Cluster Ability Characterization

8/3/2019 Cluster Ability Characterization

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Structural Balance of Signed Bipartite Graphs

Kaushik Sarkar

Arizona State UniversityP.O. Box 87-8809, Tempe, AZ, 85281 USA

[email protected]

1 Introduction

The idea of structural balance in a social network was first discussed by Heideraround 1940s. Subsequently the ides was formalized in terms of signed graphsby Cartwright and Harary. They also discussed the relationship between clus-terability and balance of social network and showed that one is the necessaryand sufficient condition for the other. But unfortunately their discussion doesnot apply to bipartite graphs. There are many kinds of social networks that cannaturally be modelled by signed bipartite graphs e.g. affiliation network, opin-ion network etc, and often we are interested in knowing if there exists a bipolarclustering in such networks. As we will see here also the question of bipolarclusterability is closely related to the idea of balance in the network. Here wedevelop the counterpart of the balance theory for signed bipartite graphs andshow that like general graphs here also it is a necessary and sufficient conditionof bipolar clusterability.

2 Balance and Clusterability in Signed Graphs

(Cartwright and Harary)For simplicity we consider complete signed graphs where each edge can haveeither a weight of 1 (denotes enmity between the participating nodes) or +1(friendship between the nodes). It is clear that if we consider only two personsthey can have either of the relationships. But if there is an unbalance in the graphthat should manifest itself when we consider cycles. The smallest possible cycle incomplete graph is of length 3 (we naturally do not consider loops). There can befollowing (Fig 1) four different configurations (upto symmetry) of positive andnegative relationship combinations. Socio-psychological considerations dictatethat configurations 1a and 1c pertain to balanced networks and the other twogives rise to unbalance in the network. With this notion of balance, we define abalanced network as follows:

Definition 1. A signed complete graph is balanced if every triangle in the net-work is isomorphic to either 1a or 1c.

Cartwright & Harary proved the following characterization of structural bal-ance.


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(a) Balanced (b) Unbalanced

(c) Balanced (d) Unbalanced

Fig. 1: Structural balance in triangles


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Theorem 1. A signed complete graph is balanced if and only if all the edgeshave positive sign or the nodes of the graph can be partitioned into two sets suchthat each edge between two nodes from the same cluster has positive sign and eachedge between two nodes belonging to two different clusters has negative sign.

3 Signed Bipartite Graphs

3.1 Balance

Many kinds of social networks can be modelled by a signed bipartite graphG = (A B,E,s), where (A,B) is a partition of the vertex set of the graphG and each edge e E is between one vertex in A and one vertex in B (forsimplicity we consider undirected graphs only). s is a mapping s : E {1, 1},1 that expresses the two different types of relationships (friendship/animosity,agreement/disagreement) that can exist between any two nodes.

We are interested to know when can such a signed bipartite network bepartitioned into two clusters such that there is mutual admiration within clusterand mutual animosity between clusters. Notice that Theorem 1 does not applyhere since we cannot have any triangle in a bipartite graph. However, we canextend the same idea to signed bipartite graphs. Most natural extension wouldbe to look for unbalance in the smallest possible cycle. In the bipartite graph thelength of the smallest cycle is 4 i.e. a quadrilateral. Figure 2 shows the possiblevariations of a signed quadrilateral (upto symmetry).

Intuitively, we can see that the configurations 2a, 2c, 2d and 2f (i.e. quadri-laterals with exactly four, two or zero positive edges) correspond to balancedsituation. We define balanced bipartite graphs in the similar line:

Definition 2. A signed complete bipartite graphG = (A B,E,s), with |A

B| > 4 is balanced if every quadrilateral in the network has exactly four, two orzero positive edges (i.e. they are isomorphic to either2a or 2c or 2d).

3.2 Perfect Bipolar Clustering

As we have already mentioned we are interested in signed bipartite graphs thatcan be cleanly clustered into two opposing groups of mutually admiring nodes.Here we formalize our idea of clean clustering in the form of perfect bipolarclustering:

Definition 3. A partition (C1, C2) of the set of vertices of a signed completebipartite graph is called perfect bipolar cluster if every edge between the nodes of

(C1

A) and (

C1

B) as well (

C2

A) and (

C2

B) are positive and every edgebetween the nodes of (C1 A) and (C2 B) as well (C2 A) and (C1 B) are

negative.

