The Wigner's Semicircle Law of Weighted Random Networks

03/31/2020 ∙ by Yusuke Sakumoto, et al. ∙ 0

The spectral graph theory provides an algebraical approach to investigate the characteristics of weighted networks using the eigenvalues and eigenvectors of a matrix (e.g., normalized Laplacian matrix) that represents the structure of the network. However, it is difficult for large-scale and complex networks (e.g., social network) to represent their structure as a matrix correctly. If there is a universality that the eigenvalues are independent of the detailed structure in large-scale and complex network, we can avoid the difficulty. In this paper, we clarify the Wigner's Semicircle Law for weighted networks as such a universality. The law indicates that the eigenvalues of the normalized Laplacian matrix for weighted networks can be calculated from the a few network statistics (the average degree, the average link weight, and the square average link weight) when the weighted networks satisfy the sufficient condition of the node degrees and the link weights.

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I Introduction

Many networks (e.g., railway network, social network, the Internet, and airport network) are often modeled as weighted networks that are composed of nodes and weighted links [1, 2, 3]

. The weighting of links is important to model a network, but it is hard for large-scale and complex networks such as social network. In the social network, nodes and links correspond to persons and their acquaintance relationships, respectively. In order to correctly give the weight for each link in the social network, the strength of the relationships among people should be accurately estimated from a huge amount of personal data (e.g., communication histories in mobile phone and social media). Due to the problem of the privacy and the computational complexity, it is unrealistic to gather such personal data, and calculate the strength of the relationships accurately.

The spectral graph theory provides an algebraical approach to investigate the characteristics of weighted networks using the eigenvalues and the eigenvectors of a matrix (e.g., normalized Laplacian matrix) that represents the structure of the network [4, 5]. In particular, the eigenvalues are important to understand the characteristics related to the entire network on the basis of the spectral graph theory. In [6], we have been clarified that the eigenvalue distribution of the normalized Laplacian matrix affects the information dissemination speed throughout the social network. In general, the eigenvalues of the matrix are calculated from all the matrix elements. For the matrix representing the structure of a weighted network, its elements are determined by the links weights. Hence, the weighting of the links is required to apply the spectral graph theory for weighted networks. This would be the barrier to apply the spectral graph theory for large-scale and complex networks due to the above-mentioned problems. However, the weighting of the links is avoidable if there is a universality that the eigenvalues are independent of the detailed structure (e.g., the weight of each link) of large-scale and complex networks. Therefore, the finding of such a universality of the eigenvalues expands the applicable region of the spectral graph theory.

The random matrix theory discusses a universality of the eigenvalues if the elements of the matrix are given by random variables 

[7, 8, 9]. In [9]

, Chung et al. have been analyzed the random matrix corresponding to the normalized Laplacian matrices for unweighted networks, and have clarified the universality (the Wigner’s semicircle law) that the eigenvalues of the normalized Laplacian matrix follow the semicircle distribution. To our knowledge, no study has clarified a universality of the eigenvalues for weighted networks.

In this paper, we clarify the universality (the Wigner’s semicircle law) of the eigenvalues for weighted networks on the basis of the discussion of  [9]. The clarified universality indicates that the eigenvalues of the normalized Laplacian matrix for weighted networks can be calculated from the a few network statistics (the average degree,the average link weight, and the square average link weight) when the weighted networks satisfy the sufficient condition of the node degrees and the link weights. Using some numerical examples, we confirm the validity of the sufficient condition.

This paper is organized as follows. In Sect. II, we describe the spectral graph theory and the random matrix theory for weighted networks. In Sect. III, we prove the Wigner’s semicircle law for weighted networks on the basis of the random matrix theory. Section IV shows some numerical examples. Finally, in Sect. V, we conclude this paper and discuss the future work.

Ii Preliminary

Ii-a Spectral Graph Theory

In the spectral graph theory, the structure of a network is represented by the matrix, and its characteristics is investigated using the eigenvalues and the eigenvectors of the matrix. In this section, we describe the spectral graph theory for weighted networks.

We denote a weighted network by where and are the sets of nodes and links, respectively. Let be the number of nodes in . The link between nodes and is denoted by . Link has the weight where and . Let be the set of adjacency nodes of node . The weighted degree of nodes is defined by

(1)

To represent the structure of links and nodes in , there are adjacency matrix and degree matrix , respectively. The -th element of adjacency matrix is defined by

(2)

Degree matrix is defined by

(3)

To represent the both structure of nodes and links in , normalized Laplacian matrix is often used. Normalized Laplacian matrix is defined by

(4)

where

is the identity matrix.

