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

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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

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Balanced triads

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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

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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

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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.

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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

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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!

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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

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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

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

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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

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Idea of proof (1) Collapse all positive connected components

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

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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!

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

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