Complexity
Volume 2018 (2018), Article ID 3467158, 6 pages
https://doi.org/10.1155/2018/3467158

Research Article

More on Spectral Analysis of Signed Networks

Guihai Yu¹ and Hui Qu1,2

¹School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, Guizhou, 550025, China
²Shandong Co-Innovation Center of Future Intelligent Computing, Yantai, Shandong, 264005, China

Correspondence should be addressed to Guihai Yu; yuguihai@126.com

Received 9 August 2018; Accepted 27 September 2018; Published 16 October 2018

Guest Editor: Piotr Brodka

Copyright © 2018 Guihai Yu and Hui Qu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Spectral graph theory plays a key role in analyzing the structure of social (signed) networks. In this paper we continue to study some properties of (normalized) Laplacian matrix of signed networks. Sufficient and necessary conditions for the singularity of Laplacian matrix are given. We determine the correspondence between the balance of signed network and the singularity of its Laplacian matrix. An expression of the determinant of Laplacian matrix is present. The symmetry about of eigenvalues of normalized Laplacian matrix is discussed. We determine that the integer is an eigenvalue of normalized Laplacian matrix if and only if the signed network is balanced and bipartite. Finally an expression of the coefficient of normalized Laplacian characteristic polynomial is present.

1. Introduction

Social networks represent a large proportion of the complex socioeconomic organization in modern society which represent social entities including countries, corporations, or people. These entities interconnected through a wide range of social ties such as political treaties, commercial trade, friend, and collaboration. To display the ally/enemy, friend/foe, and trust/distrust relationships, the social system can be well represented by a signed network in which an edge of the network is assigned to be positive if two individuals are ally, friendship, trust, and negative if they are enemy, foe, and distrust. The origin of the study of signed networks can be tracked back to the work of Heider [1]. The use of signed networks was then proposed by Cartwright and Harary [2] to model the existence of balance/unbalance in the social networks.

As we know, graphs are very useful ways of presenting information about signed networks. However, when there are many actors and/or many kinds of relations, they can become so visually complicated that it is very difficult to see patterns. It is also possible to present information about signed networks in the form of matrices. Representing the information in this way also allows the application of mathematical and computer tools to summarize and find patterns. Up to now, some matrices are employed by signed networks analysts in a number of different ways. This is the so-called spectral graph theory, which is a branch of mathematical science. Its idea is to exploit numerous relationship between the structure of a network (graph) and the spectrum of some matrix (or collection of matrices) associated with the network (graph). There are many different matrices that are employed, including adjacency matrix, Laplacian matrix, and normalized Laplacian matrix. The goal of this paper is to investigate some properties of Laplacian matrix and normalized Laplacian matrix of signed networks and exploit some relation between these matrices and signed networks.

Let be an undirected network of order with vertex set and edge set . The adjacency matrix of is defined as follows: if and are adjacent and otherwise. A signed network consists of a network , referred to as its underlying network, and a sign function . The adjacency matrix of is with , where is an element in the adjacency matrix of the underlying network and is an edge of . If all edges are signed positive, the adjacency matrix is exactly the ordinary adjacency matrix . Let be a diagonal matrix where is the degree of vertex in its underlying network. The Laplacian matrix of , denoted by , is defined as . The matrix is said to be normalized Laplacian matrix of , denoted by .

A signed --walk in a signed network is a sequence of vertices and edges such that (). An --walk is called even (odd) if is even (odd). The sign of a signed walk is and (). A signed walk is balanced (unbalanced) if (). A signed cycle is called balanced (unbalanced) if its sign is +1 (−1). A signed networks is called balanced (resp. unbalanced) if each its signed cycle is balanced (resp. unbalanced).

Suppose that is a signed network. A signed function is a switching function if is transformed to a new signed network by such that the underlying graph remains the same and the sign function is defined by for an edge . Let and be two signed networks with the same underlying graph. We call and switching equivalent and write , if there exists a switching function such that . Switching preserves some signed-graphic invariants such as the sign of cycles and spectrum of combinatorial matrices (adjacency matrix, normalized Laplacian matrix).

