Corona product of signed graphs and its application to signed network modelling
aa r X i v : . [ m a t h . C O ] A ug Corona product of signed graphs and its application tosigned network modelling
Bibhas Adhikari Amrik Singh Sandeep Kumar Yadav
Abstract.
The notion of corona of two graphs was introduced by Frucht and Harary in 1970.In this paper we generalize their definition of corona product of two graphs and introducecorona product of two signed graphs by utilizing the framework of marked graphs, whichwas introduced by Beineke and Harary in 1978. We study structural and spectral propertiesof corona product of signed graphs. Further we define signed corona graphs by consideringcorona product of a fixed small signed graph with itself iteratively, and we call the smallgraph as the seed graph for the corresponding corona product graphs. Signed corona graphscan be employed as a signed network generative model for large growing signed networks. Westudy structural properties of corona graphs that include statistics of signed links, all types ofsigned triads and degree distribution. Besides we analyze algebraic conflict of signed coronagraphs generated by specially structured seed graphs. Finally we show that a suitable choiceof a seed graph can produce corona graphs which preserve properties of real signed networks.
Keywords. signed graphs, structural balance, corona product, algebraic conflict
Signed networks represent a framework to deal with binary relationship between nodes in anetwork that has two contradictory possibilities. For example, like and dislike, love and hate,trust and distrust are considered as measures of relationships between people, whereas allianceand antagonism between two countries can be considered as contradictory binary internationalrelationships. Signed network model to represent social systems was first introduced by Hararyand Cartwright in 1956 to generalize the theory of balanced state of a social system developedby Heider in 1946 [8]. Heider rationalized his theory of balanced state by considering possiblerelationships in a system of three entities [19]. On the other hand, real world data can beput into the signed network setup to determine salient features of the data. For instance,in the trust network of Epinions, users establish binary relations to each other that reflecttrust and distrust [16,25]. Mathematically, a signed graph is a graph in which some edges aredesignated with positive sign reflecting positive relationship between the constituent pair ofnodes and other edges are assigned negative sign which represents negative relationship. Thussigned graph is an ordered tuple G = ( V, E, σ ) where V denotes the set of nodes, E ⊆ V × V the edge set, and σ : E → { + , −} is called the signature function [34]. In this paper we usethe terms signed graph and signed network interchangeably.Structural balance of signed networks has always been the central topic in the study of realworld signed networks. A signed network is called balanced or in balanced state if all triadsin it are balanced [8]. A triad is balanced if all three edges are positive, or if two are positiveand one negative. Alternatively, it is also established that a signed network is balanced ifand only if all its cycles are balanced [17]. A signed cycle is called balanced if the number ofnegative edges in it is even. Most often it is hypothesized that real networks evolve towardsbalanced state [11]. See also [10]. However, recently an excellent article including a historicalbackground about the development of the theory of balance demonstrates that this is not thecase always [12]. If a signed network is not balanced it is called unbalanced. Evidently, it is nfair to call a signed network unbalanced if only a few cycles in it are unbalanced comparedto an unbalanced network in which most of the cycles are unbalanced. Thus the concept ofdegree of balance of a signed network has emerged in literature, and several measures areproposed to estimate it [2–4, 13, 23, 33].Several signed network generative models are proposed in literature with a goal to pre-serve structural or statistical properties of real world signed networks. For example, in [21] alow-rank model based on matrix completion technique is proposed to generate signed networkswhich can inherit structural balance property of real networks. In [9], the authors propose aparametric model to preserve degree distribution, sign distribution, and balance/unbalancedtriad distribution of signed networks. This model is inspired by the popular Chung-Lu modelfor generation of unsigned networks. Recently, a model is proposed based on local pref-erential attachment to generate signed networks that have community structure and highpositive clustering coefficient [27]. To the best of the knowledge of the authors, the liter-ature lack a deterministic growing network model that can preserve sign distribution, andbalance/unbalanced triad distribution of signed networks. Besides, spectral properties of theexisting network generative models are completely unexplored.Thus how the spectral proper-ties relate to the structural properties of the networks generated by these models is unclear.Spectral property such as algebraic conflict [2, 23], the smallest signed Laplacian eigenvalueof the network can estimate the degree of balance.In this paper we first introduce the notion of corona product of signed graphs by gener-alizing the definition of corona product of unsigned graphs. Let G be an unsigned graph on n nodes, and H an unsigned graph. Then the corona product graph G ◦ H of G and H isdefined as the graph obtained by taking one copy of G and n copies of H , and then joiningthe i th vertex of G to every vertex in the i th copy of H [14]. Note that except the existingedges of G, H in G ◦ H there will be nk new edges created where k denotes the number ofnodes in H. For corona product of signed graphs, the job is to define the signs of these newedges so that when the graphs
G, H are unsigned those edge signs will be positive. We definethe signs of the new edges utilizing the formalism of marked graph which is a framework in-troduced by Beineke and Harary [6,17]. We call this a marking scheme on the node set for thedefinition of corona product. We study structural balance of G ◦ H, and provide a necessaryand sufficient condition based on the structural properties of G, H such that G ◦ H becomesbalanced. Let T i denote a triad in a signed network with i number of negative edges in it, i = 0 , , , . Obviously, the distribution of the numbers of T i s influence degree of balance ofa signed network. Some of the earliest measures of degree of balance are defined in terms ofthe number of T i s. For example, the ratio of the number of signed to unsigned triads in asigned network is considered as the degree of balance.(a) T (b) T (c) T (d) T Figure 1: Possible triads in an (undirected) signed network. Triads (a) and (c) arebalanced, and triads (b) and (d) are unbalanced. Solid and dashed edges representpositive and negative edges, respectively.
The positive and negative degree of a node u in a signed graph are defined by number ofpositive and negative edges incidental to the node u denoted by d + ( u ) and d − ( u ), respectively.The total degree and net-degree of u are given by d ± ( u ) = d + ( u ) + d − ( u ) and d ( u ) = d + ( u ) − d − ( u ), respectively. A signed graph is called net-regular if every node of it has thesame net-degree d [29]. Note that net-regular signed graphs are signed counterpart of unsignedregular graphs. The adjcency matrix A = [ a ij ] corresponding to a signed graph G = ( V, E, σ )is defined by a ij = 1 if ( i, j ) ∈ E and σ (( i, j )) = + , a ij = − σ (( i, j )) = − , and a ij = 0 therwise. The signed Laplacian matrix of G on n nodes is defined as L = D − A where D isa diagonal matrix of order n whose i th diagonal entry is the total degree of i th node of G [23].The signless Laplacian matrix of G is defined by Q = D + A. The eigenvalues of