Since normalized Laplacian matrix is symmetric , its eigenvalue  () are real numbers. Let be the eigenvector of eigenvalue where . We assign a number to in ascending order, and hence means the -th minimum eigenvalue of . The range of the eigenvalues is given by

(5)

If , weighted network is connected (i.e., there is at least one path between every pair of nodes). Then, weighted network is not bipartite graph if . We define spectral radius by . Spectral radius is in because for . Eigenvector of minimum eigenvalue is given by

(6)

where is defined by

(7)

Since eigenvector is the orthonormal basis, matrix

is the orthogonal matrix (

).

Using and , normalized Laplacian matrix is given by

(8)

From the above equation, is determined by eigenvalues and eigenvectors . Hence, weighted network can be analyzed not only with but also with their eigenvalues and eigenvectors. The spectral graph theory provides an algebraical analysis method of with eigenvalues and eigenvectors of . In particular, eigenvalues is important to understand the statistical characteristics related to the whole of weighted network . In order to calculate eigenvalues , in general, all elements must be given correctly. However, if there is a useful universality of eigenvalues , we can investigate the statistical characteristics of weighted network without all elements .

Ii-B Random Matrix Theory

The random matrix theory focuses on random matrices that the elements are given by random variables, and clarifies that a universality of the eigenvalues appears if the matrix size approaches to infinity. When the links and their weights in weighted network are randomly given with a stochastic rule, elements of normalized Laplacian matrix become random variables, and can be treated as a random matrix. Note that the universality of eigenvalues of corresponds to a characteristic of the statistical ensemble of the random networks generated with the same stochastic rule. Hence, clarifying such a universality contributes the growth of the statistical mechanics on networks.

In [9], the link between nodes and in unweighted networks is randomly generated using the stochastic rule with random variable . If , there is no link between nodes and . On the other hand, if , the link exists between nodes and

. We denote probability

by , which is given by

(9)

where and . Using Eq. (9), the expectation of the node ’s degree is given by . Let , , and be the average degree, the minimum degree, and the maximum degree. These are defined by , , and , respectively. We can write by . Similarly to [9], we assume that so that . As the scale of networks becomes large, the degrees of nodes are likely to become large. Hence, similarly to [9], we assume that diverges to infinity as . Using the above stochastic rule, special networks (e.g., networks with no links) are rarely generated. However, the probability generating such special networks is very small, and hence this is no problem to discuss a universality appearing when .

In weighted network , the weight of link is randomly set using the stochastic rule with random variable . In the stochastic rule, random variable

follows the conditional probability density function

, which is defined by

(10)

If  (i.e., link does not exist), is always , and hence where is the Dirac delta function. On the contrary, if , . Since link weights in actual networks cannot be infinity, we assume that is bounded. Using a finite value , . For the sake of convenience, we write

(11)

Let be the

-th moment of

with the condition . is defined by

(12)

Even if the weights of all links in are divided by , normalized Laplacian matrix is invariant. Hence, without loss of generality, we assume that . With this assumption, has the following properties: (a)  for , and (b) since .

Following the above stochastic rules, not only elements but also eigenvalues  () of the normalized Laplacian matrix for the weighted network become random variables depending on the set of random variables  where . Let be the random variable for eigenvalue of . We denote the conditional eigenvalue density of eigenvalues  () of by , which is defined by

(13)

where is the function of . Since eigenvalues vary stochastically, conditional eigenvalue density is also a random variable. Note that can be treated as a conditional probability density function because . Using probability , eigenvalue density of is given by

(14)

Since , can be also treated as a probability density function. Then, we denote the -th moment for using by , which is defined by

(15)

Using the approach of the random matrix theory, previous work [9] has clarified the universality (the Wigner’s semicircle law) that eigenvalue density for unweighted networks follows a certain distribution (i.e., semicircle distribution). If such a universality exists even for weighted network , eigenvalue density can be obtained without giving link weights correctly. This allows the analysis of based on spectral graph theory even if is a large-scale and complex network such as social network.

Iii Wigner’s Semicircle Law of Weighted Network

In this section, we probe the Wigner’s semicircle law for weighted network on the basis of the discussion in [9]. The Wigner’s semicircle law for is as follows:

Theorem.

