Elsevier Inc. All rights reserved.

Keywords: Adjacency tensor Hypergraph Z-eigenvalue Bound Maximum degree

1. Introduction In the current numerical multilinear algebra (or tensor computation) associative literatures, a lot of them discuss miscellaneous tensors and their properties (see [3,4,6,11–21]). Many of them investigate the eigenvalues of tensors, which extend the concept of eigenvalues of the square matrices, and form an important part of numerical multilinear algebra, and have been found some applications (see [18]) in automatic control, statistical data analysis, optimization, magnetic resonance imaging, quantum physics, higher order Markov chains, spectral hypergraph theory, Finsler geometry, etc. This work

*

Corresponding author. E-mail address: [email protected] (A. Chang). 1 The work is supported by the National Natural Science Foundation of China (Nos. 10931003 and 10871046) and the Research Project Serving the West Coast launched by Longyan University. 0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.07.016

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attracts attention of researchers from different disciplines. At the same time, there were many papers on hypergraphs with their applications in various fields (see [1,10,11,21]). As graphs are related to matrices [2,5,7–9], hypergraphs are related to tensors [11,13,16,20,21] which could reveal more higher order structures than matrices. The definition of eigenvalues of a tensor was independently proposed by Lim in [12] and Qi in [16]. Also in [16], several kinds of eigenvalues of tensors have been proposed, such as Z-eigenvalues, E-eigenvalues, H-eigenvalues and N-eigenvalues. Furthermore, the study of hypergraph via its adjacency tensor and its eigenvalues was initiated by Lim in [13]. So it is natural to study the relations between the eigenvalues of tensors and parameters of hypergraphs. Considerable interest has arisen in this work recently. Some results are based on the H-eigenvalues of tensors [6,12–14,20,21] of a uniform hypergraph. Other results are based on the Z-eigenvalues or E-eigenvalues of tensors [11,15–18] of an even uniform hypergraph. More recently, Chang et al. reveal some similarities as well as differences between the H-eigenvalues and Z-eigenvalues of a nonnegative tensor in [4]; and Cooper et al. give a detailed discussion on the properties and applications of spectra of the adjacency tensors for uniform hypergraphs in [6]. In this paper, we study the adjacency tensor and its Z-eigenvalues for a uniform hypergraph. In the next section, we introduce some notations and definitions, which imply that the adjacency tensor of a uniform hypergraph is nonnegative and symmetric. We show in Section 3 that the adjacency tensor of a uniform hypergraph has a Z-eigenvalue 0, and its largest Z-eigenvalue has a corresponding nonnegative Z-eigenvector. We investigate the properties and bounds of the smallest and the largest Z-eigenvalues of the adjacency tensor for a uniform hypergraph, and give some inequalities between these two Z-eigenvalues and other hypergraph parameters. And a short conclusion is given in the last section. 2. Preliminaries Firstly, we introduce some notations and conceptions used in this paper. Scalars are written as lowercase letters (λ, α , a, b, . . .), vectors are written as bold lowercase letters (x, y, . . .), the i-th entry of a vector x is denoted by xi , matrices and tensors correspond to italic capitals ( A , T , . . .), sets correspond to blackboard bold letters (E, X, V, . . .), and e and I are reserved for the vector of all ones and the identity matrix, respectively. In this paper, tensors refer to r order tenn them a scalar, desors. For a tensor n T with entries T i 1 i 2 ···ir and a vector x ∈ R , we associate T x x · · · x , and a vector, denoted T xr −1 , as its i 1 -th entry being noted T xr , as ir i 1 ,i 2 ,...,i r =1 i 1 i 2 ···i r i 1 i 2

n

i 2 ,...,i r =1

T i 1 i 2 ···ir xi 2 · · · xir . A tensor T is called symmetric if T i 1 i 2 ···ir = T i  i  ···ir for arbitrary permuta1 2