A, L, and Q arecalled adjacency, signed Laplacian and signless Laplacian eigenvalues of G. The eigenvaluesof these matrices inherit structural balance properties of the corresponding signed graph. Forexample, a signed graph G is balanced if and only if adjacency eigenvalues of G are same asthe adjacency eigenvalues of the underlying unsigned graph corresponding to G [1]. The leastsigned Laplacian eigenvalue is zero if and only if the corresponding graph is balanced [23].For an unbalanced graph, the least signed Laplacian eigenvalues is considered as a measureof degree of balance of the graph and it is called algebraic conflict of the graph [2, 23]. Wederive adjacency eigenvalues of G ◦ H in terms of adjacency eigenvalues of the signed graphs G and H respectively, when H is net-regular. We determine signed Laplacian eigenvaluesof G ◦ H from which the formula of signless Laplacian eigenvalues follows after a marginalmodification.Utilizing the corona product framework for unsigned graphs Sharma et al. [31,32] definedcorona graphs as a model for generating large networks. See also [28]. Given a connectedgraph G = G (0) the corona graphs are defined as G ( m ) = G ( m − ◦ G, m ≥ . (1)The graph G is called the seed graph for the corona graph G ( m ) . Adapting this idea and in-corporating the definition of corona product introduced in this paper, we define signed coronagraphs by considering the seed graph as a signed graph as in equation (1). Thus constructionof signed corona graphs provides a deterministic growing signed network generative model.Obviously, the structural balance and spectral properties of G ( m ) depend on the choice of theseed graph and the distribution of signs of the new edges which appear due to the definition ofcorona product, that is, the marking scheme on the node set of the seed graph. The modellingperspective of preserving properties of real networks using G ( m ) raises the following question.Can we choose a seed graph G so that desired sign distribution, triad distribution, degreeof balance in G ( m ) be obtained? We show that this is indeed the case. That is, given thestatistics of signs of edges and triads of a real signed network it is possible to determine a seedgraph such that the corresponding large signed corona graphs provide desired distribution ofsigned edges and triads approximately. The algebraic conflict of G ( m ) can be obtained byiterative use of the formula of signed Laplacian eigenvalues derived for corona product ofsigned graphs discussed above. Besides, we determine degree sequence of G ( m ) from whichthe positive and negative degree distribution of G ( m ) can be computed and compared withreal networks.Throughout we assume that the seed graph is a simple connected signed graph. The maincontributions of this paper are as follows(1) We introduce corona product of two signed graphs based on the framework of markedgraphs and study its structural properties.(2) We study spectra and Laplacian spectra of corona product of signed graphs, and providecomputable expressions of eigenvectors corresponding to such eigenvalues.(3) We define signed corona graphs and propose it as a signed network generative model.We provide computable formulae of number of signed edges, triads, and positive andnegative degrees of every node in a corona graph. We investigate algebraic conflict ofcorona graphs, which are generated by specially structured seed graphs. Finally weshow that corona graphs can inherit properties of real signed networks. In this section we introduce corona product G ◦ H of a pair of signed graphs G, H , andwe study the structural balance and spectral properties of G ◦ H. Note that this operation product) is non-commutative, that is, G ◦ H need not be equal to H ◦ G. First we review thenotion of marked graphs as follows.A graph is called a marked graph if every node of the graph is marked by either a positiveor negative sign. Thus a marked graph is a tuple G = ( V, E, µ ) where V is the node set, E the edge set and µ : V → { + , −} is called the marking function. An obvious way to constructa signed graph from a marked graph is be defining the sign of an edge of the marked graphas the product of signs of its adjacent vertices [18]. On the other hand, a marked graph canbe defined from a signed graph G = ( V, E, σ ) by defining the marking of a node v ∈ V as µ ( v ) = Y e ∈ E v σ ( e ) (2)where E v is the set of signed edges adjacent at v. Such a method of marking is also knownas canonical marking [6, 7]. There can be multiple ways to define a marking function for asigned graph. We consider two marking functions, canonical marking and plurality marking in this paper. We define plurality marking of a node v of a signed graph G = ( V, E, µ ) as µ ( v ) = (cid:26) + , if max { d + ( v ) , d − ( v ) } = d + ( v ) − , Otherwise (3)Hence a node is negatively marked in plurality marking scheme only when d − ( v ) > d + ( v ) . Thus from now onward we denote a signed graph as a 4-tuple G = ( V, E, σ, µ ) where σ and µ are the signature function and marking function defined on the edge set E and node set V respectively.Thus we introduce the following definition for corona product of two signed graphs. Definition 2.1.
Let G = ( V , E , σ , µ ) and G = ( V , E , σ , µ ) be signed graphs on n and k nodes respectively. Then corona product G ◦ G of G , G is a signed graph by takingone copy of G and n copies of G , and then forming a signed edge from i th node of G toevery node of the i th copy of G for all i. The sign of the new edge between i th node of G , say u and j th node in the i th copy of G , say v is given by µ ( u ) µ ( v ) where µ i is a markingscheme defined by σ i , i = 1 , . For instance, the corona product G ◦ G of signed graphs G and G is shown in Figure2. Note that canonical and plurality marking are same for the graph G . For G the markingof the nodes 1 , − and + respectively. Thus the choice of the markingfunction produce different corona product graphs. (a) G a b (b) G b a a b ba (c) G ◦ G when G , G are withcanonical marking b a a b ba (d) G ◦ G when G , G are withplurality marking Figure 2: The corona product of G ◦ G is shown in (c), (d) with canonical andplurality marking functions on G i , i = 1 , elow we discuss structural and spectral properties of the corona product of two graphs. G ◦ G We provide the statistics about the number of edges and triads in G = G ◦ G = ( V, E, σ, µ )for a given pair of signed graphs G = ( V , E , σ , µ ) and G = ( V , E , σ , µ ) as follows.Observe that the number of nodes in G ◦ G is given by | V | + | V | | V | where | V i | denotesthe number of nodes in G i , i = 1 , . Let M + i and M − i denote the number of positively andnegatively marked nodes in G and G respectively. The Table 1 describes the statistics ofedges in G ◦ G . Edges G G G ◦ G | E | | E | | E | + | V | | E | + | V | | V | | E +1 | | E − | | E +1 | + | V | | E +2 | + M +1 M +2 + M − M − − edges | E − | | E − | | E − | + | V | | E − | + M +1 M − + M − M +2 Table 1: Statistics of number of nodes and edges in G ◦ G . Now we consider counting triads in G ◦ G . Let s ∈ { + , −} . We denote | E s | + + as thenumber of edges of sign s which connect two positively marked nodes in G , | E s | ± as thenumber of edges of sign s which connect one positively marked and one negatively markednodes in G , and | E s | − − as the number of edges of sign s which connect two negatively markednodes in G , . Then the Table 2 describes the count of triads of type T i which denotes a triadhaving i number of negative edges, i = 0 , , , triads G G G ◦ G T | T ( G ) | | T ( G ) | | T ( G ) | + | V | | T ( G ) | + M +1 | E +2 | + + + M − | E +2 | − − T | T ( G ) | | T ( G ) | | T ( G ) | + | V | | T ( G ) | + M +1 ( | E +2 | ± + | E − | + + )+M − ( | E +2 | ± + | E − | − − ) T | T ( G ) | | T ( G ) | | T ( G ) | + | V | | T ( G ) | + M +1 ( | E +2 | − − + | E − | ± )+M − ( | E +2 | + + + | E − | ± ) T | T ( G ) | | T ( G ) | | T ( G ) | + | V | | T ( G ) | + M +1 | E − | − − + M − | E − | + + Table 2: Counts of triads in G ◦ G . The proof of the formulas presented in Table 1 and Table 2 directly follows from thedefinition of corona product, and easy to verify. Indeed, note that when a node i of G getslinked with all the nodes of the i th copy of G , the total number of new triads created is | E | , that is, a triad ( i, j, l ) is formed for every edge ( j, l ) in G . Now the type of this triad dependson the marking of i, j, l and the sign of the edge ( j, l ) . The signs of all three edges are givenby µ ( i ) µ ( j ) , µ ( i ) µ ( l ) and σ (( j, l )) . The total number of triads of G = G ◦ G is given byT( G ) = T( G ) + | V | (T( G ) + | E | ) (4)where T( G i ) denotes the total number of triads in G i , i = 1 , . It is evident that G ◦ G is unbalanced if one of the G i , i = 1 , G i , i = 1 , G ◦ G is balanced. For instance,the graph G ◦ G in Figure 2 is balanced while G and G are balanced, whereas the graph G ◦ G is unbalanced even if G and G are balanced in Figure 3. (a) G a b (b) G b a a b ba (c) G ◦ G when G , G are withcanonical marking b a a b ba (d) G ◦ G when G , G are withplurality marking Figure 3: The corona product of G ◦ G is shown in (c), (d) with different markingfunctions on G i , i = 1 , In the following theorem we classify when G ◦ G will be an unbalanced graph when G i is balanced, i = 1 , Theorem 2.2.
