Signed NetworksSocial and Economic Networks

Jafar Habibi

MohammadAmin Fazli

Social and Economic Networks 1

ToC

• Signed Networks• Positive & negative links

• Structural Balance theory

• Structure of balanced networks

• Some applications

• Weak Balance theory

• Generalizing the definition of structural balance

• Approximately balanced networks

• Readings:• Chapter 5 from the Kleinberg book

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Positive & Negative Links

• Signed graphs: we can label graph edges with positive (+) and negative (-) signs

• + signs show positive aspects of relationships• Friend, like, follow, ...

• - sigs show negative aspects of relationships• Antagonistic, dislike, …

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Structural Balance

• Theories in social Psychology by Heider, Cartwright and Harary

• Balanced Triangles: we can classify triads in this graph into two categories, balanced & unbalanced

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Balanced/Unbalanced Graphs

• Signed graphs are balanced iff all of its triads are balanced

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Balanced Complete Graphs

• Harary Theorem: If a labeled complete graph is balanced, then either all pairs of nodes are friends, or else the nodes can be divided into two groups, X and Y , such that every pair of nodes in X like each other, every pair of nodes in Y like each other, and everyone in X is the enemy of everyone in Y.• Proof: See the blackboard.

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Structural Balance in Political Relations

• Moore: the conflict over Bangladesh’s separation from Pakistan in 1972• [T]he United States’s somewhat surprising support of Pakistan ... becomes

less surprising when one considers that the USSR was China’s enemy, China was India’s foe, and India had traditionally bad relations with Pakistan. Since the U.S. was at that time improving its relations with China, it supported the enemies of China’s enemies. Further reverberations of this strange political constellation became inevitable: North Vietnam made friendly gestures toward India, Pakistan severed diplomatic relations with those countries of the Eastern Bloc which recognized Bangladesh, and China vetoed the acceptance of Bangladesh into the U.N.

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Structural Balance May Not Be Good

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Weak Structural Balance

• In a triangle with two positive edges, we have the problem of a person whose two friends don’t get along; in a triangle with three negative edges, there is the possibility that two of the nodes will allythemselves against the third.

• Davis et al. have argued that in many settings, the first of these factors may be significantly stronger than the second

• Weak Structural Balance Property: There is no set of three nodes such that the edges among them consist of exactly two positive edges and one negative edge.

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Weakly Balanced Complete Graphs

• If a labeled complete graph is weakly balanced, then its nodes can be divided into groups in such a way that every two nodes belonging to the same group are friends, and every two nodes belonging to different groups are enemies.• Proof: See the blackboard

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Structural Balance in Arbitrary Graphs

• How to define structural balance in arbitrary graphs?• Definition 1: it is possible to fill in all the missing labeled edges in such a way

that the resulting signed complete graph is balanced.

• Definition 2: it is possible to divide the nodes into two sets X and Y , such that any edge with both ends inside X or both ends inside Y is positive, and any edge with one end in X and the other in Y is negative.

• These two definitions are equivalent

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Structural Balance in Arbitrary Graphs

• Theorem: A signed graph is balanced if and only if it contains no cycle with an odd number of negative edges. • Proof: see the blackboard

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Structural Balance in Arbitrary Graphs

• Proof: see the blackboard

• Identifying super nodes

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Structural Balance in Arbitrary Graphs

• Proof: see the blackboard

• BFS on the reduced graph

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Approximately Balanced Networks

• The conditions of balance theory (even weak balance theory) are very extreme

• What about when 0.01% of triangles are unbalanced?• Is the graph still balanced or these few unbalanced triangles will make the

graph totally unbalanced?

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Approximately Balanced Networks

• Let 𝜖 be any number such that 1 ≤ 𝜖 ≤1

8, and define 𝛿 = 3 𝜖. If at

least 1 − 𝜖 of all triangles in a labeled complete graph are balanced, then either• there is a set consisting of at least 1 − δ of the nodes in which at least 1 − δ

of all pairs are friends, or else

• the nodes can be divided into two groups, X and Y , such that• at least 1 − δ of the pairs in X like each other,

• at least 1 − δ of the pairs in Y like each other, and

• at least 1− δ of the pairs with one end in X and the other end in Y areenemies.

• Proof: See the blackboard

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Approximately Balanced Networks

• Proof for the approximate balanced networks characterization: see the blackboard

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