Structural Balance
Positive and Negative Relations CMSC 498J: Social Media Computing Department of Computer Science University of Maryland Spring 2015 Hadi Amiri [email protected] Announcement • Survey! ▫ ▫ ▫ ▫ Timing Project Lectures HWs 2 Lecture Topics • Balanced and Unbalanced Networks • Structure of Balanced Networks ▫ Weakly Balanced Networks 3 Structural Balance • We viewed networks as having positive connotations ▫ friendship, collaboration, membership relations, etc. • But there are also negative relations in networks. Support / Oppose relations Trust / Distrust relations Friend / Foe relations How should we reason about these networks? 4 Structural Balance- Cnt. • Given a network, how can we annotate its links with positive and negative signs? • Structural Balance is one basic framework for this purpose. 5 Structural Balance- Cnt. • Let’s assume we have Complete Graph with links labeled by + and – signs. ▫ Each pair of nodes are either friends or enemies ▫ Makes sense for a group of people with mutual awareness a classroom, a small company, a sports team, a fraternity or sorority, or international relations where nodes are countries. We will relax the complete graph assumption later. 6 Structural Balance- Cnt. • Considering any 2 people in isolation, the edge between them can be labeled + or – ▫ they are either friends or enemies. + - 7 Structural Balance- Cnt. triangles with one or three +'s as balanced + + - + - + + + - - - - Balanced Unbalanced triangles with zero or two +'s are unbalanced • But for 3 people, certain configurations are socially / psychologically more plausible. 8 Structural Balance- Cnt. • Which type is more abundant in real world networks? ▫ People try to minimize unbalanced triangles in their personal relationships as they are sources of stress or psychological dissonance. 9 Structural Balance Property • A Labeled Complete Graph is balanced if every one of its triangles is balanced! • Structural Balance Property ▫ For every triangle, either all its three edges are labeled +, or else exactly one of them is labeled +. 10 Structural Balance Property- Cnt. 11 Structural Balance Property- Cnt. • Extreme definition! 12 Structure of Balanced Nets • What does a balanced network look like in general? a j b i c d 2 groups of friends (A-B and C-D), with negative relations between the 2 groups! 13 Structure of Balanced Nets- Cnt. • If a Labeled Complete Graph is balanced, then either: ▫ The network contains only positive edges, or ▫ Global division of network: Nodes can be divided into 2 groups X and Y such that: a pos link btw every pair of nodes in X, a pos link btw every pair of nodes in Y, and a neg link btw every node of X and every node of Y. + X - + Y Balance Theorem: These are the only ways to have a balanced network! 14 Structure of Balanced Nets- Cnt. • Prove balanced theorem! • Solution: ▫ Consider the network from a node’s perspective! Pick any node in the network, say node A, Let X be all of A's friends, and Y be all of A's enemies. This divides all nodes as the graph is complete! ▫ Prove: Every two nodes in X are friends, and Every two nodes in Y are friends, and Every node in X is an enemy of every node in Y. 15 Structure of Balanced Nets- Cnt. X Y 16 Structural Balance- Applications • How does such a social network, labels of edges, evolve over time? • Social networks implicitly seek out structural balance! 17 App 1. International Relations • Separation of Bangladesh from Pakistan in 1972 ▫ Why did US support Pakistan? USSR was China’s enemy China was India’s enemy India was Pakistan’s enemy US was China’s friend China vetoed Bangladesh 18 App 2. International Relations • World War I Fr: France Ru: Russia It: Italy Ge: Germany AH: AustriaHungary GB: Great Britain Structural Balance is not necessarily a good thing : hard-to-resolve opposition between two sides. 19 App 3. Trust and Distrust • Networks with positive or negative relations! ▫ slashdot.com, epinions.com • Important to understand the Nature of Relationships! ▫ A trusts B, B trusts C -> natural to expect A trust C. ▫ What if A distrusts B and B distrusts C: A should trust or to distrust C? Distrust as enemy relationship: ▫ A should trust C, otherwise a triangle with 3 negative edges. Distrust as less knowledge relationship: ▫ A should distrust C (even more strongly than B). Propagation of trust and distrust. Guha, et. al. WWW 2004. 20 Weak Structural Balance Friends of friends may try to reconcile their differences + + + + - 2 of the 3 mutual enemies ally themselves against the third one! - 21 Weak Structural Balance • Weak Structural Balance Property ▫ There is no triangle with exactly two positive edges and one negative edge. 3 Mutual enemies are allowed as there could be less of a force leading any 2 of them to become friend as compared to the first case (mutual enemies with a common friend to - - reconcile)! - 22 Structure of Weakly Balanced Nets • What does a weakly balanced network look like? ▫ If a Labeled Complete Graph is weakly balanced, then its nodes can be divided into groups such that: every 2 nodes in the same group are friends, and every 2 nodes in different groups are enemies. Weakly Balance Theorem: This is the only way to have a weakly balanced network! 23 Structure of Weakly Balanced Nets • Prove weakly balanced theorem! • Solution: ▫ Consider the network from a node’s perspective! Pick a node in the network, say node A, Let X (our 1st group) consist of A and all of A's friends, ▫ Prove: X is a group of mutual friends. A's friends are friends with each other. Nodes in X are enemies of nodes not in X. A and his friends are enemies with other nodes in graph. 24 Structure of Weakly Balanced Nets X Remove X from the graph and let it be 1st group. The smaller graph is still weakly balanced, why? Find others group in this way until all the nodes are assigned to a group. 25 Questions? 26 Reading • Ch.05 Positive and Negative Relationships [NCM] 27
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