Let G = ( V , E , σ , µ ) and G = ( V , E , σ , µ ) be balanced signed graphs.Then G ◦ G is unbalanced if and only if G contains one of the following types of edges.(i) a positive edge which connects two oppositely marked nodes(ii) a negative edge which which connects two positively marked nodes(iii) a negative edge which connects two negatively marked nodes. Proof:
The proof follows from the fact that any positively or negatively marked node of G forms a triad T in G ◦ G when there are edges of type ( i ) and/or ( ii ) in G , otherwise itforms a triad of type T when G has an edge of type ( iii ) . (cid:3) It follows from Theorem 2.2 that for two balanced signed graphs G and G , the coronaproduct graph G ◦ G is balanced if and only if every positive edge of G is incidental to apair of negatively or positively marked nodes, and every negative edge must be incidental toa pair of oppositely marked nodes.Now we focus on the spectral properties of G ◦ G . Let G = ( V , E , σ , µ ) and G =( V , E , σ , µ ) be two signed graphs with n and k number of nodes, respectively. Suppose V = { u , . . . , u n } and V = { v , . . . , v k } . Let us denote the marking vectors corresponding tovertices in G and G as µ [ V ] = (cid:2) µ ( u ) µ ( u ) . . . µ ( u n ) (cid:3) and µ [ V ] = (cid:2) µ ( v ) µ ( v ) . . . µ ( v k ) (cid:3) where µ j ( u ) = 1 if marking of u = + , otherwise µ j ( u ) = − , j = 1 , . Then with a suitable labeling of the nodes the adjacency matrix of G ◦ G is given by A ( G ◦ G ) = (cid:20) A ( G ) µ [ V ] ⊗ diag( µ [ V ]) µ [ V ] T ⊗ diag( µ [ V ]) A ( G ) ⊗ I n (cid:21) (5)where A ( G i ) denotes the adjacency matrix associated with G i , i = 1 , , ⊗ denotes the Kro-necker product of matrices, I n is the identity matrix of order n, anddiag( µ [ V ]) = µ ( u ) 0 . . . µ ( u ) . . . . . . µ ( u n ) . hus A ( G ◦ G ) is a symmetric matrix of order n + nk = n (1 + k ) . Recall that a signedgraph is called net-regular if d + ( v ) − d − ( v ) = d is same for every node v of the graph, and d iscalled the net-regularity of the graph. Then note that the net-regularity of a net-regular graphis always an eigenvalue of the graph and the all-one vector is the corresponding eigenvector.We denote the all-one vector of dimension k as k . The following theorem provides adjacencyspectra of G ◦ G when G is net-regular. Theorem 2.3.
Let G be any signed graph on n nodes and G be a net-regular signed graphon k nodes having net-regularity d. Let ( λ i , X i ) be an adjacency eigenpair of G , and ( η j , Y j ) be an eigenpair of G , i = 1 , . . . , n, j = 1 , . . . , k. Let η k = d. Then an adjacency eigenpair of G ◦ G is given by ( λ ( i ) ± , Z ( i ) ± ) , i = 1 , . . . , n where λ ( i ) ± = d + λ i ± p ( d − λ i ) + 4 k , Z ( i ) ± = X i µ ( v ) λ ( i ) ± − d diag ( µ [ V ]) X i µ ( v ) λ ( i ) ± − d diag ( µ [ V ]) X i ... µ ( v k ) λ ( i ) ± − d diag ( µ [ V ]) X i . In addition, if all the nodes in G are either positively or negatively marked, that is µ [ V ] = k or − k then (cid:18) η j , (cid:20) Y j ⊗ e i (cid:21)(cid:19) is an eigenpair of G ◦ G where j = 1 , . . . , k − , and { e i : j = 1 , . . . , n } the standard basisof R n . Proof:
Let ( λ, Z ) be an eigenpair of G ◦ G where λ ∈ R , Z = Z Z ... Z k , Z l ∈ R n , l = 0 , , . . . , k. Then setting A ( G ◦ G ) Z = λZ we obtain (cid:20) A ( G ) µ [ V ] ⊗ diag( µ [ V ]) µ [ V ] T ⊗ diag( µ [ V ]) A ( G ) ⊗ I n (cid:21) Z = λZ which yields the following system of equations. A ( G ) Z + µ ( v ) µ [ V ] Z + . . . + µ ( v k ) µ [ V ] Z k = λZ µ ( v ) µ [ V ] Z + k X j =1 [ A ( G )] j Z j = λZ ... µ ( v k ) µ [ V ] Z + k X j =1 [ A ( G )] j Z j = λZ k where [ A ( G )] xj denote the ( x, j )th entry of A ( G ) . Now adding the last k equations weobtain ( µ ( v ) + µ ( v ) + . . . + µ ( v k )) µ [ V ] Z + d ( Z + . . . + Z k ) = λ ( Z + . . . + Z k ) ⇒ ( µ ( v ) + µ ( v ) + . . . + µ ( v k )) µ [ V ] Z = ( λ − d )(( Z + . . . + Z k )) . et Z j = µ ( v j ) λ − d µ [ V ] Z j = 1 , . . . , k and putting these in the first equation of the system wehave A ( G ) Z = (cid:18) λ − kλ − d (cid:19) Z . Now setting Z = X i and λ − kλ − d = λ i we obtain the quadratic polynomial equation λ − λ ( d + λ i ) − ( k − dλ i ) = 0 solving which the roots λ ( i ) ± and its corresponding eigenvectors Z ( i ) ± follows.If µ [ V ] = k or − k then µ [ V ] Y j = 0 since k is an eigenvector of A ( G ) , and eigenvectorsof a symmetric matrix form an orthogonal set. Hence (cid:20) A ( G ) µ [ V ] ⊗ diag( µ [ V ]) µ [ V ] T ⊗ diag( µ [ V ]) A ( G ) ⊗ I n (cid:21) (cid:20) Y j ⊗ e i (cid:21) = (cid:20) µ [ V ] Y j ⊗ diag( µ [ V ]) e j η j Y j ⊗ e i (cid:21) = η j (cid:20) Y j ⊗ e i (cid:21) . This completes the proof. (cid:3)
Note that if G is net-regular with plurality marking then either µ ( v j ) = + or µ = − for all j. Hence the Theorem 2.3 provides all the eigenvalues of G ◦ G for such a signedgraph G . We mention that Barik et al. [5] gave the complete description of adjacency eigenvalues of G ◦ G when G , G are unsigned and G is regular. Theorem 2.3 generalizes their findings,and provides some insights about how signs of the edges and marking of the nodes influencethe eigenpairs for corona product graphs.Now we consider signed Laplacian spectra of corona product graphs. The signed Laplacianmatrix associated with a signed graph G on n nodes is defined as L ( G ) = D ( G ) − A ( G )where D ( G ) is a diagonal matrix of order n such that the i th diagonal entry is d + i + d − i and A ( G ) is the adjacency matrix as usual [34]. The signless Laplacian matrix is definedas Q ( G ) = D ( G ) + A ( G ) . Note that signed Laplacian matrix is a symmetric positive semi-definite matrix, and positive definite when the corresponding graph is unbalanced [20, 23].The signless Laplacian matrix associated with a signed graph G and be considered as thesigned Laplacian corresponding to the graph G which is obtained from G by converting thepositive (resp. negative) edges to negative (resp. positive) edges in G [15].First we have the following observation which will be used in sequel. Lemma 2.4.
