1. The Structure of Information Networks
Community Structure in Networks
and Structural Balance
with emphasis on information and social networks
Ymir Vigfusson

2. Florentine Families: Power
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

3. Communities, clusters, groups, modules
Network Communities
Networks of tightly connected groups
Network communities:
Sets of nodes with lots of connections inside and few to outside (the rest of the network)
Communities, clusters, groups, modules
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

4. Finding Network Communities
How to automatically find such densely connected groups of nodes?
Ideally such automatically detected clusters would then correspond to real groups
For example:
Communities, clusters, groups, modules
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

5. Micro-Markets in Sponsored Search
Find micro-markets by partitioning the “query x advertiser” graph:
query
advertiser
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

6. Social Network Data Zachary’s Karate club network:
Observe social ties and rivalries in a university karate club
During his observation, conflicts led the group to split
Split could be explained by a minimum cut in the network
Why would we expect such clusters to arise?
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

7. Group Formation in Networks
[Backstrom et al. KDD ‘06]
Group Formation in Networks
In a social network nodes explicitly declare group membership:
Facebook groups, Publication venue
Can think of groups as node colors
Gives insights into social dynamics:
Recruits friends? Memberships spread along edges
Doesn’t recruit? Spread randomly
What factors influence a person’s decision to join a group?
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

8. Group Growth as Diffusion
Analogous to diffusion Group memberships spread over the network:
Red circles represent existing group members
Yellow squares may join
Question:
How does probability of joining a group depend on the number of friends already in the group?
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

9. P(join) vs. # friends in the group
[Backstrom et al. KDD ‘06]
P(join) vs. # friends in the group
LiveJournal: 1 million users 250,000 groups
DBLP: 400,000 papers
100,000 authors
2,000 conferences
Very different data sets, same threshold sort of thing
Diminishing returns:
Probability of joining increases with the number of friends in the group
But increases get smaller and smaller
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

10. Groups: More Subtle Features
Connectedness of friends:
x and y have three friends in the group
x’s friends are independent
y’s friends are all connected
Who is more likely to join? y x
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

11. Connectedness of Friends
[Backstrom et al. KDD ‘06]
Connectedness of Friends
Competing sociological theories:
Information argument [Granovetter ‘73]
Social capital argument [Coleman ’88]
Information argument:
Unconnected friends give independent support
Social capital argument:
Safety/trust advantage in having friends who know each other x y
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

12. Connectedness of Friends
LiveJournal: 1 million users, 250,000 groups
Social capital argument wins!
Prob. of joining increases with the number of adjacent members.
Going to a party/protest
Believing a rumor
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

13. So, This Means That A person is more likely to join a group if
[Backstrom et al. KDD ‘06]
So, This Means That
A person is more likely to join a group if
she has more friends who are already in the group
friends have more connections between themselves
Thus groups form clusters of tightly connected nodes
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

14. Community Detection How to find communities?
We will work with undirected (unweighted) networks
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

15. Method 1: Strength of Weak Ties
Intuition:
Edge strengths (call volume) in real network
Edge betweenness in real network
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

16. Method 1: Girvan-Newman
Divisive hierarchical clustering based on the notion of edge betweenness:
Number of shortest paths passing through the edge
Girvan-Newman Algorithm:
Undirected unweighted networks
Repeat until no edges are left:
Calculate betweenness of edges
Remove edges with highest betweenness
Connected components are communities
Gives a hierarchical decomposition of the network
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

17. Girvan-Newman: Example12 1 33 49
Need to re-compute betweenness at every step
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

18. Girvan-Newman: Example
Step 1:
Step 2:
Hierarchical network decomposition:
Step 3:
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

19. Girvan-Newman: Results
Communities in physics collaborations
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

20. Girvan-Newman: Results
Zachary’s Karate club: hierarchical decomposition
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

21. We need to resolve 2 questions
How to compute betweenness?
How to select the number of clusters?
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

22. How to Compute Betweenness?
Want to compute betweenness of paths starting at node A
Breadth first search starting from A: 1 2 3 4
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

23. How to Compute Betweenness?
Count the number of shortest paths from A to all other nodes of the network:
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

24. How to Compute Betweenness?
Compute betweenness by working up the tree: If there are multiple paths count them fractionally
The algorithm:
Add edge flows:
-- node flow = ∑child edges
-- split the flow up based on the parent value
Repeat the BFS procedure for each starting node
1+1 paths to H
Split evenly
1+0.5 paths to J
Split 1:2
1 path to K.
Split evenly
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