This paper is organized as follows. In Section 2, we study some properties of Laplacian matrix of signed networks. Sufficient and necessary conditions for the singularity of Laplacian matrix are given. The correspondence between the balance of signed network and the singularity of its Laplacian matrix is determined. An expression of the determinant of Laplacian matrix is present. In Section 3, the symmetry about of eigenvalues of normalized Laplacian matrix is discussed. Sufficient and necessary condition for that the integer is an eigenvalue of normalized Laplacian matrix is given. An expression of all coefficients of normalized Laplacian characteristic polynomial is present.

2. Laplacian Matrix and Signed Network

Hou et al. [3] introduced the incidence matrix of a signed network as follows. Let be an matrix indexed by the vertex and the edge of signed network and

The following is immediate by the direct calculation.

Theorem 1 (see [3]). Let be a signed network. Then and is a positive semidefinite matrix.

Theorem 2. Let be a connected signed network on vertices . Then is singular if and only if any --walk has the same sign. In this case, is a simple eigenvalue with an eigenvector , where is a --walk in .

Proof. Let . Note that for any nonzero vector , if and only if . By (14), if and only if for any edge . Let be any --walk and . Suppose that . So we have

which implies that each --walk has the same sign.
Note that . Hence

This implies that is a simple eigenvalue of with an eigenvector .
Suppose that any --walk has the sign. Then for any edge , . Let be a column vector such that (). Then , i.e., . So we have

This implies that is singular.

Note that for the underlying network it is known that the multiplicity of the eigenvalue 0 of Laplacian matrix is equal to the number of components. For signed network, the following holds from the proof of Theorem 2.

Theorem 3. The multiplicity of the eigenvalue 0 of Laplacian matrix of a signed network is the number of components whose Laplacian matrix is singular.

Theorem 4 (see [4]). A signed network is balanced if and only if for each pair of distinct vertices all paths joining and have the same sign.

From Theorems 2 and 4, we have the following.

Theorem 5. A signed network is balanced if and only if is singular.

The following is immediate from Theorem 5.

Theorem 6. The Laplacian matrix of a signed network is singular if and only if the Laplacian matrix of any its cycles is singular. In particular, the Laplacian matrix of any acyclic graph is singular.

In [5], authors determined the determinant of the Laplacian matrix of mixed graphs. Here by the similar method we shall extend it to the case for signed graphs.

Theorem 7. for any signed cycle .

Proof. Let be a signed cycle with vertex set and edge set such that () and . For the incidence matrix , we expand its the first row

By directly calculation and the fact that for any edge . It follows that

So the result holds.

Theorem 8. Let be a signed unicyclic network with a cycle . Then

Proof. By Theorem 7, the results hold if is a signed cycle. Assume that has a pendant vertex, say . Let be the unique neighbor of in . Let be the edge joining and . After permutations, the first row and the first column of correspond to the vertex and the edge , respectively. Note that is a square matrix since is unicyclic. We get the determinant of by expanding along the first row as follows:

where is a signed subgraph obtained from by deleting the vertex . Hence we have

Repeating the above finite steps, we have .

Let be a connected signed network. We call a subnetwork as an essential spanning subnetwork of if either is balanced and is a spanning tree of , or else is not balanced, and every component of is a unicyclic signed network in which the unique cycle is negative. By we denote the set of all essential spanning subnetworks of .

Theorem 9. Let be a connected signed network. Then

where is the number of essential spanning subgraphs which contain unbalanced cycles and .

Proof. It is evident that the result holds if is a tree. Assume that contains some cycles. By Cauchy-Binet Theorem [6] and , we have

where is a square submatrix of .
Note that is the vertex-edge incidence matrix of a spanning subgraph of , say , with the edge set . Moreover, . Note that every component of is unicyclic and . By Theorem 8, we have

So the result holds.

The following is immediate from Theorem 9, which is coincident with the definition of balance of signed network.

Theorem 10. Let be a signed network. Then is balanced if and only if each cycle of is balanced cycle.