Let G be a signed graph on n nodes. Then n is an eigenvector correspondingto a signed Laplacian eigenvalue λ if and only if d − i = d − (= λ/ for every node i in G. Proof:
From the definition of signed Laplacian matrix, we have L ( G ) = D ( G ) − A ( G ) where[ D ( G )] ii = d + i + d − i , the i th diagonal entry of D ( G ) . Then L ( G ) n = d − d − ...2 d − n . Hence n is an eigenvector of L ( G ) if and only if d − i equals a constant, say d − for all i. Thiscompletes the proof. (cid:3)
For a pair of signed graphs G = ( V , E , σ , µ ) and G = ( V , E , σ , µ ) on n and k nodes respectively, the signed and signless Laplacian matrices of G ◦ G are given by L ( G ◦ G ) = (cid:20) L ( G ) + kI n − µ [ V ] ⊗ diag( µ [ V ]) − µ [ V ] T ⊗ diag( µ [ V ]) ( L ( G ) + I k ) ⊗ I n (cid:21) Q ( G ◦ G ) = (cid:20) Q ( G ) + kI n µ [ V ] ⊗ diag( µ [ V ]) µ [ V ] T ⊗ diag( µ [ V ]) ( Q ( G ) + I k ) ⊗ I n (cid:21) with a suitable labeling on the vertices of G ◦ G . Then we have the following theorem. heorem 2.5. Let G = ( V , E , σ , µ ) be a signed graph on n nodes and G = ( V , E , σ , µ ) be a signed graph on k nodes. Let V = { v , . . . , v k } . Let ( λ i , X i ) be a signed Laplacian eigen-pair of G , i = 1 , . . . , n. Let d − j denote the negative degree of a node v j in G . Then the rootsof the polynomial equations x − k − x − (2 d − + 1) − x − (2 d − + 1) − . . . − x − (2 d − k + 1) = λ i , (6) i = 1 , . . . , n, are signed Laplacian eigenvales of G ◦ G . An eigenvector corresponding to suchan eigenvalue x = λ corresponding to λ i in equation (6) is given by X i − µ ( v ) λ − (2 d − + 1) diag ( µ [ V ]) X i − µ ( v ) λ − (2 d − + 1) diag ( µ [ V ]) X i ... − µ ( v k ) λ − (2 d − k + 1) diag ( µ [ V ]) X i . where i = 1 , . . . , n. Proof:
The proof is similar to the proof of Theorem 2.3. Indeed, Let Z = Z Z ... Z k , Z j ∈ R n ,j = 0 , . . . , k be an eigenvector corresponding to an eigenvalue λ of L ( G ◦ G ) . Then from L ( G ◦ G ) Z = λZ we have the following system of equations( L ( G ) + kI n ) Z − µ ( v )diag( µ [ V ]) Z − . . . − µ ( v k )diag( µ [ V ]) Z k = λZ − µ ( v )diag( µ [ V ]) Z + ([ L ( G )] + 1) Z − . . . + [ L ( G )] k Z k = λZ ... − µ ( v k )diag( µ [ V ]) Z + [ L ( G )] k Z − . . . + ([ L ( G )] kk + 1) Z k = λZ k where [ L ( G )] xy is the ( x, y ) entry of L ( G ) . Then adding the last k equations we have − ( µ ( v ) + . . . + µ ( v k ))diag( µ [ V ]) Z = k X j =1 ( λ − (2 d − j + 1)) Z j . Setting Z j = − µ ( v j ) λ − (2 d − j + 1) diag( µ [ V ]) Z , j = 1 , . . . , k and putting these in the first equa-tion of the system we have L ( G ) Z = (cid:18) λ − k − λ − (2 d − + 1) − . . . − λ − (2 d − k + 1) (cid:19) Z . Hence the desired result follows by setting Z = X i . (cid:3) It is interesting to observe from Theorem 2.3 and Theorem2.5 that adjacency and signedLaplacian eigenvalues of G ◦ G depend only on the degrees of the nodes of G whereas theeigenvectors are influenced by degrees of nodes of G and the marking scheme of G . Besides,observe that the equation (6) gives the complete list of eigenvalues of G ◦ G only whenany two distinct nodes in G have distinct negative degrees. Otherwise, the degrees of thepolynomial equations (6) drop and it can not produce all the eigenvalues. In the extreme casewhen negative degrees of nodes in G are all equal then the polynomials become quadraticand produce only 2 n eigenvalues. The following theorem considers this case. heorem 2.6. Let G = ( V , E , σ , µ ) be a signed graph on n nodes and G = ( V , E , σ , µ ) be a signed graph on k nodes. Let V = { v , . . . , v k } . Let ( λ i , X i ) be a signed Laplacian eigen-pair of G , and ( η j , Y j ) are signed Laplacian eigenpairs of G , i = 1 , . . . , n, j = 1 , . . . , k. Let d − = d − j denote the negative degree of every node v j in G . Then a signed Laplacian eigenpairof G ◦ G is given by ( λ ( i ) , Z ( i ) ± ) where λ ( i ) ± = 2 d − + 1 + λ i + k ± p [(2 d − + 1) − ( λ i + k )] + 4 k Z ( i ) ± = X i − µ ( v ) λ ( i ) ± − (2 d − + 1) diag ( µ [ V ]) X i − µ ( v ) λ ( i ) ± − (2 d − + 1) diag ( µ [ V ]) X i ... − µ ( v k ) λ ( i ) ± − (2 d − + 1) diag ( µ [ V ]) X i where i = 1 , . . . , n. Let η k = 2 d − . In addition if all the nodes in G are marked either positively or negativelythen an eigenpair of G ◦ G is (cid:18) η j + 1 , (cid:20) Y j ⊗ e i (cid:21)(cid:19) , i = 1 , . . . , n, j = 1 , . . . , k − where { e i : i = 1 , . . . , n } is the standard basis of R n . Proof:
The first part follows from the proof of Theorem 2.5 by setting d − j = d − for j =1 , . . . , k. Next, if d − j = d − for all j then by Lemma 2.4 k is an eigenvector of G correspondingto the eigenvalue 2 d − = η k . Hence Tk Y j = 0 for j = 1 , . . . , k − L ( G ) is a symmetricmatrix and it has complete set of orthogonal vectors. Further, if all the nodes of G arepositively marked then µ [ V ] = Tk . Finally L ( G ◦ G ) (cid:20) Y j ⊗ e i (cid:21) = (cid:20) − µ [ V ] Y j ⊗ e i ( L ( G ) Y j + Y j ) ⊗ e i (cid:21) = ( η j + 1) (cid:20) Y j ⊗ e i (cid:21) . If the nodes of G are marked then µ [ V ] = − Tk and hence the desired result follows. (cid:3) We emphasize that the Theorem 2.6 generalizes the result on Laplacian spectra of coronaproduct of unsigned graphs obtained in [5]. Indeed, setting d − = 0 in the Theorem 2.6describes complete set of eigenpairs of corona product of unsigned graphs. Remark 2.7.
Similar results like Lemma 2.4, Theorem 2.5 and Theorem 2.6 can be obtainedfor signless Laplacian eigenpairs of G ◦ G only replacing d − j by d + j for j = 1 , . . . , k. Indeed,note that k is an eigenvector corresponding to the signless Laplacian eigenvalue d + of G ◦ G if and only if d + j = d + for j = 1 , . . . , k. Corona product of unsigned graphs has been used as a framework to develop models forcomplex networks in literature [30,32]. Inspired by this idea we propose signed corona graphsas a model for generating large signed networks. In this approach, we consider a smallconnected signed graph G = G (0) as a seed graph for generation of corona graphs given by G ( m ) = G ( m − ◦ G, m ≥ ith a suitable choice of marking function which will be used in every step i ≤ m − G ( m ) . We explore different structural and spectral properties of G ( m ) whichcontribute to investigate how close does it preserve properties of real signed networks. Finallywe consider the problem of structural reconstruction of real signed networks using G ( m ) by asuitable choice of seed graph that can inherit properties of real networks.Obviously the structural properties of the seed graph G influence the structural andspectral properties of G ( m ) . For example, if the edges in G are of the type mentioned in( i ) , ( ii ) , ( iii ) of Theorem 2.2 and all triads in G are unbalanced then all the triads in G ( m ) will be unbalanced. Thus in order to have balanced triads in G ( m ) , G must have edgeswhich does not fall in the category of ( i ) , ( ii ) , ( iii ) of Theorem 2.2 or at least one balancedtriad in G itself. Below we do a thorough analysis of structural and spectral propertiesof corona graphs generated by following the canonical marking scheme. We mention thatthe analytical expression of different properties of corona graphs corresponding to pluralitymarking is complicated and cumbersome. G ( m ) First we have the following observations about G ( m ) , m ≥ G on n nodes with k edges.1. Total number of nodes in G ( m ) is n ( n + 1) m [32].2. The total number of copies of the seed graph in G ( m ) is ( n + 1) m . This follows fromthat fact that at each step 1 ≤ i ≤ m the number of seed graphs added during theformation of G ( i ) is the number of nodes in G ( i − . Thus the total number of copies ofthe seed graph is1 + n + n ( n + 1) + . . . + n ( n + 1) m − = 1 + n m − X j =0 ( n + 1) j = ( n + 1) m .
3. The total number of edges added in the formation of G ( i +1) from G ( i ) , ≤ i ≤ m − G ( i ) and edgesin the newly appeared copies of seed graphs is given by n ( n + 1) i . This follows fromthat fact that for each node v in G ( i ) , n new edges will be created, and the number ofnodes in G ( i ) is n ( n + 1) i .
