- Email: [email protected]
Transitivity Model on Signed Graphs
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Electronic Notes in Discrete Mathematics 63 (2017) 455–460 www.elsevier.com/locate/endm
Transitivity Model on Signed Graphs Deepa Sinha
1
Department of Mathematics South Asian University, Akbar Bhawan Chanakyapuri, New Delhi-110021, India.
Deepakshi Sharma
2
Department of Mathematics South Asian University, Akbar Bhawan Chanakyapuri, New Delhi-110021, India.
Abstract In this paper, we generalize the already established iterated local transitivity model for online social networks in signed networks. In this model, at each time step t and for already existing vertex x, a new vertex(clone) x is added which joins to the neighbors of x. The sign of new edge xx is the marking on x. We also discuss the properties such as balance, clusterability, sign-compatibility and consistency. The signed networks focus on the type of relations (friendship and enmity) between the vertices(members of online social network). The ILT model for signed network gives an insight on how the network reacts to the addition of clone vertex. Also the properties like balance and clusterability helps establish a natural balance in society by providing a possible formation of group of vertices in society for a peaceful co-existence and smooth functioning of social system. Keywords: social network, signed social network, local transtivity model, marked singed graph, neighborhood, balance, sign-compatibility, clusterability, algorithm.
https://doi.org/10.1016/j.endm.2017.11.043 1571-0653/© 2017 Elsevier B.V. All rights reserved.
456
1
D. Sinha, D. Sharma / Electronic Notes in Discrete Mathematics 63 (2017) 455–460
Introduction
In this paper, we extend the work on the theoretic aspect of friendship and enmity for online social networks which grow very fast. We give sign to edges where a positive sign indicates a friendly relation and a negative sign a hostile relation. Further, a positive vertex indicates that a person befriends most of his/her neighbors and a negative vertex states a has hostile relations with their acquaintances. A signed graph [4] is an ordered pair Σ = (Σu , σ), where Σu is a graph G = (V, E), called the underlying graph of Σ and σ : E → {+, −} is a function from the edge set E of Σu into the set {+, −}, called the signature (or sign in short) of Σ. A signed graph is all-positive (respectively, all-negative) if all its edges are positive (negative); further, it is said to be homogeneous if it is either allpositive or all-negative and heterogeneous otherwise. The positive (negative) degree of a vertex v ∈ Σ denoted by deg + (v)(deg − (v)) is the number of positive (negative) edges incident on the vertex v and deg(v) = deg + (v) + deg − (v). Let v be an arbitrary vertex of a graph G. We denote the set consisting of all the vertices of G adjacent to v by N (v). This set is called the open neighborhood set of v and sometimes we call it as open neighborhood of v. The set consisting of all the vertices of G adjacent to v along with v itself is called the closed neighborhood set or closed neighborhood of v. It is denoted by N [v]. A marked signed graph is an ordered pair Σμ = (Σ, μ) where Σ = (Σu , σ) is a signed graph and μ : V (Σu ) → {+, −} is a function from the vertex set V (Σu ) of Σu into the set {+, −}, called a marking of Σ. Henceforth the vertex receiving ‘+’ mark will be called positive vertex and the vertex receiving ‘-’ mark is called negative vertex. In this model the marking on a vertex denotes its nature of bond with its neighbors. A positive vertex indicates that a person befriends most of its neighbors and a negative vertex states that it has hostile relations with its acquaintances. A new clone vertex added makes his relations according to these markings of already existing vertices. A cycle in a signed graph S is said to be positive if the product of the signs of its edges is positive or, equivalently, if the number of negative edges in it is even. A cycle which is not positive is said to be negative A signed graph is balanced if all its cycles are positive. The partition criterion to characterize the balance property of a signed graph is given by Harary [2]. The balance in ILT model for signed networks partitions our network into two groups so 1 2
Email: deepa [email protected] Email: [email protected]
D. Sinha, D. Sharma / Electronic Notes in Discrete Mathematics 63 (2017) 455–460
457
that the relations of friendship are in the same group and that of enmity goes across between the two partitions. The balance in online social network is required for the smooth functioning of the society. For terminology and notation in graph theory we refer the reader to [3] and for signed graph we refer the reader to [4].