tion (i 1 , i 2 , . . . , i r ) of (i 1 , i 2 , . . . , i r ). Throughout the paper, we focus on r-uniform hypergraphs with r  3. By an r-uniform hypergraph, we mean a hypergraph G = (V, E) with vertices set V = {1, . . . , n} of size n  r and edges set E = {E1 , . . . , Em } with size m and |Ei | = r for every i ∈ {1, . . . , m}. Here | · | means the cardinality of a set. A finite path from vertex i to vertex j is a finite sequence of vertices with its start vertex i and its end vertex j such that from each of its vertices there is an edge to the next vertex. Two vertices are called connected if there is a finite path between them. A connected component X of G is a subset of V such that any two vertices in X are connected and no other vertex in V \ X is connected to any vertex in X. Given two hypergraphs G = (V, E) and G  = (V , E ), if V ⊆ V and E ⊆ E, then G  is said to be a subgraph of G . A set of vertices S ⊂ V(G ) is said to induce the subgraph G [S] = (S, E ∩ P (S)), where P (S) is the power set of S, i.e., the set composed of all subsets of S. Definition 1. (See [11].) For every i ∈ V, the degree of vertex i, denoted as di , is defined as the cardinality of the set D := {E p ∈ E | i ∈ E p }. The vertex i is called isolated if di = 0. Let A (G ) be the adjacency matrix of a 2-uniform hypergraph (i.e. graph) G = (V, E), then for any x ∈ Rn

x T A (G )x =

 {i , j }∈E

2xi x j ,

(1)

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as in [2]. Then, A (G ) is a nonnegative and symmetric matrix. A natural generalization of (1) to r order is as the following way. For an r-uniform hypergraph G = (V, E), its adjacency tensor T A corresponds to the following form:

T A xr :=



A (E p )xr ,

∀x ∈ Rn

(2)

E p ∈E

with

A (E p )xr = rx p 1 x p 2 · · · x pr ,

in which E p := { p 1 , p 2 , . . . , p r } ⊆ V,

(3)

here A (E p ) is a tensor associated to edge E p . It is easy to see that T A i 1 ···ir = 0 if two of {i 1 , . . . , i r } n are the same and |E| = 1r i =1 d i as those for 2-uniform hypergraphs (see [2]). Definition 2. (See [13,6].) Given an r-uniform hypergraph G = (V, E), we associate it an r-order |V|-dimensional nonnegative symmetric tensor T A , called the adjacency tensor of G , as:



T A i 1 i 2 ···ir :=

1

(r −1)! ,

0,

{i 1 , i 2 , . . . , i r } = Ei ∈ E, otherwise.

(4)

It is a direct computation to see that the adjacency tensor T A of an r-uniform hypergraph defined by Definition 2 indeed satisfies (2). 3. Z-eigenvalues of the adjacency tensors In this section, we introduce Z-eigenvalues of the adjacency tensors for r-uniform hypergraphs and discuss some of their properties and bounds. The concept of Z-eigenvalues is important for the sequel discussion, which is defined as follows. Definition 3. (See [16,12].) For a tensor T , a pair (λ, x) is a Z-eigenpair of T if the following hold:



T xr −1 = λx, λ ∈ R, x ∈ Rn ,

x T x = 1.

(5)

λ is called a Z-eigenvalue and x is the associated Z-eigenvector. From Definition 3 and the fact that the gradient of T xr with respect to x is rT xr −1 when T is symmetric, the following lemma is easy to get. One can see the Introduction in [12] or the proofs for Theorems 3 and 5 in [16] or Theorem 9 in [11]. Lemma 1. The Z-eigenvectors of a symmetric adjacency tensor T A and the critical points of the following maximization problem have a one-to-one correspondence:



max T A xr s.t. x 2 = 1, x ∈ Rn .

(6)

Here · 2 represents 2-norm in Rn . Furthermore, if x is a Z-eigenvector of T A , then the corresponding Zeigenvalue is T A xr . Since the maximization problem (6) is maximizing a continuous function on a compact set, it must have at least one critical point. Hence, there is at least one Z-eigenpair for a symmetric adjacency tensor T A . From (3), the following proposition is easy to get. Proposition 1. Let E p = { p 1 , p 2 , . . . , p r } ∈ E. Then A (E p )xr = 0 if and only if at least one of x p 1 , x p 2 , . . . , x pr is equal to 0.

