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

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

Balanced triads

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

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

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.

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

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!

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

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

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

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

Idea of proof (1) Collapse all positive connected components

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

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!

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