25. How to select number of clusters?
Define modularity to be Q = (number of edges within groups) – (expected number within groups) Actual number of edges between i and j is Expected number of edges between i and j is
Ki and Kj are degrees of vertices
And m is the total number of edges in the network
Aij = adjacency matrix
Q can be +ve or –ve
+ve values means presence of community structure
m…number of edges
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

26. Modularity: Definition
Q = (number of edges within groups) – (expected number within groups)
Then:
Modularity lies in the range [−1,1]
It is positive if the number of edges within groups exceeds the expected number
0.3<Q<0.7 means significant community structure
m … number of edges
Aij … 1 if (i,j) is edge, else 0
ki … degree of node i
ci … group id of node i
(a, b) … 1 if a=b, else 0
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

27. Modularity: Number of clusters
Modularity is useful for selecting the number of clusters: Q
HW: Question – come up with a graph where GIRVAN-NEWMAN MODULARITY IS NOT UNIMODAL
Why not optimize modularity directly?
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

28. Signed networks
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

29. Today: Signed Networks- +
Networks with positive and negative relationships
Our basic unit of investigation will be signed triangles
First we will talk about undirected nets then directed
Plan for today:
Model: Consider two soc. theories of signed nets
Data: Reason about them in large online networks
Application: Predict if A and B are linked with + or - - +
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

30. Signed Networks Networks with positive and negative relationships
Consider an undirected complete graph
Label each edge as either:
Positive: friendship, trust, positive sentiment, …
Negative: enemy, distrust, negative sentiment, …
Examine triples of connected nodes A, B, C
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

31. Theory of Structural Balance
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
Consistent with “friend of a friend” or “enemy of the enemy” intuition
Inconsistent with the “friend of a friend” or “enemy of the enemy” intuition
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

32. Balanced/Unbalanced Networks
Graph is balanced if every connected triple of nodes has:
all 3 edges labeled +, or
exactly 1 edge labeled +
Unbalanced
Balanced
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

33. Local Balance  Global Factions
Balance implies global coalitions [Cartwright-Harary]
Claim: If all triangles are balanced, then either:
The network contains only positive edges, or
Nodes can be split into 2 sets where negative edges only point between the sets + L R - +
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

34. – + – + + – + Analysis of Balance B D A C E R L
Every node in L is enemy of R – B D + – +
Any 2 nodes
in L are friends +
Any 2 nodes
in R are friends A – + R C E L
Friends of A
Enemies of A
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

35. Example: International Relations
Positive edge: alliance
Negative edge: animosity
Separation of Bangladesh from Pakistan in 1971: US supports Pakistan. Why?
USSR was enemy of China
China was enemy of India
India was enemy of Pakistan
US was friendly with China
China vetoed Bangladesh from U.N. B – –? – P I + – +? C + – U R
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

36. 12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

37. 12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

38. 12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

39. 12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

40. 12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

41. 12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

42. Balance in General Networks
Def 1: Local view
Fill in the missing edges to achieve balance
Def 2: Global view
Divide the graph into two coalitions
The 2 defs. are equivalent! - +
Balanced?
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

43. 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 + edges
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
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

44. Signed Graph: Is it Balanced?
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

45. Positive Connected Components
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

46. Reduced Graph on Super-Nodes
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

47. BFS on Reduced Graph 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 R
Unbalanced!
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

48. Real Large Signed Networks
[Leskovec et al. CHI ‘10]
Real Large Signed Networks
Each link AB is explicitly tagged with a sign:
Epinions: Trust/Distrust
Does A trust B’s product reviews?
(only positive links are visible)
Wikipedia: Support/Oppose
Does A support B to become Wikipedia administrator?
Slashdot: Friend/Foe
Does A like B’s comments?
Other examples:
Online multiplayer games + –
MAKE IT DIRECTED
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

49. Balance in Real Network Data
[Leskovec et al. CHI ‘10]
Balance in Real Network Data
Does structural balance hold? + x –
Triad
Epinions
Wikipedia
Balance
P(T)
P0(T)
0.87
0.62
0.70
0.49 
0.07
0.05
0.21
0.10
0.32
0.08
0.007
0.003
0.011
0.010  + - +
Real data + – + - x x - x x
P(T) … probability of a triad
P0(T)… triad probability if the signs would be random x
Shuffled data
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

50. Global Structure of Signed Nets
Intuitive picture of social network in terms of densely linked clusters
How does structure interact with links?
Embeddedness of link (A,B): Number of shared neighbors
Suggest the “coalitions” structure of networks

51. Global Factions: Embeddedness
[CHI ‘10]
Global Factions: Embeddedness
Epinions
Embeddedness of ties:
Positive ties tend to be more embedded
Positive ties tend to be more clumped together
Public display of signs (votes) in Wikipedia further attenuates this
Wikipedia
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