2
The ILT model for signed networks
In the Iterated Local Transitivity (ILT) model [1], at each time step and for every existing vertex, a new vertex appears which joins to the closed neighbor set of vertices. Here we present the model which generates a simple, undirected signed graph Σt : t ≥ 0. At each time step t, the vertices are marked positive or negative in Σt . We begin with a connected signed graph Σ0 which is fixed. Let Σt be a signed graph generated at the time t such that it is an induced subsigned graph of signed graph Σt+1 at the time t + 1. The vertex v at the time t is denoted by vt and at the t + 1 as vt+1 . Σt+1 is constructed as follows: For each vertex x of Σt , its clone x is added and is connected to all vertices in Nt [x]. The sign of edge σt+1 (x y) = μt (y) for each y ∈ Nt [x], where the marking μ : V (Σ) → {+, −} is such that + if degt+ (xt ) ≥ degt (xt )/2 μ(xt ) = (1) − if degt− (xt ) > degt (xt )/2. Proposition 2.1 If x is a positive vertex, x be its clone then at time step t + 1: + degt+1 (x) = 2degt+ (x) + 1 + (x ) = |Nt+ (x)| + 1 degt+1 − (x) = 2degt− (x) degt+1 − (x ) = |Nt− (x)|. degt+1 If x is negative then: − (x) = 2degt− (x) + 1 degt+1 − (x ) = |Nt− (x)| + 1 degt+1 + (x) = 2degt+ (x) degt+1 + (x ) = |Nt+ (x)|. degt+1 Here degt+ (xt )(degt− (xt )) is the positive(negative) degree of each vertex vt at time step t and Nt+ (vt ) = {vt ∈ Nt (vt ) : μ(vt ) = +}.
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D. Sinha, D. Sharma / Electronic Notes in Discrete Mathematics 63 (2017) 455–460
Balance in the ILT model
In this section we discuss the property of balance in the ILT model for a given signed graph Σ0 . Theorem 3.1 For a given signed graph Σ0 , Σt is balanced for all t if and only if for each positive edge uv in Σ0 , end vertices u and v possess same marking and for each negative edge uv in Σ0 end vertices are oppositely marked in Σ0 . 3.1
Algorithm
We give Algorithm 1 which detects if, for a given initial signed graph Σ0 in ILT model the signed graph Σt at each time step t, is balanced or not. We take the adjacency matrix of Σ0 along with its number of vertices given by n. The variables count positive and count negative gives the positive degree and negative degree of each vertex i, respectively. Array sign gives the signs of each vertex i. The given algorithm emerges from Theorem 3.1, by taking the adjacency matrix A of signed graph Σ0 and the number of vertices n as input. After taking the inputs, we compute the number of positive and negative edges for each vertex i by using count positive and count negative in Step 3 to Step 5. Next we assign a positive mark to vertex i if count positive >= count negative by employing array sign[i] and giving it value 1, otherwise negative mark by providing −1 in sign[i]. The final process involves checking the marks on the end vertices. If for a positive edge ij, sign[i] = sign[j] or if for negative edge sign[i] = sign[j] we put t = 1, and exit the loop. Now if t = 1 then clearly the ILT model can not be balanced. 3.2
Implementation of Algorithm
We use an adjacency matrix A to give a implementation of Algorithm 1. Let us first consider its adjacency matrix A and signed graph Σ0 in Figure 1. ⎤ ⎡ 0 −1 −1 ⎥ ⎢ ⎥ ⎢ ⎢−1 0 1 ⎥ ⎦ ⎣ −1 1 0 For the given signed graph Σ0 , we first give the marking on its vertices in accordance with Proposition 2.1. After initialising t = 0, we enter the loop in step 3 to count the number of positive and negative edges incident to each vertex. Clearly vertex 1 receives a negative mark, whereas, 2 and 3 obtain the
D. Sinha, D. Sharma / Electronic Notes in Discrete Mathematics 63 (2017) 455–460
459
