CS224W Social and Information Network Analysis Jure Leskovec Stanford University httpcs224wstanfordedu Today Signed Networks Networks with positive and negative relationships ID: 757282
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Slide1
Networks with Signed Edges
CS224W: Social and Information Network AnalysisJure Leskovec, Stanford Universityhttp://cs224w.stanford.eduSlide2
Today: Signed NetworksNetworks with
positive and negative relationshipsOur basic unit of investigation will
be
signed triangles
First we will talk about undirected nets then directedPlan for today:Model: Consider two soc. theories of signed netsData: Reason about them in large online networksApplication: Predict if A and B are linked with + or -
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
2
-
-
+
-
-
+Slide3
Signed NetworksNetworks with
positive and negative relationshipsConsider an undirected complete graphLabel each edge as either:
Positive
:
friendship, trust, positive sentiment, …Negative: enemy, distrust, negative sentiment, …Examine triples of connected nodes A, B, C10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
3Slide4
Theory of Structural Balance
Start with the intuition [Heider ’46]:Friend of my
friend
is my
friendEnemy of enemy is my friendEnemy of friend is my enemyLook at connected triples of nodes:
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
4
+
+
+
-
-
+
+
+
-
-
-
-
Unbalanced
Balanced
Consistent
with “friend of a friend” or “enemy of the enemy” intuition
Inconsistent
with the “friend of a friend” or “enemy of the enemy” intuitionSlide5
Balanced/Unbalanced Networks10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu5
Balanced
Unbalanced
Graph is
balanced
if every connected
triple of nodes has:
all 3 edges labeled +, or
exactly 1 edge labeled +Slide6
Local Balance Global Factions
Balance implies global coalitions [Cartwright-Harary]
If
all triangles are balanced
, then either:The network contains only positive edges, orNodes can be split into 2 sets where negative edges only point between the sets10/12/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
6
+
+
L
+
R
-Slide7
Analysis of Balance10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu7
B
+
C
D
E
+
–
–
Friends of A
Enemies of A
Every node in L is enemy of R
+
+
–
A
Any 2 nodes
in L are friends
Any 2 nodes
i
n R are friends
L
RSlide8
Example: International Relations
International relations:Positive edge: allianceNegative edge: animositySeparation of Bangladesh from Pakistan in 1971: U
S supports
P
akistan. Why?USSR was enemy of ChinaChina was enemy of IndiaIndia was enemy of
PakistanUS was friendly with C
hinaChina vetoed
Bangladesh from U.N.
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
8
P
R
I
C
U
+
+
–
–
–
+?
B
–?
–Slide9
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
9Slide10
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
10Slide11
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
11Slide12
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
12Slide13
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
13Slide14
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
14Slide15
Balance in General Networks
Def 1: Local viewFill in the missing edges to achieve balance
Def 2:
Global viewDivide the graph into two coalitionsThe 2 defs. are equivalent!10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
15
Balanced?
-
+Slide16
Is a Signed Network Balanced?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 + edgesFor each component create a super-nodeConnect components A and B if there is a negative edge between the membersAssign super-nodes to sides using BFS10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
16Slide17
Signed Graph: Is it Balanced?10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu17Slide18
Positive Connected Components10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu18Slide19
Reduced Graph on Super-Nodes
10/12/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu19Slide20
BFS on Reduced Graph
Using BFS assign each node a sideGraph is unbalanced if any two
super-nodes are assigned the
same side
10/12/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu20
L
R
R
L
L
L
R
Unbalanced!Slide21
Real Large Signed NetworksEach link
AB is explicitly tagged with a sign:
E
pinions
: Trust/DistrustDoes A trust B’s product reviews?(only positive links are visible)Wikipedia: Support/OpposeDoes A support B to becomeWikipedia administrator?
Slashdot: Friend/Foe
Does A like B’s comments?Other examples: Online multiplayer games
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
21
+
+
+
+
+
+
+
+
–
–
–
–
–
–
–
[CHI ‘10]Slide22
Balance in Our Network DataDoes structural balance hold?
