Positive and Negative Relations

Transcription

1 Positive and Negative Relations CMSC 498J: Social Media Computing Department of Computer Science University of Maryland Spring 2016 Hadi Amiri

2 Lecture Topics Balanced and Unbalanced Networks Structure of Balanced Networks Weakly Balanced Networks 2

3 Structural Balance We viewed networks as having positive relations or links friendship, collaboration, membership relations, etc. But there are networks with negative relations. Support / Oppose relations Trust / Distrust relations Friend / Foe relations How should we reason about these networks? 3

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. 4

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. 5

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 triangles with one or three +'s as balanced Structural Balance- Cnt. But for 3 people, certain configurations are socially / psychologically more plausible Balanced Unbalanced triangles with zero or two +'s are unbalanced 7

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. 8

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 +. 9

10 Structural Balance Property- Cnt. 10

11 Structural Balance Property- Cnt. Unrealistic definition! 11

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! 12

13 Structure of Balanced Nets- Cnt. If a Labeled Complete Graph is balanced, then either: 1. The network contains only positive edges, or 2. 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 Balance Theorem: These are the only ways to have a balanced network! Y 13

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. 14

15 Structure of Balanced Nets- Cnt. X Y 15

16 Structural Balance- Applications How does such a social network, labels of edges, evolve over time? Social networks implicitly seek out structural balance! 16

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 17

18 App 2. International Relations World War I Fr: France Ru: Russia It: Italy Ge: Germany AH: Austria- Hungary GB: Great Britain Structural Balance is not necessarily a good thing : hard-to-resolve opposition between two sides. 18

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

20 Weak Structural Balance Friends of friends may try to reconcile their differences 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! 22

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 1 st 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. 23

24 Structure of Weakly Balanced Nets X Remove X from the graph and let it be 1 st group. The smaller graph is still weakly balanced, why? Find others group in this way until all the nodes are assigned to a group. 24

25 Questions? 25

26 Reading Ch.05 Positive and Negative Relationships [NCM] 26