4. The total number of edges in G ( m ) is k ( n +1) m + P m − j =0 n ( n +1) j = ( k + n )( n +1) m − n. We denote n ( i )+ and n ( i ) − the number of positive and negative nodes in G ( i ) , ≤ i ≤ m − G ( i ) = n ( n + 1) i (= n ( i )0 + n ( i ) − = n ( i ) , say).We denote the vertex set which consists of the marked nodes in G ( i ) as V ( i ) , ≤ i ≤ m . Letthe number of positive and negative links in G (0) be denoted as e (0)+ and e (0) − respectively.Consider counting the number of signed edges in G ( m ) . Note that the number of copies ofthe seed graph added during the formation of G ( i +1) from G ( i ) is n ( n + 1) i (which is exactlythe number of nodes in G ( i ) ), 0 ≤ i ≤ m − . Then in G ( i +1) the number of new positive andnegative links are created as e ( i +1)+ = e (0)+ n ( n + 1) i + n ( i )+ n (0)+ + n ( i ) − n (0) − (7) e ( i +1) − = e (0) − n ( n + 1) i + n ( i )+ n (0) − + n ( i ) − n (0)+ (8)respectively. The first term in the expression of e ( i +1)+ counts the number of positive linkswhich are part of the newly added copies of seed graph, the second term is the number of linkswhich are formed by joining positively marked nodes in V ( i ) and positively marked nodes inthe copies of the seed graph, and the third term is the number of positive edges formed by xisting negatively marked nodes in G ( i ) and the newly appeared negatively marked nodes.Similarly, the expression of e ( i +1) − follows.Now let us count the number of positively and negatively marked nodes in G ( i +1) afterits formation from G ( i ) . There can be two types of nodes in G ( i +1) , the newly appeared nodes v ∈ V ( i +1) \ V ( i ) and the existing set of nodes V ( i ) of G ( i ) . First, consider the nodes in V ( i +1) \ V ( i ) . The number of positively and negatively marked nodes among such nodes shallbe n ( i )+ n and n ( i ) − n respectively. This follows from the fact that when a new node (positively/negatively marked) gets linked to an existing positively (resp. negatively) marked node in V ( i ) then it becomes positively (resp. negatively) marked due to the definition of canonicalmarking. Now let us consider the marking of nodes in V ( i +1) ∩ V ( i ) . Note that any node v ∈ V ( i +1) ∩ V ( i ) was already marked during the formation of G ( i ) . When G ( i +1) is formed,each such node forms links to all the nodes in a copy of a newly appeared seed graph. If itwas µ ( v ) = + then it remains the same marking when n (0) − is even and n (0)+ is even or odd,whereas it changes its marking only when n (0) − is odd and n (0)+ is either even or odd. Similarly,if it was µ ( v ) = − then it remains the same marking for even value of n (0)+ and any value of n (0) − , whereas it changes its marking when n (0)+ is odd and any value of n (0) − . Thus finally wehave the following. The number of total positively and negatively marked nodes in G ( i +1) isdescribed in Table 3. Marking Condition on G in V ( i +1) \ V ( i ) in V ( i +1) ∩ V ( i ) in G ( i +1) + n (0)+ odd n ( i )+ n n ( i ) − n ( i ) − + n n ( i )+ n (0) − odd n (0)+ even 0 n ( i )+ nn (0) − odd n (0)+ even n ( i )+ n ( i )+ (1 + n ) n (0) − even n (0)+ odd n ( i ) n ( i ) + n ( i )+ nn (0) − even − n (0)+ odd n ( i ) − n n ( i )+ n ( i )+ + n n ( i ) − n (0) − odd n (0)+ even n ( i ) n ( i ) − n + n ( i ) n (0) − odd n (0)+ even n ( i ) − n ( i ) − (1 + n ) n (0) − even n (0)+ odd 0 n ( i ) − nn (0) − evenTable 3: The relation between the number of marked nodes in G ( i +1) and G ( i ) . Thelast column displays the values of n ( i +1)+ and n ( i +1) − . Now we are in a position to estimate the number of signed links in G ( m ) by utilizing quations (7), (8) and the fact that n ( i )+ = n ( i ) − n ( i ) − .e ( m )+ = e (0)+ + m − X i =0 e ( i +1)+ = e (0)+ n m − X i =0 ( n + 1) i ! + n (0)+ m − X i =0 n ( i )+ + n (0) − m − X i =0 n ( i ) − = e (0)+ ( n + 1) m + n (0)+ [( n + 1) m −
1] + ( n (0) − − n (0)+ ) m − X i =0 n ( i ) − where the values of n ( i ) − can be computed from the recurrence relation given in Table 3 for all i. Similarly, e ( m ) − = e (0) − ( n + 1) m + n (0) − [( n + 1) m −
1] + ( n (0)+ − n (0) − ) m − X i =0 n ( i ) − . It is evident from the above expressions that marking of nodes in the seed graph playsthe key role. Besides, the number of positive and negative edges become equal when thenumber of positively and negatively marked nodes are same in the seed graph. Now we focuson couting the number of triads of type T i , i = 0 , , , G ( m ) . First observe that therecan be six types of links in a signed graph. A positive/negative link can be incidental topositively marked nodes, oppositely marked nodes or negatively marked nodes. When suchnodes in a triad in a copy of the seed graph get linked during the formation of each step ofthe corona graph G ( i +1) from G ( i ) , differnt types of triads are created. The Figure 4 depictsall the possible cases. ( a, +) ( b, +)( v, +) (a) T ( a, +) ( b, − )( v, +) (b) T ( a, − ) ( b, − )( v, +) (c) T ( a, +) ( b, +)( v, +) (d) T ( a, +) ( b, − )( v, +) (e) T ( a, − ) ( b, − )( v, +) (f) T ( a, +) ( b, +)( v, − ) (g) T ( a, +) ( b, − )( v, − ) (h) T ( a, − ) ( b, − )( v, − ) (i) T ( a, +) ( b, +)( v, − ) (j) T ( a, +) ( b, − )( v, − ) (k) T ( a, − ) ( b, − )( v, − ) (l) T Figure 4: Formation of signed triads in every step of the corona product. Each signededge consisiting nodes ( a, p ) , ( b, q ) joins an exiting node ( v, r ) where p, q, r ∈ { + , −} denote the marking of the nodes. 13 e denote | E (0) s | p q as the number of links in G (0) having sign s that is adjacent to nodeswith markings p and q. For example, | E (0) − | + − denotes the number of negative links in G (0) adjacent to oppositely marked nodes. Then the Table 2 gives the number of triads of T j , j =0 , , , G ( i +1) from G ( i ) due to the new signed edges createdbetween the nodes in V ( i ) and V ( i +1) \ V ( i ) , ≤ i ≤ m − . We denote this number as T ( i +1) j . j T ( i +1) j j = 0 n ( i )+ | E (0)+ | + + + n ( i ) − | E (0)+ | − − j = 1 n ( i )+ (cid:18) | E (0)+ | + − + | E (0) − | + + (cid:19) + n ( i ) − (cid:18) | E (0)+ | + − + | E (0) − | − − (cid:19) j = 2 n ( i )+ (cid:18) | E (0)+ | − − + | E (0) − | + − (cid:19) + n ( i ) − (cid:18) | E (0)+ | + + + | E (0) − | + − (cid:19) j = 3 n ( i )+ | E (0) − | − − + n ( i ) − | E (0) − | + + Table 4: The number of T j type triads between the nodes in V ( i ) and V ( i +1) \ V ( i ) created during the formation of G i +1 from G ( i ) . Note that if any type of triad is present in a seed graph G = G (0) , it will appear everytime a copy of the seed graph gets attached to the formation of G ( m ) . Let T (0) j denote thenumber of triads of type T j that exist in G (0) . Then the total number of T j j = 0 , , , j ( G ( m ) ) in G ( m ) can be calculated as follows.T ( G ( m ) ) = T (0)0 n m − X i =0 ( n + 1) i ! + m − X i =0 T ( i +1)0 = T (0)0 ( n + 1) m + | E (0)+ | + + m − X i =0 n ( i )+ + | E (0)+ | − − m − X i =0 n ( i ) − = T (0)0 ( n + 1) m + | E (0)+ | + + [( n + 1) m −
1] + (cid:18) | E (0)+ | − − − | E (0)+ | + + (cid:19) m − X i =0 n ( i ) − where the first term includes the number of T type triads which are in the copies of the seedgraph in G ( m ) , and the remaining terms are due to the new triads which are born during theformation of the corona product in each step of the formation of G ( m ) . The values of n ( i ) − canbe found from Table 3 for each i. Similarly we have the following.T ( G ( m ) ) = T (0)1 ( n + 1) m + (cid:18) | E (0)+ | + − + | E (0) − | + + (cid:19) [( n + 1) m −
1] + (cid:18) | E (0) − | − − − | E (0) − | + + (cid:19) m − X i =0 n ( i ) − T ( G ( m ) ) = T (0)2 ( n + 1) m + (cid:18) | E (0)+ | − − + | E (0) − | + − (cid:19) [( n + 1) m −
1] + (cid:18) | E (0)+ | + + − | E (0)+ | − − (cid:19) m − X i =0 n ( i ) − T ( G ( m ) ) = T (0)3 ( n + 1) m + | E (0) − | − − [( n + 1) m −
1] + (cid:18) | E (0) − | + + − | E (0) − | − − (cid:19) m − X i =0 n ( i ) − where the values of n ( i ) − can be computed by using the recurrence relation given in Table 3.Note from the above formulas that the number of T j type triads in G ( m ) can be controlledby choosing the seed graph with an appropriate distribution of signed edges whose adjacentnodes are marked in a particular fashion. .2 Degrees of nodes of G ( m ) Note that after the appearance of a node in a particular step during the formation of G ( m ) ,from next step onward the total degree of the node increases by n at every step until theprocess stops. The dynamics of the change in positive and negative degrees depend on theexisting marking of the node and the marking of the nodes which get linked to it at everystep. Since the new nodes which get linked to an existing node are the nodes in the copyof the seed graph, the change in degrees of the existing node depends on the marking of thenodes in the seed graph. We first derive the positive and negative degrees of the nodes in G = G (0) , and then we consider nodes in G ( i ) that is which appear at the i th step during theformation of G ( m ) , ≤ i ≤ m. The initial marking of a node v is denoted by µ ( v ) . By initialmarking we mean once the node v joins the existing graph.We have the following theorem which provides positive and negative degree of a node inthe seed graph G (0) after the formation of G ( m ) . Theorem 3.1.