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Proposition 2. Given an r-uniform hypergraph G = (V, E), let T A be its adjacency tensor, k denote the number of connected components containing at least one edge in G . If r |E| < |V|, T A must have Z-eigenvalue 0 with multiplicity at least |V| − (r − 1)|E| − k. Proof. Suppose that the i-th (1  i  k) connected component containing at least one edge in G has k ni  r vertices and mi  1 edges. Then we know that i =1 mi = |E| and ni  mi (r − 1) + 1. So these k

k

k

connected components containing at most i =1 ni vertices. Hence G contains at least |V| − i =1 ni isolated vertices. That is, G contains at least |V| − (r − 1)|E| − k isolated vertices. Obviously, one isolated vertex corresponds with Z-eigenvalue 0 at least once. Hence the proposition follows. 2

Lemma 2. Given an n-vertex r-uniform hypergraph G , let T A be its adjacency tensor and ek (1  k  n) be the k-th n-dimensional coordinate vector. Then any normalized vector x, which is linear combination of at most r − 2 different ek , is a Z-eigenvector of T A corresponding to Z-eigenvalue 0. Proof. For any normalized vector x which is linear combination of at most r − 2 different ek , one can get that the multiplication of any r − 1 entries in vector x is equal to 0. Then

T A xr −1 = 0 = 0x. By Definition 3, the lemma follows.

2

From Definition 3, we know that all Z-eigenvalues of the symmetric adjacency tensor T A for an r-uniform hypergraph G are real. Thus, we could order all the Z-eigenvalues of T A with multiplicity as:

λb  λb−1  · · ·  λ2  λ1 . It is easy to see that

λb = min T A xr s.t. x 2 = 1, x ∈ Rn ,

(7)

λ1 = max T A xr s.t. x 2 = 1, x ∈ Rn .

(8)

By Lemma 2 and [3], we know that

1b

(r − 1)n − 1 . r−2

So it is not vacuous to talk about λ1 and λb . Hence, we introduce the following concept. Definition 4. We call λb the smallest Z-eigenvalue of the adjacency tensor T A for r-uniform hypergraph G , denoted as λb (G ). We call λ1 the largest Z-eigenvalue of the adjacency tensor T A for r-uniform hypergraph G , denoted as λ1 (G ). From Definition 4, the following lemma is easy to get. One can see the proof of Theorem 3.10 in [4]. Lemma 3. Given an r-uniform hypergraph G = (V, E) with |V| = n, let T A be its adjacency tensor. Then there exists a nonnegative Z-eigenvector x of T A corresponding to Z-eigenvalue λ1 (G ) such that λ1 (G ) = T A xr .

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Example 1. Let G = (V, E) be a 4-vertex 3-uniform hypergraph with V = {1, 2, 3, 4} and E = {{1, 2, 3}, {2, 3, 4}}, T A be its adjacency tensor. By Definition 2, one can get that T A 123 = T A 132 = T A 213 = T A 231 = T A 312 = T A 321 = 1 , 2

T A 243 = T A 324 = T A 342 = T A 423 = T A 432 = and T A i jk = 0 elsewhere. Then the Z-eigenvalue problem is to solve:

1 , 2

T A 234 =

⎧ x2 x3 = λx1 , ⎪ ⎪ ⎪ ⎪ x1 x3 + x3 x4 = λx2 , ⎨ x1 x2 + x2 x4 = λx3 , ⎪ ⎪ x2 x3 = λx4 , ⎪ ⎪ ⎩ 2 x1 + x22 + x23 + x24 = 1.

Using the software of Mathematica 5.0, we can calculate the all three different Z-eigenvalues of T A to be:

2.

3.

