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 G be an undirected network of order n with vertex set V(G)={v1,v2,…,vn} and edge set E(G). The adjacency matrix A(G)=(aij)n×n of G is defined as follows: aij=1 if vi and vj are adjacent and aij=0 otherwise. A signed network Γ=(G,σ) consists of a network G=(V,E), referred to as its underlying network, and a sign function σ:E→{+,-}. The adjacency matrix of Γ is A(Γ)=(aijσ) with aijσ=σ(vivj)aij, where aij is an element in the adjacency matrix of the underlying network G and vivj is an edge of G. If all edges are signed positive, the adjacency matrix A(G,σ) is exactly the ordinary adjacency matrix A(G). Let DΓ=diagd1,d2,…,dn be a diagonal matrix where di is the degree of vertex vi in its underlying network. The Laplacian matrix of Γ, denoted by L(Γ), is defined as D(Γ)-A(Γ). The matrix D-1/2L(Γ)D-1/2 is said to be normalized Laplacian matrix of Γ, denoted by L(Γ).
A signed i1-ik-walk W in a signed network Γ is a sequence of vertices and edges W:vi1e12vi2e23vi3⋯vik-1e(k-1)kvik such that es(s+1)=visvis+1∈E(Γ) (s=1,2,…,k-1). An i1-ik-walk W is called even (odd) if k is even (odd). The sign of a signed walk W=v1e12v2e23⋯vl is sgn(W)=a12σa23σ⋯a(l-1)lσ and ei(i+1)=vivi+1 (i=1,2,…,l-1). A signed walk W is balanced (unbalanced) if sgn(W)=1 (sgn(W)=-1). 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 Γ=(G,σ) is a signed network. A signed function θ:V→{+1,-1} is a switching function if Γ is transformed to a new signed network Γθ=(G,σθ) by θ such that the underlying graph remains the same and the sign function is defined by σθ(e)=θ(vi)σ(e)θ(vj) for an edge e=vivj. Let Γ1=(G,σ1) and Γ2=(G,σ2) be two signed networks with the same underlying graph. We call Γ1 and Γ2 switching equivalent and write Γ1~Γ2, if there exists a switching function θ such that Γ2=Γ1θ. 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.
2. Laplacian Matrix and Signed Network Hou et al. [3] introduced the incidence matrix of a signed network as follows. Let S(Γ)=(sij) be an n×m matrix indexed by the vertex and the edge of signed network Γ and (1)sij=+1if vi is the head of ej+1if vi is the tail of ej and σej=+-1if vi is the tail of ej and σej=-0otherwise.
The following is immediate by the direct calculation.
Theorem 1 (see [3]). Let Γ be a signed network. Then L(Γ)=S(Γ)ST(Γ) and L(Γ) is a positive semidefinite matrix.
Theorem 2. Let Γ be a connected signed network on vertices v1,v2,…,vn. Then L(Γ) is singular if and only if any 1-i-walk has the same sign. In this case, 0 is a simple eigenvalue with an eigenvector α=(1,sgnW2,sgnW3,…,sgn(Wn))T, where Wi is a 1-i-walk in Γ.
Proof. Let xT=(x1,x2,…,xn)∈Cn. Note that for any nonzero vector x, L(Γ)=0 if and only if ST(Γ)x=0. By (14), ST(Γ)x=0 if and only if xi=aijσxj for any edge e=vivj. Let Wi=u1u2⋯ui be any 1-i-walk and u1=v1,ui=vi. Suppose that L(Γ)=0. So we have (2)x1=a12σx2=a12σa23σx3=⋯=a12σa23σ⋯ai-1iσxi=sgnWixi,which implies that each 1-i-walk has the same sign.
Note that sgn-1Wi=sgn(Wi). Hence (3)xT=x1,x2,…,xn=x1,sgnW2x1,sgnW3x1,…,sgnWnx1=x1αT.This implies that 0 is a simple eigenvalue of L(Γ) with an eigenvector α.