If weighted network satisfies degree condition

(16)

of normalized Laplacian matrix converges to semicircle distribution as . Semicircle distribution is given by

(17)

where is the limit value of spectral radius as , and is given by

(18)
Remark.

When the links in is not weighted, is always if , and hence . By substituting them into Eqs. (16) and (18), we obtain the Wigner’s semicircle law for unweighted networks shown in [9]. Therefore, it can be generalized to weighted network .

Proof.

If eigenvalue density is given by semicircle distribution , even moment

and odd moment

for are given by

(19)
(20)

Since , the above moments for correspond to the central moments of . The followings are equivalent:

  1. As , eigenvalue density converges to semicircle distribution .

  2. As , even moment and odd moment converge to Eqs. (19) and (20), respectively.

In order to prove the Wigner’s semicircle law for weighted network , we show that 2. is fulfilled if the degree condition (16) is satisfied.

For the sake of convenience, we use matrix removing the effect of minimum eigenvalue from normalized Laplacian matrix . Matrix is defined by

(21)

where is the matrix where all elements are given by , and corresponds to the adjacency matrix for the complete graph including the self-loops at all nodes. Matrix has nonzero eigenvalues, and -th largest eigenvalue is given by . Hence

(22)

By substituting the above equation into Eq. (15), we obtain

(23)

Since is finite, weighted degree of the node converges to its expected value as . is given by

(24)

Note that we used to derive the above equation. By substituting into Eq. (21), we obtain matrix , which is given by

(25)

where . As , matrix converges to matrix since . Since the Wigner’s semicircle law discusses the limit theorem where , there is no problem if we prove it using instead of .

Element of matrix is a random variable, and is given by

(26)

Let be the -th moment of . Specifically, 1st moment is given by

(27)

For , -th moment is bounded by

(28)
(29)

To derive the above equation, we left the term with in the sum since . Note that is a function that increases faster than or equal to as . Then, we use as

(30)

-th moment is given by

(31)

where represents a cycle with length in the complete graph with nodes, and is the set of the cycles with length . In cycle , is the number of disjoint links, is the -th link, and is the occurrence number of link . In particular, is satisfied with

(32)

where and are treated as the same link in a cycle when counting since is a symmetric matrix.

Figure LABEL:fig:complete_graph shows the example of the complete graph with nodes. For example, includes . In cycle , , and .

[.15]complete_graphThe complete graph with 4 nodes

From Eq. (23) and , -th moment is approximated by

(33)

As , the error of the approximation approaches to . Hence, we investigate in order to prove that converges to Eqs. (19) and (20) as .

To show the convergence of even moment , we investigate . Using the independence of , even moment is bounded by

(34)

where is a set of cycles with length and disjoint nodes. Note that . For example, is included in . The cycles included in are composed of at least disjoint links. To derive the second right-hand side of the above equation, we first divide all cycles in into the sets of the cycles that the number of disjoint links is . Each set has cycles. Using Eq. (29), for , we obtain

(35)

since that is the largest in -th moments for . In the first right-hand side of Eq. (34), the reason why we do not consider the cycles longer than is as follows. First, . Hence, is 0 except if for . Since for , the number of links in the cycle is at most .

To extract the main term of , we rewrote Eq. (34) with

(36)

where is

(37)

In the above equation, is given by

(38)

where is the number of permutations by selecting nodes from nodes, and is given by

(39)

Then, is the number of combinations that each appears more than once in the cycle with length , and is given by

(40)

In the above equation, since there is no restriction except that each must appear at least twice, it is multiplied by . Moreover, is the number of the second appearance positions for , and is given by

(41)

Note that the right side of the above equation is the Catalan number.

By substituting Eq. (38) into Eq. (37), is given by

(42)

Then, in (36) is bounded by

(43)

To obtain the upper bound of , we used the Stirling’s approximation, which is

(44)

According to the following discussion, if the degree condition is satisfied the right-hand side of Eq. (43) for becomes . First, for , we derive

(45)

To prove the above equation, we should show

(46)

Similarly to [9], using , we obtain condition

(47)

to satisfy Eqs. (45) and (46). The condition (47) is the same as the degree condition (16).

Therefore, if the degree condition (16) is satisfied, for , we obtain

(48)

By substituting the above equation into Eq. (36), the upper bound of is given by

(49)

On the other hand, the lower bound of