√

√

√

√

√

√

√

6 with its corresponding Z-eigenvectors ( 66 , 33 , 33 , 66 ) T , ( 66 , − 33 , 3√ √ √ √ √ √ √ √ √ √ 3 6 T 6 3 3 − 3 , 6 ) , (− 6 , 3 , − 3 , − 66 )T and (− 66 , − 33 , 33 , − 66 )T . √ √ √ √ √ The Z-eigenvalue λb = − 36 with its corresponding Z-eigenvectors (− 66 , − 33 , − 33 , − 66 ) T , √ √ √ √ √ √ √ √ √ √ √ √ (− 66 , 33 , 33 , − 66 )T , ( 66 , − 33 , 33 , 66 )T and ( 66 , 33 , − 33 , 66 )T . The Z-eigenvalue λ = 0 with its corresponding Z-eigenvectors (x1 , 0, 0, ± 1 − x21 ) T , (x1 ,

1. The Z-eigenvalue λ1 =

± 1 − 2x21 , 0, −x1 )T and (x1 , 0, ± 1 − 2x21 , −x1 )T , where −1  x1  1.

In this example, we see that the largest Z-eigenvalue of G is λ1 (G ) = √

√

√

√

√

6 , 3

which has a correspond-

6 , 33 , 33 , 66 )T ; 6

and any ek (1  k  4) is a Z-eigenvector of T A ing nonnegative Z-eigenvector ( corresponding to Z-eigenvalue 0, as indicated by Lemma 2. Theorem 1. Let T A be the adjacency tensor of an n-vertex r-uniform hypergraph G = (V, E). Then we have

λb (G )  0 

d¯ n

r −2 2

 λ1 (G )  ,

where the average degree d¯ :=

1 n

n

i =1 d i

=

(9) r |E| n

and the maximum degree  := max1i n di .

Proof. By Lemma 2 and Definition 4, λb (G )  0. And it is obvious that 0  d¯

to prove n

r −2 2

d¯

n

r −2 2

. Hence, we only have

 λ1 (G )  .

On the one hand, let 1 be the n order vector with all entries equal to √1 , we see that n

r

1 nd¯ d¯ λ1 (G )  T A 1r = r |E| √ = r = r −2 . n

n2

n

2

On the other hand, let y be a nonnegative Z-eigenvector of T A corresponding to Z-eigenvalue λ1 (G ) and y i (0 < y i  1) be an entry of y with maximum value. Then the i-th eigenvalue equation gives that

λ1 (G ) y i =

n 

T A ii 2 ···ir y i 2 · · · y ir =

i 2 ,...,i r =1





{i ,i 2 ,...,i r }∈E



y i 2 · · · y ir

{i ,i 2 ,...,i r }∈E

y ri −1 = di y ri −1  di y i   y i .

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That is

λ1 (G )  . Therefore, the theorem follows.

2

Remark 1. When G is a 2-uniform hypergraph, the result of Theorem 1 is

λn (G )  0  d¯  λ1 (G )  , which is a well-known result as we can see from Proposition 3.1.2 in [2]. Theorem 2. Let G = (V(G ), E(G )) be an r-uniform hypergraph and G  be its subgraph. Then,

λ1 G   λ1 (G ).

(10)

Furthermore, if G  is the induced subgraph of

G , then

λb (G )  λb G   λ1 G   λ1 (G ).

(11)

Proof. Let T A be the adjacency tensor of G , and T A be the adjacency tensor of G  . By Lemma 3, 

there exists a nonnegative Z-eigenvector x = (x1 , x2 , x3 , . . . , x|V(G  )| ) T ( x 2 = 1, x ∈ R|V(G )| ) of T A corresponding to Z-eigenvalue λ1 (G  ) such that λ1 (G  ) = T A xr . Since G  is a subgraph of G , V(G  ) ⊆ V(G ) and E(G  ) ⊆ E(G ). Now we construct a nonnegative vector

y = (x1 , x2 , x3 , . . . , x|V(G  )| ,

0, . . . , 0 ) T .   

|V(G )|−|V(G  )|

Then y ∈ R|V(G )| , y 2 = 1 and T A yr  T A xr . By the definition of λ1 (G ), λ1 (G )  T A yr . Hence, we have

λ1 (G )  T A yr  T A xr = λ1 G  .