10/12/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
22
Triad
Epinions
Wikipedia
Balance
P(T)
P
0
(T)
P(T)
P
0(T)
0.87
0.62
0.70
0.49
0.07
0.05
0.210.10
0.05
0.32
0.08
0.49
0.007
0.003
0.011
0.010
-
-
+
+
+
-
-
-
-
+
+
+
P(T) … probability of a triad
P
0
(T)… triad probability if the
signs would be random
Real data
Shuffled data
+
x
x
+
–
–
–
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
–
–
–
[CHI ‘10]
x
x
x
x
xSlide23
Global Structure of Signed NetsIntuitive picture of social
network in terms of densely linked clustersHow does structure interact with links?
Embeddedness
of
link (A,B): Number of shared neighbors23Slide24
Global Factions: Embeddedness
Embeddedness of ties:Positive ties tend to be more embedded
Positive ties
tend to be more
clumped togetherPublic display of signs (votes) in Wikipedia further attenuates this10/12/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
24
Epinions
Wikipedia
[CHI ‘10]Slide25
Global Structure of Signed Nets
Clustering:+net: More clustering than baseline–net: Less clustering than baselineSize of connected component:+/–net: Smaller than the baseline
25
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
[CHI ‘10]
+
+
-
-
+
+
+
+
+
-
+
+
+Slide26
Evolving Directed NetworksNew setting:
Links are directed and created over time
How many
are now explained by balance?Only half (8 out of 16)Is there a better explanation?
Yes. Status.
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
26
16 *2 signed directed triads
-
+
-
+
+
+
+
+
-
-
--
[CHI ‘10]
B
X
A
Slide27
B
B
Alternate Theory: Status
Links are
directed and created over time
Status theory
[Davis-
Leinhardt
‘68, Guha et al. ’04,
Leskovec et al. ‘10]Link A B means: B has
higher status than ALink A B means: B has
lower status than AStatus and balance give different
predictions:10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu27
A
X
-
-
A
X
+
+
+
–
[CHI ‘10]
Balance:
+
Status:
–
Balance:
+
Status:
–Slide28
B
Theory of Status
Edges are
directed
Edges are created over timeX has links to A and BNow, A links to B (triad A-B-X)How does sign of A-B depend
signs of X?We need to formalize:Links are
embedded in triads:Provides context for signs
Users are heterogeneous
in their
linking behavior10/12/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
28
A
X
+
+
?
B
X
A
[CHI ‘10]Slide29
16 Types of Context29
Link (A,B) appearsin the context(A,B; X)
16 different
contextualized
links:
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
[CHI ‘10]Slide30
Generative (Receptive) Surprise
Surprise:
How much behavior of user
deviates
from baseline in context t:
(A1, B1; X
1),…, (An, B
n; Xn) …
instances of contextualized link t
k of them closed with a pluspg(Ai)… generative baseline of Ai empirical prob. of A
i giving a plusThen: generative surprise of triad type
t:10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu30
Vs.
B
X
A
B
X
-
-
A
[CHI ‘10]
Std. rnd. var.:
Give a better explanation of what we really
do (
2 slides):
For every node compute the baseline
Identify all the edges that close same type of triads
Compute surpriseSlide31
Status: Two Examples
Two basic examples:
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
31
B
X
-
-
B
X
+
+
A
A
Gen. surprise of A:
—
Rec. surprise of B:
—
Gen. surprise of A:
—
Rec. surprise of B:
—Slide32
ENDEND (when I spent 15 min for finishing up the previous lecture)
10/12/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
32Slide33
Joint Positive EndorsementX positively endorses A and BNow A links to B
A puzzle:In our data we observe:Fraction of positive links deviatesAbove generative baseline of A
Below receptive baseline of B
Why?
B
X
+
+
?
A
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
10/12/2011
33
[CHI ‘10]Slide34
A Story: Soccer Team
Ask every node: How does skill of B compare to yours?Build a signed directed networkWe haven’t asked A about B But we know that X thinks
A and B are both better than him
What can we infer about A’s answer?
B
X
+
+
?
A
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
10/12/2011
34
[CHI ‘10]Slide35
A Story: Soccer Team
A’s viewpoint:Since B has positive evaluation, B is high statusThus, evaluation A gives ismore likely to be positive than the baseline
B
X
+
+
?
Y
B
How does A evaluate B?
A
A is evaluating someone who is better than avg.