3. Normalized Laplacian Matrix and Signed Network

For a signed network , the normalized Laplacian matrix is symmetric and positive semidefinite [7], so its eigenvalues are real and nonnegative, denoted by . Firstly we recall some properties of normalized Laplacian matrix.

Lemma 11 (see [7]). Let be a signed network on vertices with normalized Laplacian eigenvalues . Then .

Lemma 12 (see [3, 7]). Let and be two signed networks with the same underlying network. Then if and only if and are signature similar.

In [8], the symmetry about of eigenvalues for bipartite signed network was present as follows. Here we present a stronger result.

Theorem 13 (see [8]). Let be a bipartite signed network. If is an eigenvalue of , then is also an eigenvalue of .

Theorem 14. Let be a connected signed network. Then is bipartite if and only if all eigenvalues of are symmetric about 1 (including multiplicities); i.e., for each eigenvalue , is also an eigenvalue of .

Proof. It suffices to verify that and have the same spectrum. Note that . is bipartite if and only if can be expressed as . It is evident that

This yields to the result.

From Lemma 11, the integer is the upper bound of normalized Laplacian eigenvalues. In this sequel, we give a sufficient and necessary condition for that the integer is an eigenvalue of normalized Laplacian matrix.

Theorem 15. Let be a connected signed network. Then is an eigenvalue of if and only if is a balanced bipartite signed network.

Proof. By Courant-Fischer theorem, we have

Assume that is an eigenvalue of with nonzero eigenvector . By Lemma 11 and (14), for any edge incident to and . So can be partitioned into two parts such that no edge existing between any two vertices in every part. This means that is bipartite. For any even cycle , we have

Moreover, . So and is balanced. This implies that is balanced.
If is balanced bipartite, then is an eigenvalue of . By Theorem 14 and Lemma 12, is an eigenvalue of .

As we know, the coefficients of characteristic polynomial of adjacency (Laplacian) matrix are related to the graph structure. In [9], expressions of coefficients of (Laplacian) characteristic polynomial was present. We would present the expression of the coefficients of normalized Laplacian characteristic polynomial. Firstly, we recall the Sachs formula for the coefficients of adjacency characteristic polynomial of signed networks. Here some definitions are needed. An elementary figure is the graph or the cycle. A basic figure is the disjoint union of elementary figures.

Lemma 16 (see [9]). Let and be a signed network and its adjacency characteristic polynomial, respectively. Then

where is the set of basic figures on vertices in , is the number of components of , and is the set of cycles in and .

Let be the normalized Laplacian characteristic polynomial of . By the definition of normalized Laplacian matrix, we have

Theorem 17. Let be a signed network on vertices and be its normalized Laplacian characteristic polynomial. Then

where is the set of basic figures on vertices in , is the number of components of , is the set of cycles in , , , and is the degree of in .

Proof. Note that

Set

So . Moreover, equals to the sum of all minors of . Then is the sum of all minors of . It is evident that each such minor of is the product of the corresponding minors of , , and , respectively. Furthermore, any minor of is the determinant of adjacency matrix of an induced subgraph of with vertices. So this result holds from Lemma 16.

4. Conclusion

Recently, there are some results on the spectral theory of signed graphs [10–18]. In this paper we investigate some properties of (normalized) Laplacian matrix of signed network and present a correspondence between the balance of signed networks and the singularity of Laplacian matrix. Moreover, we give the expressions of determinant of Laplacian matrix and coefficients of normalized Laplacian characteristic polynomial, respectively. Actually there are some other aspects of spectrum of signed graphs, which can be investigated. It will be left to our future study. In addition, there are many spectrum-based invariants, which are widely investigated, such as graph energy (e.g., graph theory [19, 20], incidence energy [21], and matching energy [22, 23]), HOMO-LUMO index [24, 25], and inertia [26–29]. In the future, we would like to study some properties of these spectrum-based indices of signed networks.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of China (nos. 11301302, 11861019), the Natural Science Foundation of Shandong (no. BS2013SF009), and Foundation of Shandong Provincial Education Department (no. J17KA165).

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