That is

λ1 G   λ1 (G ). Furthermore, if G  is the induced subgraph of G , then



V G  ⊂ V(G ) and E G  = E(G ) ∩ P V G  . By Definition 4 and Lemma 1, then there exists a Z-eigenvector x = (x1 , x2 , x3 , . . . , x|V(G  )| ) T

 ( x 2 = 1, x ∈ R|V(G )| ) of T A corresponding to Z-eigenvalue λb (G  ) such that λb (G  ) = T A x r . Now we construct a vector



y = x1 , x2 , x3 , . . . , x|V(G  )| ,

0, . . . , 0   

T

.

|V(G )|−|V(G  )|

Then y ∈ R|V(G )| , y 2 = 1 and T A y r = T A x r . By the definition of λb (G ), λb (G )  T A y r . Hence, we have

λb (G )  T A y r = T A x r = λb G  .

(12)

By Definition 4, it is obvious that λb (G  )  λ1 (G  ), which, together with (10) and (12), implies (11).

2

Remark 2. The result of Theorem 2 is true when G is a 2-uniform hypergraph (see Propositions 3.1.1 and 3.2.1 in [2]).

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Theorem 3. Let T A be the adjacency tensor of an n-vertex r-uniform hypergraph G = (V, E). Then we have





λ1 (G ) 

n−1

n

r−1

2−r 2

(13)

,

and equality holds if and only if G is an n-vertex r-uniform complete hypergraph with Z-eigenvector ee 2 corresponding to Z-eigenvalue λ1 (G ). Proof. By Lemma 3, there exists a nonnegative Z-eigenvector x (x  0, x 2 = 1) of T A corresponding to Z-eigenvalue λ1 (G ) such that λ1 (G ) = T A xr . By (2) and (3), we have that

λ1 (G ) = T A xr =

 E p ∈E

xi 1 xi 2 · · · xi r

1i 1


r r

n r

x1 + x2 + · · · + xn



r

n

 n(x2 + x2 + · · · + x2 ) r n n 1 2 n

r

 √ r =r

rx p 1 x p 2 · · · x pr

E p ∈E



r



A (E p )xr =

n

n

n

r

 =r

n r

n

−r 2

=

n−1



r−1

n

2−r 2

, n

where the first inequality follows from the fact that |E|  r and equality holds if and only if G is an n-vertex r-uniform complete hypergraph, the second inequality from the Maclaurin’s inequality



1i 1
· · · xi r

 1r



x1 + x2 + · · · + xn n

for x  0 and equality holding if and only if x1 = x2 = · · · = xn , the third inequality from the Cauchy– Schwarz inequality



(x1 + x2 + · · · + xn )2  n x21 + x22 + · · · + xn2 and equality holding if and only if x1 = x2 = · · · = xn , the fourth equality from the fact that x21 + x22 + · · · + xn2 = x 22 = 1. This completes the proof. 2 Remark 3. When G is a 2-uniform hypergraph, the result of Theorem 3 is λ1 (G )  n − 1, and equality holds if and only if G is an n-vertex 2-uniform complete hypergraph (with adjacency eigenvector ee 2 corresponding to adjacency eigenvalue n − 1), i.e., n-vertex complete graph, which is a well-known result as we can see from Theorem 0.13 in [7]. Theorem 4. Let T A be the adjacency tensor of r-uniform hypergraph G = (V(G ), E(G )) with at least one edge. Then we have

λ1 (G )  r

2−r 2



3−r 2

.

