http://cs224w.stanford.edu Start with the intuition [Heider 46]: - PowerPoint PPT Presentation

CS224W: Analysis of Networks Jure Leskovec, Stanford University http://cs224w.stanford.edu

¡ Start with the intuition [Heider ’46]: § Friend of my friend is my friend § Enemy of enemy is my friend § Enemy of friend is my enemy ¡ Look at connected triples of nodes: + - + - - + - + - + - + Unbalanced Balanced Inconsistent with the “friend of a friend” Consistent with “friend of a friend” or or “enemy of the enemy” intuition “enemy of the enemy” intuition 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 2

¡ So far we talked about complete graphs Def 1: Local view Fill in the missing edges to achieve balance - - Def 2: Global view + Divide the graph into - - two coalitions The 2 definitions are equivalent! Balanced? 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 3

¡ Graph is balanced if and only if it contains no cycle with an odd number of negative edges ¡ How to compute this? – § Find connected components on +edges – – – § If we find a component of nodes on +edges Even length that contains a –edge Þ Unbalanced cycle § For each component create a super-node – § Connect components A and B if there is a – – negative edge between the members – – § Assign super-nodes to sides using BFS Odd length cycle 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 4

10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 5

10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 6

10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 7

¡ Using BFS assign each node a side ¡ Graph is unbalanced if any two connected super-nodes are assigned the same side L R R L L L û R Unbalanced! 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 8

¡ Each link A Š B is explicitly tagged with a sign: § E pinions: Trust/Distrust § Does A trust B’s product reviews? + – – (only positive links are visible to users) + + – + – – + § W ikipedia: Support/Oppose – – + + § Does A support B to become + Wikipedia administrator? § S lashdot: Friend/Foe § Does A like B’s comments? § Other examples: § Online multiplayer games 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 10

[CHI ‘10] ¡ Does structural balance hold? + + – x – § Compare frequencies of signed triads + + in real and “shuffled” signs + – x + + Epinions Wikipedia Consistent + + Triad + with Balance? P(T) P 0 (T) P(T) P 0 (T) ü + Real data + 0.87 0.62 0.70 0.49 Balanced + ü + - + - 0.07 0.05 0.21 0.10 x + x – + + ü Unbalanced + + + 0.05 0.32 0.08 0.49 + – - + x x – û - - 0.007 0.003 0.011 0.010 + + x - + P(T) … fraction of a triads P 0 (T)… triad fraction if the signs would appear at random Shuffled data 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 11

[CHI ‘10] Edge sign according to the balance theory. ¡ New setting: Links are Do people close triad X with the “balanced” edge? directed , created over time B û ü ü ü § Node A links to B X A A § Directions and signs of links - + + - û û ü ü from/to X provide context - + - + û û ü ü X × × + - + - û û û ü A B 16 signed directed triads ¡ How many r are now ( in directed networks people explained by balance? traditionally applied balance by ignoring edge directions) § Only half (8 out of 16) 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 12

[CHI ‘10] ¡ Status in a network [Davis-Leinhardt ’68] + § A Š B :: B has higher status than A § A Š B :: B has lower status than A – § Note: Here the notion of status is now implicit and governed by the network (rather than using the number of edits of a user as a proxy for status as we did before) § Apply status principle transitively over paths + – § Can replace each A Š B with A B Š § Obtain an all-positive network with same status interpretation 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 13

B ? + ? + A X ? ? - - ? ? - - ? ? + + ¡ Status does not make predictions for all the triads (denoted by ?) 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 14

[CHI ‘10] X - X + - + A B A B Balance: + Balance: + Status: – Status: – Status and balance give different predictions! 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 15

At a global level (in the ideal case): ¡ Status ⇒ Hierarchy 3 + § All-positive directed network + + 2 should be approximately acyclic 1 ¡ Balance ⇒ Coalitions § Balance ignores directions and - - implies that subgraph of negative + + - edges should be approximately bipartite 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 16

[CHI ‘10] ¡ Edges are directed: X + § X has links to A and B + § Now, A links to B (triad A-B-X) § How does sign of A Š B depend signs from/to X? A B ? + + P(A Š B | X) vs. P(A Š B) ¡ We need to formalize: Vs. § 1) Links are embedded in triads: A B Triads provide context for signs § 2) Users are heterogeneous in their linking behavior 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 17

[CHI ‘10] ¡ Link A Š B appears in context X: A Š B | X ¡ 16 possible contexts: Note: Context of a red link is uniquely determined by the directions and signs of links from/to X 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 18

¡ Users differ in frac. of + links they give/receive ¡ For a user U: § Generative baseline: Frac. of + given by U § Receptive baseline: Frac. of + received by U Basic question: ¡ How do different link contexts cause users to deviate from their baselines ? § Link contexts as modifiers on a person’s predicted behavior § Def: Surprise : How much behavior of A/B deviates from his/her baseline when A/B is in context X 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 19

[CHI ‘10] ¡ Intuition: How much behavior of user A in context X deviates from his/her baseline behavior § Baseline: For every user A : Context X: X - p g (A i ) … generative baseline of A i § Fraction of times A i gives a plus - § Context: (A 1 , B 1 | X 1 ),…, (A n , B n | X n ) … all instances of triads in context X A B § ( A i , B i , X i ) … an instance where when Vs. user A i links to user B i the triad of type X is created. A B § Say k of those triads closed with a plus + § k out of n times: A i Š B i 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 20

[CHI ‘10] ¡ Surprise: How much behavior of user A in Context X: X - context X deviates from his/her baseline behavior § Generative surprise of context X : - n å - k p ( A ) g i = = A B s ( X ) i 1 g n å Vs. - p ( A )( 1 p ( A )) g i g i = i 1 A B § p g (A i ) … generative baseline of A i § Context X: (A 1 , B 1 | X 1 ),…, (A n , B n | X n ) § k of instances of triad X closed with a plus edges § Receptive surprise is similar, just use p r (A i ) 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 21

[CHI ‘10] X + ¡ Surprise: How much behavior of user deviates from baseline when in context X - § Generative surprise of context X= _ u A B – + z + + n + å - k p ( A ) v g i - – = = s ( X ) i 1 g n y q + å + - p ( A )( 1 p ( A )) g i g i + w = i 1 We have 3 triads of context X: (z,u,v), (y,v,w), (q,v,w) They all close with a plus: So k =3 P g (u)=1/2=0.5 P g (v)=2/2=1 S g (X)=(3-2.5)/√ (0.5*0.5+1*0+1*0) = 1 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 22

¡ Assume status theory is at work ¡ What sign does status predict for edge A Š B? § We have to look at this separately from the viewpoint of A and from the viewpoint of B X - X + - + A B A B Gen. surprise of A: – Gen. surprise of A: – Rec. surprise of B: – Rec. surprise of B: – 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 23

[CHI ‘10] ¡ X positively endorses A and B X + ¡ Now A links to B + A puzzle: A B ¡ In our data we observe: ? Fraction of positive links deviates § Above generative baseline of A: S g (X) >0 § Below receptive baseline of B: S r (X) < 0 ¡ Why? 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 24

X + ¡ A’s viewpoint: + § Since B has a positive evaluation, B is likely of high status § Thus, evaluation A gives is A B ? more likely to be positive than A’s baseline behavior 10/12/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 25