Suppose that any 1-i-walk has the sign. Then for any edge eij∈E(Γ), sgn(Wj)=sgnWi·aijσ. Let xT=(x1,x2,…,xn) be a column vector such that xi=sgn(Wi)x1 (i=2,3,…,n). Then xj=sgn(Wj)x1=sgn(Wi)·aijσx1=xiaijσ, i.e., xi=aijσxj. So we have (4)xTLΓx=∑eij∈EΓxi-aijσxj2=∑eij∈EΓaijσxj-aijσxj2=0.This implies that L(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.
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 v1,v2 all paths joining v1 and v2 have the same sign.
From Theorems 2 and 4, we have the following.
Theorem 5. A signed network Γ is balanced if and only if L(Γ) 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. det L ( C ) = 2 [ 1 - sgn ( C ) ] for any signed cycle C.
Proof. Let C be a signed cycle with vertex set V(C)={v1,v2,…,vn} and edge set E(C)={e1,e2,…,en} such that ei=vivi+1 (1≤i≤n-1) and en=vnv1. For the incidence matrix S(C), we expand its the first row (5)detSC=∏i=1nsiei+-1n+1s1en∏i=2nsiei-1.By directly calculation and the fact that sieisjej=-σ(e)aij=-aijσ for any edge ei=vivi+1. It follows that (6)detLC=detSC·detSTC=2-2sgnC.So the result holds.
Theorem 8. Let Γ be a signed unicyclic network with a cycle C. Then (7)detLΓ=detLC=21-sgnC.
Proof. By Theorem 7, the results hold if Γ is a signed cycle. Assume that Γ has a pendant vertex, say u. Let v be the unique neighbor of u in Γ. Let e be the edge joining u and v. After permutations, the first row and the first column of S(Γ) correspond to the vertex u and the edge e, respectively. Note that S(Γ) is a square matrix since Γ is unicyclic. We get the determinant of S(Γ) by expanding along the first row as follows:(8)detSΓ=sue·detSΓ′,where Γ′ is a signed subgraph obtained from Γ by deleting the vertex u. Hence we have (9)detLΓ=detSΓ·detSTΓ=detSΓ′·detSTΓ′=detLΓ′.Repeating the above finite steps, we have detL(Γ)=detL(C).
Let Γ be a connected signed network. We call a subnetwork H as an essential spanning subnetwork of Γ if either Γ is balanced and H is a spanning tree of Γ, or else Γ is not balanced, V(Γ)=V(H) and every component of H 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.
Theorem 9. Let Γ be a connected signed network. Then(10)detLΓ=∑l=04lbl,where bl is the number of essential spanning subgraphs which contain l unbalanced cycles and b0=0.
Proof. It is evident that the result holds if Γ is a tree. Assume that Γ contains some cycles. By Cauchy-Binet Theorem [6] and L(Γ)=S(Γ)·ST(Γ), we have (11)detLΓ=∑E′⊆EΓ;E′=VΓdetSVΓ,E′·detSTVΓ,E′,where S[V(Γ),E′] is a square submatrix of S(Γ).
Note that S[V(Γ),E′] is the vertex-edge incidence matrix of a spanning subgraph of Γ, say HE′, with the edge set E′=VΓ. Moreover, detLHE′=S[V(Γ),E′]·ST[V(Γ),E′]. Note that every component of HE′ is unicyclic and HE′∈E(Γ). By Theorem 8, we have (12)detLΓ=∑E′⊆EΓ;E′=VΓdetSVΓ,E′·detSTVΓ,E′=∑E′⊆EΓ;E′=VΓdetLHE′=∑H∈EΓdetLH=∑H∈EΓ ∏i=1bl21-sgnCiH where sgnCiH=-1=∑l=04lbl.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 Γ be a signed network. Then Γ is balanced if and only if each cycle of Γ is balanced cycle.
3. 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≤λ1≤λ2≤⋯≤λn. Firstly we recall some properties of normalized Laplacian matrix.
Lemma 11 (see [7]). Let Γ be a signed network on n vertices with normalized Laplacian eigenvalues λ1≤λ2≤⋯≤λn. Then λn≤2.
Lemma 12 (see [3, 7]). Let Γ1 and Γ2 be two signed networks with the same underlying network. Then Γ1~Γ2 if and only if L(Γ1) and L(Γ2) are signature similar.
