More on Spectral Analysis of Signed Networks

Spectral graph theory plays a key role in analyzing the structure of social (signed) networks. In this paperwe continue to study some properties of (normalized) Laplacianmatrix 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 1 of eigenvalues of normalized Laplacian matrix is discussed. We determine that the integer 2 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.


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.
Suppose that Γ = (, ) is a signed network.A signed function  :  → {+1, −1} is a switching function if Γ is transformed to a new signed network Γ  = (,   ) by  such that the underlying graph remains the same and the sign function is defined by   () = (V  )()(V  ) for an edge  = V  V  .Let Γ 1 = (,  1 ) and Γ 2 = (,  2 ) be two signed networks with the same underlying graph.We call Γ 1 and Γ 2 switching equivalent and write Γ 1 ∼ Γ 2 , if there exists a switching function  such that Γ 2 = Γ  1 .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 1 of eigenvalues of normalized Laplacian matrix is discussed.Sufficient and necessary condition for that the integer 2 is an eigenvalue of normalized Laplacian matrix is given.An expression of all coefficients of normalized Laplacian characteristic polynomial is present.

Laplacian Matrix and Signed Network
Hou et al. [3] introduced the incidence matrix of a signed network as follows.Let (Γ) = (  ) be an × matrix indexed by the vertex and the edge of signed network Γ and The following is immediate by the direct calculation.eorem (see [3]).Let Γ be a signed network.en (Γ) = (Γ)  (Γ) and (Γ) is a positive semidefinite matrix.
Suppose that any 1--walk has the sign.Then for any edge This implies that (Γ) 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.

eorem . e multiplicity of the eigenvalue of Laplacian matrix of a signed network is the number of components whose
Laplacian matrix is singular.

eorem (see [4]). A signed network is balanced if and only if for each pair of distinct vertices
From Theorems 2 and 4, we have the following.

eorem . A signed network Γ is balanced if and only if 𝐿(Γ) is singular.
The following is immediate from Theorem 5.
eorem .e 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.
eorem .Let Γ be a signed unicyclic network with a cycle .en Proof.By Theorem 7, the results hold if Γ is a signed cycle.
Assume that Γ has a pendant vertex, say .Let V be the unique neighbor of  in Γ.Let  be the edge joining  and V.After permutations, the first row and the first column of (Γ) correspond to the vertex  and the edge , respectively.Note that (Γ) is a square matrix since Γ is unicyclic.We get the determinant of (Γ) by expanding along the first row as follows: det where Γ  is a signed subgraph obtained from Γ by deleting the vertex .Hence we have Repeating the above finite steps, we have det (Γ) = det ().
Let Γ be a connected signed network.We call a subnetwork  as an essential spanning subnetwork of Γ if either Γ is balanced and  is a spanning tree of Γ, or else Γ is not balanced, (Γ) = () and every component of  is a unicyclic signed network in which the unique cycle is negative.By E(Γ) we denote the set of all essential spanning subnetworks of Γ.
eorem .Let Γ be a connected signed network.en where   is the number of essential spanning subgraphs which contain  unbalanced cycles and  0 = 0.
Note that [(Γ),   ] is the vertex-edge incidence matrix of a spanning subgraph of Γ, say    , with the edge set Note that every component of    is unicyclic and    ∈ E(Γ).By Theorem 8, we have where sgn So the result holds.
The following is immediate from Theorem 9, which is coincident with the definition of balance of signed network.
eorem .Let Γ be a signed network.en Γ is balanced if and only if each cycle of Γ is balanced cycle.

Normalized Laplacian Matrix and Signed Network
For a signed network Γ, the normalized Laplacian matrix L(Γ) is symmetric and positive semidefinite [7], so its eigenvalues are real and nonnegative, denoted by 0 ≤  1 ≤  2 ≤ ⋅ ⋅ ⋅ ≤   .Firstly we recall some properties of normalized Laplacian matrix.
In [8], the symmetry about 1 of eigenvalues for bipartite signed network was present as follows.Here we present a stronger result.
eorem .Let Γ be a connected signed network.en Γ is bipartite if and only if all eigenvalues of L(Γ) are symmetric about (including multiplicities); i.e., for each eigenvalue   , 2 −   is also an eigenvalue of L(Γ).
From Lemma 11, the integer 2 is the upper bound of normalized Laplacian eigenvalues.In this sequel, we give a sufficient and necessary condition for that the integer 2 is an eigenvalue of normalized Laplacian matrix.
eorem .Let Γ be a connected signed network.en 2 is an eigenvalue of L(Γ) if and only if Γ is a balanced bipartite signed network.
Proof.By Courant-Fischer theorem, we have Assume that 2 is an eigenvalue of L(Γ) with nonzero eigenvector   = ( 1 ,  2 , . . .,   ).By Lemma 11 and (14),   = −(V  V  )  for any edge  incident to V  and V  .So (Γ) can be partitioned into two parts such that no edge existing between any two vertices in every part.This means that Γ is bipartite.For any even cycle Moreover, ) and  2 is balanced.This implies that Γ is balanced.If Γ is balanced bipartite, then 0 is an eigenvalue of L(Γ).By Theorem 14 and Lemma 12, 2 is an eigenvalue of L(Γ).
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  2 or the cycle.A basic figure is the disjoint union of elementary figures.Lemma (see [9]).Let Γ = (, ) and (Γ, ) =   + 1  −1 + ⋅ ⋅ ⋅ +   be a signed network and its adjacency characteristic polynomial, respectively.en where B  is the set of basic figures on  vertices in , () is the number of components of , and () is the set of cycles in  and () = ∏ ∈() sgn().

Conclusion
Recently, there are some results on the spectral theory of signed graphs [10][11][12][13][14][15][16][17][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][27][28][29].In the future, we would like to study some properties of these spectrum-based indices of signed networks.