A is
more positive than average
Y… average node
A
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
10/12/2011
35Slide36
A Story: Soccer Team
B’s viewpoint:Since A has positive evaluation, A is high statusThus, evaluation B receivesis less likely to be positive than
the baseline
B
X
+
+
?
A
Y
A
How is B evaluated by A?
B is evaluated by someone better than average.
They will be
more negative to B than average
Y… average node
B
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
Sign of A
B deviates in different directions depending on the viewpoint!
10/12/2011
36Slide37
Consistency with Status
Determine node status:Assign X status 0Based on signs and directionsof edges set status of A and BSurprise is status-consistent, if:
G
en. surprise is status-consistent
if it has same sign as status of BRec. surprise is status-consistent if it has the opposite
sign from the status of ASurprise is balance-consistent, if:
If it completes a balanced triad
10/12/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
37
Status-consistent if:
Gen. surprise > 0
Rec. surprise < 0
B
X
+
+
A
+1
+1
0
[CHI ‘10]Slide38
Status vs. Balance (Epinions)
Predictions:10/12/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
38
t
14
t
15
t
16
t
3
t
2
[CHI ‘10]
S
g
(
t
i
)
S
r
(
t
i
)
B
g
B
r
S
g
S
r
Slide39
From Local to Global StructureBoth theories make predictions about the global structure of the network
Structural balance – FactionsFind coalitionsStatus theory – Global StatusFlip direction and sign of minus edgesAssign each node a unique status
so that edges point from low to high
10/12/2011
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu39
[WWW ‘10]
+
+
-
3
1
2Slide40
From Local to Global StructureFraction of edges of the network that satisfy Balance and Status?
Observations:No evidence for global balance beyond the random baselinesReal data is 80% consistent vs. 80% consistency under random baseline
Evidence for global status
beyond the random baselines
Real data is 80% consistent, but 50% consistency under random baselineJure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu[WWW ‘10]
10/12/2011
40Slide41
Predicting Edge Signs
Edge sign prediction problemGiven a network and signs on all but one edge, predict the missing sign
Machine Learning Formulation:
Predict sign of edge (
u,v)Class label: +1: positive edge-1: negative edgeLearning method:Logistic regression
Dataset:Original: 80%
+edgesBalanced: 50% +edges
Evaluation:Accuracy and ROC curvesFeatures for learning:
Next slide
u
v
+
+
?
+
+
+
+
+
–
–
–
–
–
–
–
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
10/12/2011
41
[WWW ‘10]Slide42
Features for Learning
For each edge (
u,v
) create features:
Triad counts (16):
Counts of signed triads
edge u
v takes part in
Node degree (7 features):Signed degree:
d+out(u), d-out(u), d+
in(v), d-in(v)Total degree: dout(u), d
in(v)Embeddedness of edge (u,v)
u
v
-
+
+
+
-
-
+
-
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
10/12/2011
42
[WWW ‘10]Slide43
Edge Sign Prediction
Classification Accuracy:Epinions: 93.5%Slashdot: 94.4%
Wikipedia: 81%
Signs can be modeled from local network structure alone
Trust propagation model of [Guha et al. ‘04] has 14% error on EpinionsTriad features perform less well for less embedded edgesWikipedia is harder to model:Votes are publicly visible
Epin
Slash
Wiki
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
10/12/2011
43
[WWW ‘10]Slide44
Balance and Status: Complete Model
44
+
+
+
-
-
+
-
-
+
+
+
-
-
+
-
-
+
+
+
-
-
+
-
-
+
+
+
-
-
+
-
-Slide45
Generalization
Do people use these very different linking systems by obeying the same principles?How generalizable are the results across the datasets?Train on row “dataset”, predict on “column”
Nearly
perfect generalization
of the models even though networks come from very different applications
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
10/12/2011
45Slide46
Concluding RemarksSigned networks provide insight into how social computing systems are used:
Status vs. BalanceDifferent role of reciprocated linksRole of embeddedness and public displaySign of relationship can be reliably predicted
from the
local network context
~90% accuracy sign of the edge46Slide47
Concluding RemarksMore evidence that
networks are globally organized based on statusPeople use signed edges consistently regardless of particular applicationNear perfect generalization of models across datasetsMany further directions:Status difference of nodes
A and B
[ICWSM ‘10]
:
A<B A=B A>B
Status difference (A-B)