In [8], the symmetry about 1 of eigenvalues for bipartite signed network was present as follows. Here we present a stronger result.
Theorem 13 (see [8]). Let Γ=(G,σ) be a bipartite signed network. If λ is an eigenvalue of L(Γ), then 2-λ is also an eigenvalue of L(Γ).
Theorem 14. Let Γ be a connected signed network. Then Γ is bipartite if and only if all eigenvalues of L(Γ) are symmetric about 1 (including multiplicities); i.e., for each eigenvalue λi, 2-λi is also an eigenvalue of L(Γ).
Proof. It suffices to verify that I-L(Γ) and -(I-L(Γ)) have the same spectrum. Note that I-L(Γ)=D-1/2A(Γ)D-1/2. Γ is bipartite if and only if D-1/2A(Γ)D-1/2 can be expressed as 0BBT0. It is evident that (13)-I00I0-B-BT0-I00I=0BBT0.This yields to the result.
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.
Theorem 15. Let Γ be a connected signed network. Then 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(14)λn=supf≠0∑u~vfu-σuvfv2∑vf2vdv.Assume that 2 is an eigenvalue of L(Γ) with nonzero eigenvector yT=(y1,y2,…,yn). By Lemma 11 and (14), yi=-σ(vivj)yj for any edge e incident to vi and vj. So V(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 C2k=v1v2⋯v2kv1, we have (15)y1=-σv1v2y2=σv1v2σv2v3y3=-σv1v2σv2v3σv3v4y4=⋯=-σv1v2σv2v3⋯σv2k-1v2ky2k.Moreover, y1=-σ(v1v2k)y2k. So σ(v1v2)σ(v2v3)⋯σ(v2k-1v2k)=σ(v1v2k) and C2k 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 K2 or the cycle. A basic figure is the disjoint union of elementary figures.
Lemma 16 (see [9]). Let Γ=(G,σ) and ϕ(Γ,x)=xn+a1xn-1+⋯+an be a signed network and its adjacency characteristic polynomial, respectively. Then (16)ai=∑B∈Bi-1pB2cBσB,where Bi is the set of basic figures on i vertices in G, p(B) is the number of components of B, and c(B) is the set of cycles in B and σ(B)=∏C∈c(B)sgn(C).
Let ψ(Γ,x) be the normalized Laplacian characteristic polynomial of Γ. By the definition of normalized Laplacian matrix, we have (17)ψΓ,x=detxI-LΓ=detxI-D-1/2LΓD-1/2=detxI-I+D-1/2AΓD-1/2=detx-1I+D-1/2AΓD-1/2=x-1n+c1x-1n-1+⋯+cn-1x-1+cn.
Theorem 17. Let Γ be a signed network on n vertices and ψ(Γ,x)=(x-1)n+c1(x-1)n-1+⋯+cn-1(x-1)+cn be its normalized Laplacian characteristic polynomial. Then (18)ck=∑B∈Bk-1pB2cBσB1Dk,where Bk is the set of basic figures on k vertices in G, p(B) is the number of components of B, c(B) is the set of cycles in B, σ(B)=∏C∈cBsgn(C), Dk=∏vi∈V(B)di, and di is the degree of vi in Γ.
Proof. Note that (19)ψΓ,x=detx-1I+D-1/2AΓD-1/2=-1ndet1-xI-D-1/2AΓD-1/2.Set (20)det1-xI-D-1/2AΓD-1/2=x-1n+c1′x-1n-1+⋯+cn-1′x-1+cn′.So ck=(-1)kck′. Moreover, (-1)kck′ equals to the sum of all k×k minors of D-1/2A(Γ)D-1/2. Then ck is the sum of all k×k minors of D-1/2A(Γ)D-1/2. It is evident that each such k×k minor of D-1/2A(Γ)D-1/2 is the product of the corresponding k×k minors of D-1/2, A(Γ), and D-1/2, respectively. Furthermore, any k×k minor of A(Γ) is the determinant of adjacency matrix of an induced subgraph of Γ with k 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.