Algorithm 1 Step 1 Input n, initialize t=0 for i=1 to n, repeat Step 2. Step 2 For j=1 to n enter A[i][j]. Step 3 For i=1 to n, repeat Step 4 to Step 6 Step 4 count positive = 0 count negative = 0 for j=1 to n, repeat Step 5. Step 5 Check if A[i][j] = 1 if true count positive = count positive + 1 else if A[i][j] = -1 count positive = count positive + 1 Step 6 Next check if count positive >= count negative if true then sign[i] = 1 else sign[i] = −1 Step 7 For i= 1 to n for j=1 to n, repeat Step 8 and Step 9 Step 8 If A[i][j]=1 check if sign[i] = sign[j] if true then t = 1 and goto Step 10. Step 9 If A[i][j]=-1 check if sign[i] = sign[j] if true then t = 1 and goto Step 10. Step 10 if t=1 Print ”Model is not balanced” else Print ”Model is balanced” positive marks. The marks are saved in array sign where sign[1] = 1 which represents a positive mark and sign[2] = −1 = sign[3] where −1 represents a negative mark on the vertices. Next we enter into loops in step 7 and 8 to check if each positive edge incident to vertices i and j is such that sign[i] = sign[j], this is done by inspecting if the condition does not hold i.e sign[i] = sign[j] and changing t by giving its value 1. Since, in case of the positive edge between vertices 2 and 3, both these vertices are positive and thus we move directly to step 9. In step 9, we check for each negative edge whether the incident
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D. Sinha, D. Sharma / Electronic Notes in Discrete Mathematics 63 (2017) 455–460
1
2
3
Fig. 1. The adjacency matrix A and signed graph Σ0
vertices marked oppositely. For a balanced ILT model, the end vertices of each negative edge should be marked oppositely. For the given signed graph Σ0 , this is true as for the negative edge 1 − 2 and 1 − 3 (1 is negatively marked while 2 and 3 are marked positive). Thus, t = 0. By aforementioned Theorem 3.1, Σt is balanced, which is shown in the Figure 2 for t = 0, 1. We use two loops in Step 2, Step 3 and again Step 7, thus making the complexity of the algorithm of order n2 .
-
+
+
t=0
t=1
Fig. 2. The marked signed graph Σ0 and signed graph Σt at t = 1
References [1] Bonato A, N. Hadi , P. Horn, P. Praat , and C. Wang, Models of online social networks, Internet Mathematics, 6(3) (2009):285-313. [2] Harary F., On the notion of balance of a signed graph, Michigan Mathematical Journal, 2(1953), 143-146. [3] West D.B., “Introduction to graph theory”. Upper Saddle River: Prentice hall; 2001. [4] Zaslavsky T., Signed graphs, Discrete Mathematics, 4(1982), 47-74.
Electronic Notes in Discrete Mathematics 63 (2017) 455–460 www.elsevier.com/locate/endm
Transitivity Model on Signed Graphs Deepa Sinha
1
Department of Mathematics South Asian University, Akbar Bhawan Chanakyapuri, New Delhi-110021, India.
Deepakshi Sharma
2
Department of Mathematics South Asian University, Akbar Bhawan Chanakyapuri, New Delhi-110021, India.
Abstract In this paper, we generalize the already established iterated local transitivity model for online social networks in signed networks. In this model, at each time step t and for already existing vertex x, a new vertex(clone) x is added which joins to the neighbors of x. The sign of new edge xx is the marking on x. We also discuss the properties such as balance, clusterability, sign-compatibility and consistency. The signed networks focus on the type of relations (friendship and enmity) between the vertices(members of online social network). The ILT model for signed network gives an insight on how the network reacts to the addition of clone vertex. Also the properties like balance and clusterability helps establish a natural balance in society by providing a possible formation of group of vertices in society for a peaceful co-existence and smooth functioning of social system. Keywords: social network, signed social network, local transtivity model, marked singed graph, neighborhood, balance, sign-compatibility, clusterability, algorithm.
https://doi.org/10.1016/j.endm.2017.11.043 1571-0653/© 2017 Elsevier B.V. All rights reserved.