Let G ( m ) , m ≥ be the signed corona graph corresponding to a seed graph G = G (0) on n > nodes defined by the canonical marking µ. Let the positive and negativedegree of v in G (0) be denoted by d +0 ( v ) and d − ( v ) respectively. Suppose G (0) has n + positivelymarked nodes and n − negatively marked nodes. Then for any v ∈ V (0) the positive degree d + ( v ) and negative degree d − ( v ) of v in G ( m ) are given by the Table 5. Condition µ ( v ) d + ( v ) d − ( v ) Conditionon G on mn + odd, + d +0 ( v ) + m ( n + + n − ) d − ( v ) + m ( n + + n − ) even n − odd d +0 ( v ) + m +12 n + + m − n − d − ( v ) + m +12 n − + m − n + odd − d +0 ( v ) + m ( n + + n − ) d − ( v ) + m ( n + + n − ) even d +0 ( v ) + m +12 n − + m − n + d − ( v ) + m +12 n + + m − n − odd n + even, + d +0 ( v ) + n + + ( m − n − d − ( v ) + n − + ( m − n + even/odd n − odd − d +0 ( v ) + mn − d − ( v ) + mn + even/odd n + even, + d +0 ( v ) + mn + d − ( v ) + mn − even/odd n − even − d +0 ( v ) + mn − d − ( v ) + mn + even/odd n + odd, + d +0 ( v ) + mn + d − ( v ) + mn − even/odd n − even − d +0 ( v ) + n − + ( m − n + d − ( v ) + n + + ( m − n − even/oddTable 5: Positive and negative degree of nodes of the seed graph G in G ( m ) Proof:
Let n + and n − be odd. If a node v in G is positively marked then at every step ofthe formation of G ( m ) the marking of the node will change since at every step it will generateodd number of negative links to the newly appeared nodes in a copy of the seed graph. If µ ( v ) = + (resp. µ ( v ) = − ) at any step then at the next step it will generate n − (resp. n + ) egative links. If m is even then at m/ n − and n + nodes. Thus the desired result follows. A similar argument can prove the case for m beingodd. If µ ( v ) = − it’s marking shall change at every step and hence the desired result follows.Let n + be even and n − odd. If µ ( v ) = + then next step the v will generate n − negativelinks and since n − is odd, its marking become − . From next step onward it will generate evennumber n + of negative links at every step as n + is even. Hence µ ( v ) remains − for all thesteps till m. Thus the desired follows. If µ ( v ) = − , it remains negatively marked for all thesteps since it generates even number of negative links. This completes the proof.Let n + and n − both be even. If µ ( v ) = +, it remains positively marked for all thesteps since every time it generates n − number of negative links and n − is even. Similarly, if µ ( v ) = − , it remains negatively marked as every times it creates even number of new negativelinks. Hence the proof.Finally let n + be odd and n − even. If µ ( v ) = + it remains + since at every step it createseven number n − of negative links. If µ ( v ) = − then at the next step the marking becomes +since it creates n + number of negative links and n + is odd. Next step onward it creates evennumber n − of negative links and hence the marking of v remains + for ever until m th step.Hence the desired result follows. (cid:3) Now we consider the positive and negative degrees of a node v which appears at the i thstep, that is, during the formation of G ( i ) , 1 ≤ i ≤ m and gets linked with an existing node u in G ( i − . Obviously u is a node in a copy of the seed graph and it has a marking µ ( v ) asa node in the seed graph but as soon as it joins the node u in G ( i − , the marking of v maychange depending on the marking of the node u. Let us denote the degree of v as d ± ( v ) whenwe consider it as a node of the seed graph and the positive or negative degree increases by1 when it joins u in G ( i − depending on the marking µ ( u ) . The following theorem providesthe positive and negative degree of such a node v . Theorem 3.2.