(14)

Proof. Let V(G ) = {1, 2, . . . , n}. Without loss of generality, we can suppose that 1 is a vertex satisfying

d1 = 

(15)

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in G . Let G1 be the edge-induced subgraph of G with all of those edges containing vertex 1. Then we can suppose that

   E(G1 ) = E j  E j := {1, v j 2 , v j 3 , . . . , v jr } ⊆ {1, 2, . . . , n}, 1  j  d1 ,

V(G1 ) =

d1 

E j,

j =1

in which, maybe some of v j s (2  s  r) and v kt (k = j, 1  k  d1 and 2  t  r) represent the same vertex. Let G1 be the graph such that d1  V G1 = Ej ,

     E G1 = Ej  Ej := 1, v j 2 , v j 3 , . . . , v jr , 1  j  d1 ,

j =1

in which, any two elements of set {1} ∪ { v j | 1  j  d1 , 2  s  r } are different vertices. s Let T 1 be the adjacency tensor of G1 . By Lemma 3, there exists a nonnegative Z-eigenvector x  ( x = 1, x ∈ R|V(G1 )| ) of T  corresponding to Z-eigenvalue λ (G  ) such that 2

1

1

1

d1  λ1 G1 = T 1 x r = rx1 xj 2 xj 3 · · · xjr . j =1

Now we construct a surjection



σ from V(G1 ) to V(G1 ) such that

σ (1) = 1 and σ v j s = v j s (1  j  d1 , 2  s  r ). And construct a nonnegative vector x = (x1 , x2 , x3 , . . . , x|V(G1 )| ) T such that

x1 = x1

xi =

and

v

max

js



∈σ −1 (i )

xj s







2  i  V(G1 ) .

It is easy to see that 0 < x 2  1. Let T 1 be the adjacency tensor of G1 . Then

λ1 (G1 )  T 1 =

d1 

x

x 2

r  T 1 xr

rx1 x j 2 x j 3 · · · x jr

j =1



d1  j =1



rx1 xj 2 xj 3 · · · xjr = λ1 G1 .

Furthermore, since G1 is the edge-induced subgraph of G , by Theorem 2, we have

λ1 (G )  λ1 (G1 ). Hence,

λ1 (G )  λ1 (G1 )  λ1 G1 .

(16)

Now in graph G1 , we construct a nonnegative vector z = ( z1 , z12 , z13 , . . . , zd1r ) T corresponding to V(G1 ) = {1, v 12 , v 13 , . . . , v d 1r } such that

1 z1 = √ r

and

1 z js = √ rd1

(1  j  d1 , 2  s  r ).

J. Xie, A. Chang / Linear Algebra and its Applications 439 (2013) 2195–2204

2203

Then, we have z 2 = 1 and d1  λ1 G1  T 1 zr = rz1 z j 2 z j 3 · · · z jr j =1

 d1

=

j =1

1 r√ r 1



rd1



= rd1 √

r

r −1

1

√

r −1

1

√

rd1

=r

2−r 2

3−r

d1 2 ,

which, together with (15) and (16), implies (14).

2 √

Remark 4. When G is a 2-uniform hypergraph, the result of Theorem 4 is λ1 (G )  , which is a well-known result as we can see from Result 2 on page 112 of [7]. And when G is a star, the

inequality in (14) is sharp. Furthermore, when d¯ 2    nr , one can get that

d¯

n

r −2 2

r

2−r 2



3−r 2

. In this

case the lower bound of λ1 (G ) in Theorem 4 is better than that in Theorem 1; When d¯ 2    nr , one d¯

can get that n

r −2 2

r

2−r 2



3−r 2

. In this case the lower bound of λ1 (G ) in Theorem 1 is better than that

in Theorem 4. 4. Conclusions In the present paper, we study the adjacency tensor and its Z-eigenvalues for a uniform hypergraph, and obtain some fundamental properties of the adjacency tensor and its Z-eigenvalues for a uniform hypergraph. Some bounds on the smallest and the largest Z-eigenvalues of the adjacency tensors for uniform hypergraphs are presented. In fact, most of the results in this paper are also true for 2-uniform hypergraphs. Acknowledgement The authors would like to thank the referees for their insightful and valuable comments on our paper, which help us to improve the presentation, quality, and clarity of the paper greatly. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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