456
1
D. Sinha, D. Sharma / Electronic Notes in Discrete Mathematics 63 (2017) 455–460
Introduction
In this paper, we extend the work on the theoretic aspect of friendship and enmity for online social networks which grow very fast. We give sign to edges where a positive sign indicates a friendly relation and a negative sign a hostile relation. Further, a positive vertex indicates that a person befriends most of his/her neighbors and a negative vertex states a has hostile relations with their acquaintances. A signed graph [4] is an ordered pair Σ = (Σu , σ), where Σu is a graph G = (V, E), called the underlying graph of Σ and σ : E → {+, −} is a function from the edge set E of Σu into the set {+, −}, called the signature (or sign in short) of Σ. A signed graph is all-positive (respectively, all-negative) if all its edges are positive (negative); further, it is said to be homogeneous if it is either allpositive or all-negative and heterogeneous otherwise. The positive (negative) degree of a vertex v ∈ Σ denoted by deg + (v)(deg − (v)) is the number of positive (negative) edges incident on the vertex v and deg(v) = deg + (v) + deg − (v). Let v be an arbitrary vertex of a graph G. We denote the set consisting of all the vertices of G adjacent to v by N (v). This set is called the open neighborhood set of v and sometimes we call it as open neighborhood of v. The set consisting of all the vertices of G adjacent to v along with v itself is called the closed neighborhood set or closed neighborhood of v. It is denoted by N [v]. A marked signed graph is an ordered pair Σμ = (Σ, μ) where Σ = (Σu , σ) is a signed graph and μ : V (Σu ) → {+, −} is a function from the vertex set V (Σu ) of Σu into the set {+, −}, called a marking of Σ. Henceforth the vertex receiving ‘+’ mark will be called positive vertex and the vertex receiving ‘-’ mark is called negative vertex. In this model the marking on a vertex denotes its nature of bond with its neighbors. A positive vertex indicates that a person befriends most of its neighbors and a negative vertex states that it has hostile relations with its acquaintances. A new clone vertex added makes his relations according to these markings of already existing vertices. A cycle in a signed graph S is said to be positive if the product of the signs of its edges is positive or, equivalently, if the number of negative edges in it is even. A cycle which is not positive is said to be negative A signed graph is balanced if all its cycles are positive. The partition criterion to characterize the balance property of a signed graph is given by Harary [2]. The balance in ILT model for signed networks partitions our network into two groups so 1 2
Email: deepa [email protected] Email: [email protected]
D. Sinha, D. Sharma / Electronic Notes in Discrete Mathematics 63 (2017) 455–460
457
that the relations of friendship are in the same group and that of enmity goes across between the two partitions. The balance in online social network is required for the smooth functioning of the society. For terminology and notation in graph theory we refer the reader to [3] and for signed graph we refer the reader to [4].
2
The ILT model for signed networks
In the Iterated Local Transitivity (ILT) model [1], at each time step and for every existing vertex, a new vertex appears which joins to the closed neighbor set of vertices. Here we present the model which generates a simple, undirected signed graph Σt : t ≥ 0. At each time step t, the vertices are marked positive or negative in Σt . We begin with a connected signed graph Σ0 which is fixed. Let Σt be a signed graph generated at the time t such that it is an induced subsigned graph of signed graph Σt+1 at the time t + 1. The vertex v at the time t is denoted by vt and at the t + 1 as vt+1 . Σt+1 is constructed as follows: For each vertex x of Σt , its clone x is added and is connected to all vertices in Nt [x]. The sign of edge σt+1 (x y) = μt (y) for each y ∈ Nt [x], where the marking μ : V (Σ) → {+, −} is such that + if degt+ (xt ) ≥ degt (xt )/2 μ(xt ) = (1) − if degt− (xt ) > degt (xt )/2. Proposition 2.1 If x is a positive vertex, x be its clone then at time step t + 1: + degt+1 (x) = 2degt+ (x) + 1 + (x ) = |Nt+ (x)| + 1 degt+1 − (x) = 2degt− (x) degt+1 − (x ) = |Nt− (x)|. degt+1 If x is negative then: − (x) = 2degt− (x) + 1 degt+1 − (x ) = |Nt− (x)| + 1 degt+1 + (x) = 2degt+ (x) degt+1 + (x ) = |Nt+ (x)|. degt+1 Here degt+ (xt )(degt− (xt )) is the positive(negative) degree of each vertex vt at time step t and Nt+ (vt ) = {vt ∈ Nt (vt ) : μ(vt ) = +}.
