The Structure of Networks

1. The Structure of Networks with emphasis on information and social networks T-214-SINE Summer 2011 Chapter 3 Ýmir Vigfússon

2. Positive and negative relationships • People are either mutual friends (+) or enemies (-) • For now assume a complete signed graph • Everyone is aware of everyone else • Later we will relax this • When we consider three people, some triads are inherently more stable • We call these triads balanced

3. Balanced triads

4. Balance theorem • Def: Structural balance property: • For every set of three nodes, if we consider the three edges connecting them, either all three of these edges are labeled +, or else exactly one of them is labeled +. • Settings that follow this property • Everybody is friends  • Two mutually distrusting coalitions • Balance theorem: [Harary ´53] • These are the only possibilities

5. Balance theorem • More formally: • Let‘s prove this. • Easy when everyone is friends, so assume some negative edges exist • Pick any node A. • Every node is either a friend or an enemy of A • Must find groups X and Y of mutual friends such that everyone in X dislikes everyone in Y

6. Balance theorem • Candidate solution • X = A´s friends • Y = A´s enemies • This is the correct solution! • (1) Must show that everyone in X is friends • Take two people B,C in X. They‘re friends with A. If they were enemies, we would violate structural balance. So must be friends.

7. Balance theorem • (2) Must show that D,E in Y are friends • A is enemies with both, would violate structural balance otherwise. So D,E are enemies

8. Balance theorem • (3) Must show that B,D are mutual enemies • A friends with B, enemies with D. Can‘t have friendship between B,D • So we‘re done!

9. Weak balance • There are two processes at work for strong structural balance • It is hard to maintain a positive relationship with each of mutual enemies • For three mutual enemies, two tend to team up against the third • The last property may not be strong • Suggest we weaken our definition

10. Weak balance • Now we get an analogous theorem • What changes in the proof? • We again take a node A and explore it • Now we can‘t take step (2) • Since we don‘t know anything about the relationship between D,E

11. Balance in general graphs • So far we assumed all edges exist • But that‘s a strong assumption

12. Balance in general graphs • In the advanced section, we prove the following theorem • Relate a local property (structural balance) with a global one (no cycles) • Thm: A general signed graph is balanced if and only if it contains no cycle with an odd number of negative edges

13. Idea of proof • (1) Collapse all positive connected components

14. Idea of proof • (2) If any negative edges inside the component, there is a negative cycle

15. Idea of proof • (3) Look at collapsed graph (only negative edges between components) • Try to label nodes into two components • We can do this using BFS!

16. Idea of proof • (4) If none found, the graph is balanced!