Let v be a node which joins at the i th step, ≤ i ≤ m of the formation of thecorona graph G ( m ) generated by a seed graph G = G (0) which contains n + positively markedand n − negatively marked nodes. Let v gets attached with a node u in G ( i − during its birthin G ( i ) . Then the positive degree d + ( v ) and negative degree of d − ( v ) in G ( m ) are given by theTable 6. We denote the marking of v as µ ( v ) when we consider it as a node in the copy ofa seed graph which joins the graph G ( i − , whereas µ ( u ) denotes the marking of u as a nodein G ( i − . ondition ( µ ( u ) , µ ( v )) d + ( v ) d − ( v ) m i on Gn + odd, (+ , +) d +0 ( v ) + 1 + m − i n d − ( v ) + m − i n odd odd n − odd even even d +0 ( v ) + 1 + m − i +12 n + + m − i − n − d − ( v ) + m − i +12 n − + m − i − n + odd eveneven odd(+ , − ) d +0 ( v ) + m − i n d − ( v ) + 1 + m − i n odd oddeven even d +0 ( v ) + m − i +12 n + + m − i − n − d − ( v ) + 1 + m − i +12 n − + m − i − n + odd eveneven odd( − , +) d +0 ( v ) + m − i n d − ( v ) + 1 + m − i n odd oddeven even d +0 ( v ) + m − i +12 n − + m − i − n + d − ( v ) + 1 + m − i +12 n + + m − i − n − odd eveneven odd( − , − ) d +0 ( v ) + 1 + m − i n d − ( v ) + m − i n odd oddeven even d +0 ( v ) + 1 + m − i +12 n − + m − i − n + d − ( v ) + m − i +12 n + + m − i − n − odd eveneven odd n + even, (+ , +) d +0 ( v ) + 1 + n + + ( m − i − n − if m > i d − ( v ) + n − + ( m − i − n + if m > i any any d +0 ( v ) + 1 if m = i d − ( v ) if m = in − odd (+ , − ) d +0 ( v ) + n + + ( m − i − n − if m > i d − ( v ) + 1 + n − + ( m − i − n + if m > i any any d +0 ( v ) if m = i d − ( v ) + 1 if m = i ( − , +) d +0 ( v ) + ( m − i ) n − d − ( v ) + 1 + ( m − i ) n + any any( − , − ) d +0 ( v ) + 1 + ( m − i ) n − d − ( v ) + ( m − i ) n + any any n + even, (+ , +) d +0 ( v ) + 1 + ( m − i ) n + d − ( v ) + ( m − i ) n − any any n − even (+ , − ) d +0 ( v ) + ( m − i ) n + d − ( v ) + 1 + ( m − i ) n − any any( − , +) d +0 ( v ) + ( m − i ) n − d − ( v ) + 1 + ( m − i ) n + any any( − , − ) d +0 ( v ) + 1 + ( m − i ) n − d − ( v ) + ( m − i ) n + any any n + odd, (+ , +) d +0 ( v ) + 1 + ( m − i ) n + d − ( v ) + ( m − i ) n − any any n − even (+ , − ) d +0 ( v ) + ( m − i ) n + d − ( v ) + 1 + ( m − i ) n − any any( − , +) d +0 ( v ) + n − + ( m − i − n + if m > i d − ( v ) + 1 + n + + ( m − i − n − if m > i any any d +0 ( v ) if m = i d − ( v ) + 1 if m = i ( − , − ) d +0 ( v ) + 1 + n − + ( m − i − n + if m > i d − ( v ) + n + + ( m − i − n − if m > i any any d +0 ( v ) + 1 if m = i d − ( v ) if m = i Table 6: Positive and negative degree of a node which appears at the i th step duringthe formation of G ( m ) , ≤ i ≤ m Proof:
The proof follows by observing the change of marking of a node v due to the appear-ance of n new nodes which connect to v in every step after it appears in the i th step of theformation of G ( m ) . Let v be a node which appears in G ( i ) , 1 ≤ i ≤ m. Recall that a nodeis positively marked if and only if it is attached to even number of negatively signed links,otherwise the marking of a node is negative.First assume that n − and n + are odd. Let µ ( u ) = + and µ ( v ) = + . Note that sign ofthe edge between u and v is µ ( u ) × µ ( v ) = + . Hence the marking of v in G ( i ) is + as thenumber of negative edges adjacent to it does not increase by 1 and µ ( v ) = + implies it wasattached to even number of negative edges. At the ( i + 1)th step v forms n − negative edgesand n − is odd. Hence µ ( v ) = − in G ( i +1) . Proceeding similarly, the sequence of marking of v will be + , − , + , − , . . . . Thus the positive degree of v increases as follows. At G ( i ) the degreeof v becomes d +0 ( v ) + 1 . At ( i + 1)th step the positive degree of v becomes d +0 ( v ) + 1 + n + , and the negative degree becomes d − ( v ) + n − . Since marking of v is − at ( i + 1)th step, itwill create n − positive edges and n + negative edges at the next step. Thus the positive andnegative degree will be ( d +0 ( v ) + 1) + n + + n − + n + + . . . and the negative degree will be d − ( v ) + n − + n + + n − + . . . , and both the series stop after m − i steps. The desired resultfollows by considering different cases of m and i being even and/ odd.If µ ( u ) = + and µ ( v ) = − the pattern of change of markings of v starting from the i th step is + , − , + , − , . . . . The increase of positive and negative degree are given by d +0 ( v ) + n + + n − + n + + n − + . . . and ( d − + 1) + n − + n + + n − + n + + . . . . Thus the result followsby considering different cases for m and i. imilarly it can be easily verified that the positive and negative degree of v will be d +0 ( v )+ n − + n + + n − + . . . and ( d − ( v ) + 1) + n + + n − + n + + . . . respectively when µ ( u ) = − and µ ( v ) = + . Finally if µ ( u ) = − and µ ( v ) = − then the positive and negative degree of v willbe ( d +0 ( v ) + 1) + n − + n + + n − + . . . and d − + n + + n − + n + + . . . respectively. Thus thedesired result follows.Following a similar procedure for different values of n − and n + all the desired result canbe verified. This completes the proof. (cid:3) It is evident from the expressions of the positive and negative degree from Table 5 andTable 6 that as i increases the degree decreases. This means a degree of node is higher thanthe degree of another node if it joins the network before than the another one. Finally thedegrees of the nodes in G (0) are the highest ones.For the computation of degree distribution of corona graphs the frequency or total numberof occurrence of a node with a particular degree need to be determined. Thus the questionis how many nodes do have a particular degree given in the tables above? It is obvious fromthe construction of the corona graph that the number of nodes which appear at step i witha given degree d ± ( v ) with ( µ ( u ) , µ ( v )) = (+ , +) is n ( i )+ n + ; ( µ ( u ) , µ ( v )) = (+ , − ) is n ( i )+ n − ;( µ ( u ) , µ ( v )) = ( − , +) is n ( i ) − n + ; and ( µ ( u ) , µ ( v )) = ( − , − ) is n ( i ) − n − , for 1 ≤ i ≤ m. Belowwe compute positive and negative degree distributions of corona graphs G ( i ) , ≤ i ≤ G , G .(a) G (b) G − P ( n ( m ) − ≥ d − ) m=0m=1m=2m=3m=4m=5 (c) Negative degree distribution of G ( m )1 − P ( n ( m ) − ≥ d − ) m=0m=1m=2m=3m=4m=5 (d) Negative degree distribution of G ( m )2 + P ( n ( m ) + ≥ d + ) m=0m=1m=2m=3m=4m=5 (e) Positive degree distribution of G ( m )1 + P ( n ( m ) + ≥ d + ) m=0m=1m=2m=3m=4m=5 (f) Positive degree distribution of G ( m )2 Figure 5: Degree distributions of G ( m ) i , i = 1 , .3 Algebraic conflict of corona graphs Recall that algebraic conflict of a signed graph is the least signed Laplacian eigenvalue of thegraph [2, 23]. Theorem 2.5 gives the complete list of signed Laplacian eigenvalues only whendistinct nodes of the corresponding graph have distinct negative degrees. However for a largegraph the computation of these eigenvalues by calculating roots of high degree polynomials iscomputationally challenging. Due to the exponential growth of corona graphs, withing a fewsteps the corona graphs contain huge number of nodes even if the seed graph has considerablysmall number of nodes. Thus we consider Theorem 2.6 which gives the collection of all signedLaplacian eigenvalues of the signed graph G ◦ G when marking of all the nodes of G areeither + or − , and negative degree of all the nodes in G are same. Hence setting a seedgraph G = G (0) having these properties we provide expression of signed Laplacian eigenvaluesof G ( m ) as follows by iterative use of Theorem 2.6.First we have the following lemma which will be used in sequel. Lemma 3.3.
Let n, m be positive integers. Then m n + m − X i =1 i n ( n − n + 1) m − i − + n ( n − n + 1) m − = n ( n + 1) m . Proof:
The proof follows from Lemma 1 in [32]. (cid:3)
Consider the function f : R → R defined by f ( x ) = 2 d − + 1 + x + n ± p [(2 d − + 1) − ( x + n )] + 4 n d − is a nonnegative integer and n is a positive integer. Denote f i as i number ofcompositions of f with itself. For example, f = f and f = f ◦ f. Then we have thefollowing theorem.
Theorem 3.4.