1 In general, s : E R, but for our purpose the sign of the edge is the only thing thatis important


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(a) Balanced (b) Unbalanced (c) Balanced

(d) Balanced (e) Unbalanced (f) Balanced

Fig. 2: Structural balance in quadrilaterals

Next, we show that balance, as we have defined it in the previous section, isa necessary and sufficient condition for the existence of a perfect bipolar clusterin a signed complete bipartite network.

Theorem 2. A signed complete bipartite graphG = (AB,E,s), with|AB| 4, has a perfect bipolar cluster if and only if it is balanced.

Proof. The necessity part is quite easy. If a signed complete bipartite graphG

= (A

B,E,s

) with |A

B

|>

4 has a perfect bipolar cluster then for everyquadrilateral that is completely contained within a cluster the sign of all its edgesare positive (corresponds to the balanced configuration 2a). Every quadrilateralwhich has three of its vertices in the same cluster and one on the other has exactlytwo positive edges (corresponds to Fig 2d) . Similarly, every quadrilateral thathas 2 nodes in each cluster also has either two positive edges 2c or no positiveedge at all 2f. Hence the signed bipartite graph is balanced.

To prove the sufficiency part we start with a balanced graph and show aperfect bipolar cluster. Consider a node a A. We define two sets:

C1 = +(a) (b+(a)

+(b)) (b(a)(b))

C2

= (a) (b(a)+(b)) (b+

(a)(b))

where, +(a) and (a) denotes the set of nodes connected to a by positiveand negative edges respectively. Clearly (C1, C2) is a partition of the set of nodesofG. Next we show that (C1, C2) is indeed a perfect bipolar cluster of G.

To show the previous claim we need to prove the following four subclaims:


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Fig. 3: Perfect bipolar cluster

(i) Every node in (C1 A) is connected by a positive edge with every node

of (C1 B). Suppose a (C1 A) and b (C1 B). Since a (C1 A), w.l.g. we can assume that there exists a node b +(a), s.t. a +(b) (See Fig 4). Now, consider the quadrilateral abab. By constructions({a, b}) = 1, s({b, a}) = 1 and s({b, a}) = 1. Now ifs({a, b}) = 1, thenabab has three positive edges and one negative edge i.e. G is not balanced- contradiction. So s({a, b}) = 1. The same could be proved if we hadassumed that there exists a node b (a), s.t. a (b)

Fig. 4: Case 1

(ii) Every node in (C2 A) is connected by a positive edge with every nodeof (C2 B). Suppose a (C2 A) and b (C2 B). Since a (C2 A), w.l.g. we can assume that there exists a node b (a), s.t. a

+(b) (See Fig 5). Now, consider the quadrilateral abab. By constructions({a, b}) = 1, s({b, a}) = 1 and s({b, a}) = 1. Now if s({a, b}) = 1,then abab has one positive edges and three negative edges i.e. G is notbalanced - contradiction. So s({a, b}) = 1. The same could be proved if wehad assumed that there exists a node b +(a), s.t. a (b)


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Fig. 5: Case 2

(iii) Every node in (C1 A) is connected by a negative edge with every nodeof (C2 B). Suppose a (C1 A) and b (C2 B). Since a (C1

A), w.l.g. we can assume that there exists a node b +(a), s.t. a +(b) (See Fig 6). Now, consider the quadrilateral abab. By constructions({a, b}) = 1, s({b, a}) = 1 and s({b, a}) = 1. Now ifs({a, b}) = 1, thenabab has three positive edges and one negative edge i.e. G is not balanced- contradiction. So s({a, b}) = 1. The same could be proved if we hadassumed that there exists a node b (a), s.t. a (b)

Fig. 6: Case 3

(iv) Every node in (C2 A) is connected by a negative edge with every node of(C1 B). Suppose a (C2 A) and b (C1 B). Since a (C2 A), w.l.g.we can assume that there exists a node a b (a), s.t. a +(b) (See Fig7). Now, consider the quadrilateral abab. By construction s({a, b}) = 1,

s({b, a}) = 1 and s({b, a}) = 1. Now ifs({a, b}) = 1, then abab has threepositive edges and one negative edge i.e. G is not balanced - contradiction.So s({a, b}) = 1. The same could be proved if we had assumed that thereexists a node b +(a), s.t. a (b)


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Fig. 7: Case 4

4 Conclusion

Conclusion and future works.

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Cluster Ability Characterization