458
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D. Sinha, D. Sharma / Electronic Notes in Discrete Mathematics 63 (2017) 455–460
Balance in the ILT model
In this section we discuss the property of balance in the ILT model for a given signed graph Σ0 . Theorem 3.1 For a given signed graph Σ0 , Σt is balanced for all t if and only if for each positive edge uv in Σ0 , end vertices u and v possess same marking and for each negative edge uv in Σ0 end vertices are oppositely marked in Σ0 . 3.1
Algorithm
We give Algorithm 1 which detects if, for a given initial signed graph Σ0 in ILT model the signed graph Σt at each time step t, is balanced or not. We take the adjacency matrix of Σ0 along with its number of vertices given by n. The variables count positive and count negative gives the positive degree and negative degree of each vertex i, respectively. Array sign gives the signs of each vertex i. The given algorithm emerges from Theorem 3.1, by taking the adjacency matrix A of signed graph Σ0 and the number of vertices n as input. After taking the inputs, we compute the number of positive and negative edges for each vertex i by using count positive and count negative in Step 3 to Step 5. Next we assign a positive mark to vertex i if count positive >= count negative by employing array sign[i] and giving it value 1, otherwise negative mark by providing −1 in sign[i]. The final process involves checking the marks on the end vertices. If for a positive edge ij, sign[i] = sign[j] or if for negative edge sign[i] = sign[j] we put t = 1, and exit the loop. Now if t = 1 then clearly the ILT model can not be balanced. 3.2
Implementation of Algorithm
We use an adjacency matrix A to give a implementation of Algorithm 1. Let us first consider its adjacency matrix A and signed graph Σ0 in Figure 1. ⎤ ⎡ 0 −1 −1 ⎥ ⎢ ⎥ ⎢ ⎢−1 0 1 ⎥ ⎦ ⎣ −1 1 0 For the given signed graph Σ0 , we first give the marking on its vertices in accordance with Proposition 2.1. After initialising t = 0, we enter the loop in step 3 to count the number of positive and negative edges incident to each vertex. Clearly vertex 1 receives a negative mark, whereas, 2 and 3 obtain the
D. Sinha, D. Sharma / Electronic Notes in Discrete Mathematics 63 (2017) 455–460
459
Algorithm 1 Step 1 Input n, initialize t=0 for i=1 to n, repeat Step 2. Step 2 For j=1 to n enter A[i][j]. Step 3 For i=1 to n, repeat Step 4 to Step 6 Step 4 count positive = 0 count negative = 0 for j=1 to n, repeat Step 5. Step 5 Check if A[i][j] = 1 if true count positive = count positive + 1 else if A[i][j] = -1 count positive = count positive + 1 Step 6 Next check if count positive >= count negative if true then sign[i] = 1 else sign[i] = −1 Step 7 For i= 1 to n for j=1 to n, repeat Step 8 and Step 9 Step 8 If A[i][j]=1 check if sign[i] = sign[j] if true then t = 1 and goto Step 10. Step 9 If A[i][j]=-1 check if sign[i] = sign[j] if true then t = 1 and goto Step 10. Step 10 if t=1 Print ”Model is not balanced” else Print ”Model is balanced” positive marks. The marks are saved in array sign where sign[1] = 1 which represents a positive mark and sign[2] = −1 = sign[3] where −1 represents a negative mark on the vertices. Next we enter into loops in step 7 and 8 to check if each positive edge incident to vertices i and j is such that sign[i] = sign[j], this is done by inspecting if the condition does not hold i.e sign[i] = sign[j] and changing t by giving its value 1. Since, in case of the positive edge between vertices 2 and 3, both these vertices are positive and thus we move directly to step 9. In step 9, we check for each negative edge whether the incident
460
D. Sinha, D. Sharma / Electronic Notes in Discrete Mathematics 63 (2017) 455–460
1
2
3
Fig. 1. The adjacency matrix A and signed graph Σ0
vertices marked oppositely. For a balanced ILT model, the end vertices of each negative edge should be marked oppositely. For the given signed graph Σ0 , this is true as for the negative edge 1 − 2 and 1 − 3 (1 is negatively marked while 2 and 3 are marked positive). Thus, t = 0. By aforementioned Theorem 3.1, Σt is balanced, which is shown in the Figure 2 for t = 0, 1. We use two loops in Step 2, Step 3 and again Step 7, thus making the complexity of the algorithm of order n2 .
-
+
+
t=0
t=1
Fig. 2. The marked signed graph Σ0 and signed graph Σt at t = 1
References [1] Bonato A, N. Hadi , P. Horn, P. Praat , and C. Wang, Models of online social networks, Internet Mathematics, 6(3) (2009):285-313. [2] Harary F., On the notion of balance of a signed graph, Michigan Mathematical Journal, 2(1953), 143-146. [3] West D.B., “Introduction to graph theory”. Upper Saddle River: Prentice hall; 2001. [4] Zaslavsky T., Signed graphs, Discrete Mathematics, 4(1982), 47-74.