Let G be a signed graph on n nodes such that d − is the fixed negative degree ofeach node of G and every node of G is either positively or negatively marked. Suppose η j , j =1 , . . . , n are the signed Laplacian eigenvalues of G. Let η n = 2 d − . Then signed Laplacianeigenvalues of G ( m ) , m ≥ defined by the seed graph G are given by(a) f m ( η j ) with multiplicity , j = 1 , . . . , n (b) f i ( η j + 1) with multiplicity n ( n + 1) m − i − , ≤ i ≤ m − , j = 1 , . . . , n − , and m > (c) η j + 1 with multiplicity n ( n + 1) m − , j = 1 , . . . , n − . Proof:
First note that 2 d − is a signed Laplacian eigenvalue of G due to Lemma 2.4. Theexpression of signed laplacian eigenvalues of G ( m ) follows from iterative use of Theorem 2.6.Using Lemma 3.3 it can be easily verified that this is the complete list of signed Laplacianeigenvalues of G ( m ) . (cid:3) The algebraic conflict of corona graphs defined by a seed graph satisfying the propertiesof Theorem 3.4 can be calculated by taking the minimum of the signed Laplacian eigenvaluesof G ( m ) as given in Theorem 3.4. However it is interesting to verify that 2 d − is a fixed pointof the function f ( x ) = 2 d − + 1 + x + n − p [(2 d − + 1) − ( x + n )] + 4 n n. This implies that if 2 d − is the algebraic conflict of a seed graph G with every nodehaving negative degree d − , d − remains the algebraic conflict of G ( m ) for any m ≥ . Anexample of a seed graph having algebraic conflict 2 d − is given by G with algebraic conflict 2 where n = 4 and d − = 1 . Otherwise, for seed graphs G (0) if 2 d − is not a signed Laplacian eigenvalue , we conjecturethat η min + 1 ≤ f i ( η j + 1) , ≤ i ≤ m − , ≤ j ≤ n − η min + 1 ≤ f m ( η j ) , ≤ j ≤ n, andhence algebraic conflict of G ( m ) is algebraic conflict of G (0) plus one. We verify the same forall possible signed graphs on 3 , , G ( m ) is notaffected by values of m and large signed corona graphs can have small algebraic conflict whichis at the order of the algebraic conflict of the seed graph. We consider a few seed graphs on3 , , G ( m ) , ≤ m ≤ G (b) G (c) G (d) G Figure 7: Sample seed graphs on 3 , , λ ( G ( m ) ) G1G2G3G4
Figure 8: Algebraic conflicts of corona graphs generated by seed graphs listed inFigure 7. λ ( G ( m ) ) denotes algebraic conflict. As mentioned in the introduction, we now explore how signed corona graphs can gain usto preserve properties of real signed networks. We consider all the data sets of real signednetworks, see Table 7, that are reported in [25] and [9], and available in SNAP projectwebpage [26]. One remarkable characteristic about these real networks is that each dataset contains very high percentage of positive links than negative links, and balance triadsdominates in number in all these networks. In particular, the number of T is almost 70 − etwork Nodes Edges p ( E + ) Triad p ( T )Epinions 119,217 814,200 0.85 T T T T T T T T T T T T T T T T T T T T p ( E + ) denotes the proportion of positive edges in the entire network. p ( T ) isthe fraction of triads of type T in the entire network. It can be observed from the the expression of number of each type of triads and signededges in G ( m ) obtained in Section 3.1 that the distribution of T j s and signed edges in G ( m ) are highly influenced by number of T j , signed links, number of nodes and marking of nodesin the seed graph. Besides, the number of nodes in G ( m ) only can be of the form n ( n + 1) m where n denotes the number of nodes in the seed graph. Hence it is evident that G ( m ) cannot have any desired number of nodes in it, but given a total number of nodes in a syntheticnetwork to be produced, one can determine suitable choices of n, m such that the number ofnodes in G ( m ) approximately matches with the desired number of nodes. Then the goal willbe to choose a seed graph on n number of nodes that can approximately catch up the desirednumber of signed edges and triads T j in G ( m ) . An another guide from Theorem 2.2 is thatthe marking of nodes which are adjacent to a positive/negative link in the seed graph mustfollow some conditions in order to produce all types of signed triads in the network.Structural properties of corona graphs corresponding to some seed graphs on 4 or 5 nodesfor which all type of triads exist, are described in Table 8. It follows that these coronagraphs fail to preserve properties of real networks. Evidently, a corona graph G ( m ) for some m ≥ ≫ T j , j = 1 , , e + ≫ e − . Thusseed graph must contain high number of positive edges and T , and negative edges in the seed raph must be connected in such a way that the generation of T j , j = 1 , , m G (0) N E p ( E + ) Triad p ( T l )6 62,500 1,24,996 0.625 T T T T T T T T T T T T T T T T T T T T G ( m ) definedby seed graph G (0) . N and E denote the total number of nodes and links in G ( m ) .p ( E + ) and p ( T j ) denote the fraction of positive links and triads of type T j in G ( m ) . Recall that the total number of triads T j of type T j , j = 0 , , , G ( m ) generated by aseed graph G = G (0) is given byT = T (0)0 k + | E (0)+ | + + k + (cid:18) | E (0)+ | − − − | E (0)+ | + + (cid:19) k T = T (0)1 k + (cid:18) | E (0)+ | + − + | E (0) − | + + (cid:19) k + (cid:18) | E (0) − | − − − | E (0) − | + + (cid:19) k T = T (0)2 k + (cid:18) | E (0)+ | − − + | E (0) − | + − (cid:19) k + (cid:18) | E (0)+ | + + − | E (0)+ | − − (cid:19) k T = T (0)3 k + | E (0) − | − − k + (cid:18) | E (0) − | + + − | E (0) − | − − (cid:19) k , and the total number of positive and negative edges in G ( m ) are given by e + = e (0)+ k + n (0)+ k + ( n (0) − − n (0)+ ) k e − = e (0) − k + n (0) − k + ( n (0)+ − n (0) − ) k respectively, where k = ( n + 1) m , k = ( n + 1) m − , k = P m − i =0 n ( i ) − , the values of n ( i ) − canbe found from Table 3. T (0) j denotes the number of triads of type T j in G. n (0) − and n (0)+ enote the number of negatively and positively marked nodes in G , and e (0) − and e (0)+ denotethe number of negative and positive edges in G respectively. However there are six types ofedges in the seed graph based on the marking of the adjacent nodes. In order to obtain acompact expression of T j s in the corona graph we set some of the values of the parametersto be zero that is we exclude certain types of edges in G. This helps to set values of the otherparameters so that finally the desired distribution of T j can be obtained. Graph m G (0)
N E p ( E + ) Triad p ( T l )1 3 9,000 25,991 0.734 T T T T T T T T T T T T G ( m ) which preserves characteristics of signed links and triadsof real networks. N and E denote the total number of nodes and links in G ( m ) .p ( E + ) and p ( T j ) denote the fraction of positive links and triads of type T j in G ( m ) respectively. Setting | E (0)+ | + − = | E (0)+ | − − = | E (0) − | − − = 0 and T (0)3 = 0 , we obtainT = T (0)0 k + | E (0)+ | + + ( k − k )T = T (0)1 k + | E (0) − | + + ( k − k )T = T (0)2 k + | E (0) − | + − k + | E (0)+ | + + k T = | E (0) − | + + k . The reason behind choosing T (0)3 = 0 is that we want the number of T to be minimum amongall triads. Besides, note that k ≫ k . Finally considering e (0)+ ≫ e (0) − , T (0)0 ≫ T (0) j , j = 1 , G ( m ) defined by the seed graphs given in Table 9 areplotted in Figure 9. However do not compare it with real networks since the number of nodesand links in G ( m ) do not match the same for real networks. − P ( n ( m ) − > d − ) m=0m=1m=2m=3 (a) Negative degree distribution of G m gener-ated by Graph 1 + P ( n ( m ) + > d + ) m=0m=1m=2m=3 (b) Positive degree distribution of G m gener-ated by Graph 1 − P ( n ( m ) − > d − ) m=0m=1m=2m=3 (c) Negative degree distribution of G m gener-ated by Graph 2 + P ( n ( m ) + > d + ) m=0m=1m=2m=3 (d) Positive degree distribution of G m gener-ated by Graph 2 − P ( n ( m ) − > d − ) m=0m=1m=2m=3m=4 (e) Negative degree distribution of G m gener-ated by Graph 3 + P ( n ( m ) + > d + ) m=0m=1m=2m=3m=4 (f) Positive degree distribution of G m gener-ated by Graph 3 Figure 9: (a) and (c) ((b) and (d)) shows the fraction of nodes in G ( m )1 and G ( m )2 having degree greater than or equal to d − ( d + ), respectively. We defined corona product of two signed graphs and study their structural and spectralproperties. A signed network generative model is proposed based on corona product bytaking the corona product of a small graph with itself iteratively. This small graph is calledseed graph for the resulting signed graphs, which are called corona graphs. We derivedfundamental properties of corona graphs that include number of signed links, signed triads,degree distribution and algebraic conflict of the corona graphs. Finally we show that a seedgraph can be chosen for which the corresponding corona graphs can preserve properties ofreal signed networks.We believe that the framework of corona product introduced in this paper can be extendedto define corona product of gain graphs and weighted graphs. Besides, generalized corona roduct of signed graphs can be defined alike generalized corona product for unsigned graphs[24]. References [1]
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