52. Global Structure of Signed Nets
[CHI ‘10]
Global Structure of Signed Nets
Clustering:
+net: More clustering than baseline
–net: Less clustering than baseline
Size of connected component:
+/–net: Smaller than the baseline + -
Suggest tha “coalitions” structure of networks
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

53. Evolving Directed Networks
[CHI ‘10]
Evolving Directed Networks
New 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.     - + - +     B X A  + - +  -    - + - +    
16 *2 signed directed triads
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

54. Alternate Theory: Status
[CHI ‘10]
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 A
Link A  B means: B has lower status than A
Status and balance give different predictions: + – X X - + - + A B A B
Balance: +
Status: –
Balance: +
Status: –
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

55. + + ?   Theory of Status Edges are directed
[CHI ‘10]
Theory of Status
Edges are directed
Edges are created over time
X has links to A and B
Now, 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 behavior X + + A B ? X   A B
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

56. 16 Types of Context Link (A,B) appears in the context (A,B; X)
[CHI ‘10]
16 Types of Context
Link (A,B) appears in the context (A,B; X)
16 different contextualized links:
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

57. Generative (Receptive) Surprise
[CHI ‘10]
Generative (Receptive) Surprise
Surprise: How much behavior of user deviates from baseline in context t:
(A1, B1; X1),…, (An, Bn; Xn) … instances of contextualized link t
k of them closed with a plus
pg(Ai)… generative baseline of Ai
empirical prob. of Ai giving a plus
Then: generative surprise of triad type t: B X - A
Vs.
what we really do (2 slides):
For every node compute the baseline
Identify all the edges that close same type of triads
Compute surprise B X A 
Std. rnd. var.:
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

58. - + - + Status: Two Examples Two basic examples: X X A B A B
Gen. surprise of A: —
Rec. surprise of B: —
Gen. surprise of A: —
Rec. surprise of B: —
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

59. END END (when I spent 15 min for finishing up the previous lecture)
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

60. Consistency with Status
[CHI ‘10]
Consistency with Status
Determine node status:
Assign X status 0
Based on signs and directions of edges set status of A and B
Surprise is status-consistent, if:
Gen. surprise is status-consistent if it has same sign as status of B
Rec. surprise is status-consistent if it has the opposite sign from the status of A
Surprise is balance-consistent, if:
If it completes a balanced triad X + + +1 A B +1
Status-consistent if: Gen. surprise > 0
Rec. surprise < 0
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

61. Status vs. Balance (Epinions)
[CHI ‘10]
Status vs. Balance (Epinions)
Predictions:
Sg(ti)
Sr(ti) Bg Br Sg Sr t3
t15 t2
t14
t16
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

62. From Local to Global Structure
[WWW ‘10]
From Local to Global Structure
Both theories make predictions about the global structure of the network
Structural balance – Factions
Find coalitions
Status theory – Global Status
Flip direction and sign of minus edges
Assign each node a unique status so that edges point from low to high + + - 3 2 1
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

63. From Local to Global Structure
[WWW ‘10]
From Local to Global Structure
Fraction of edges of the network that satisfy Balance and Status?
Observations:
No evidence for global balance beyond the random baselines
Real 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 baseline
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

64. Predicting Edge Signs Edge sign prediction problem
[WWW ‘10]
Predicting Edge Signs u v + ? –
Edge sign prediction problem
Given 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 edge
Learning method:
Logistic regression
Dataset:
Original: 80% +edges
Balanced: 50% +edges
Evaluation:
Accuracy and ROC curves
Features for learning:
Next slide
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

65. Features for Learning For each edge (u,v) create features:
[WWW ‘10]
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), din(v)
Embeddedness of edge (u,v) - + + + u v - - + -
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

66. Edge Sign Prediction Classification Accuracy:
[WWW ‘10]
Edge Sign Prediction
Epin
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 Epinions
Triad features perform less well for less embedded edges
Wikipedia is harder to model:
Votes are publicly visible
Slash
Wiki
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

67. Balance and Status: Complete Model+ + - - + - + + - - + - + + - - + - + + - - + -

68. 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
12/2/2018
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis,

69. Concluding Remarks
Signed networks provide insight into how social computing systems are used:
Status vs. Balance
Different role of reciprocated links
Role of embeddedness and public display
Sign of relationship can be reliably predicted from the local network context
~90% accuracy sign of the edge

70. Concluding Remarks
More evidence that networks are globally organized based on status
People use signed edges consistently regardless of particular application
Near perfect generalization of models across datasets
Many further directions:
Status difference of nodes A and B [ICWSM ‘10]:
A<B A=B A>B
Status difference (A-B)