4. Strength of Weak Ties
NETWORKS, CROWDS AND MARKETS - CHAPTER 3
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5. Granovetter’s experiment
▪How do people find out about new jobs?
Ans: People find the information through personal contacts
▪But contacts were often acquaintances rather than close friends
Why is it that acquaintances are
most helpful?
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6. Triadic Closure
Reasons for triadic closure:
If B and C have a friend D in common,
then:
▪B is more likely to meet C
(since they both spend time with D)
▪B and C trust each other
(since they have a friend in common)
▪D has incentive to bring B and C together
(since it is hard for D to maintain two disjoint
relationships)
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8. Local Bridge – Global property
An edge joining two nodes A and B in a
graph is a local bridge if its endpoints A
and B have no friends in common –
deleting the edge would increase the
distance between A and B to a value
strictly more than 2.
▪Span of a local bridge is the distance
its endpoints would be from each
other if the edge were deleted
▪An edge is a local bridge when it does
not form the side of any triangle in the
graph
▪Brings new information / trend
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9. Strong and Weak Ties – Local property
Friend–Acquaintance
dichotomy
Strong – Weak ties
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10. Strong Triadic Closure property
▪If the node has strong ties to two neighbors, then these neighbors must have at
least a weak tie between them
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11. Local Bridge & Weak Ties
If a node A in a network satisfies the Strong Triadic Closure property and is
involved in at least two strong ties, then any local bridge it is involved in must be
a weak tie
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12. Strength of Weak Ties
Answer
Two perspectives on links/friendships:
▪Structural: Friendships span different
parts of the network
▪Interpersonal: Friendship between
two people is either strong or weak
Explanation
▪Structure: Local bridges spanning
different parts of the network are
socially weak
▪Information: Local bridge allow you to
gather information from different parts
of the network
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13. Granovetter’s experiment
Granovetter’s theory leads to the following
conceptual picture of networks
▪Networks are composed of tightly connected
sets of nodes
▪Sets of nodes with lots of internal connections
and few external ones
▪ Communities
▪ Clusters
▪ Groups
▪ Modules
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14. How to find network communities?
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16. Graph Partitioning
Input: Undirected graph 𝐺(𝑉, 𝐸)
Bi-partitioning task:
Output: Divide vertices into two disjoint groups 𝑨, 𝑩
Questions:
◦ How can we define a “good” partition of 𝑮?
◦ How can we efficiently identify such a partition?
1
3
2
5
4 6
A B
1
3
2
5
4
6
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17. Graph Partitioning
What makes a good partition?
▪ Maximize the number of within-group
connections
▪ Minimize the number of between-group
connections
How to identify good partition?
▪Divisive methods – top down
approach
▪Agglomerative methods – bottom up
approach
17
1
3
2
5
4
6
A B
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19. Community Detection – Divisive
Approach
2. What principles lead us to remove the 7-8
edge first?
1. Which edge will be removed first?
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20. Community Detection – Divisive
Approach
Edge betweenness: Number of shortest paths passing over the edge
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25. Girvan-Newman Algorithm
▪Divisive hierarchical clustering based on the notion of edge betweenness
▪Undirected unweighted networks
▪Algorithm:
• Repeat until no edges are left:
o Calculate betweenness of edges
o Remove edges with highest betweenness
▪ Connected components are communities
▪ Gives a hierarchical decomposition of the network
[Girvan-Newman ‘02]
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28. We need to resolve 2
questions
1. HOW TO COMPUTE BETWEENNESS?
2. HOW TO SELECT THE NUMBER OF CLUSTERS?
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29. How to Compute Betweenness?
Want to compute
betweenness of paths
starting at node 𝐴
Step 1: Breath first
search starting from 𝐴
0
1
2
3
4
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30. How to Compute Betweenness?
Step 2: Count the number of shortest paths from 𝐴 to all other nodes of the network:
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31. How to Compute Betweenness?
1 path to K.
Split evenly
1+0.5 paths to J
Split 1:2
1+1 paths to H
Split evenly
Algorithm:
•Add edge flows:
-- node flow = 1+∑child edges
-- split the flow up based on the parent value
• Repeat the BFS procedure for each starting node
𝑈
Step 3: Determine the amount of flow from A to all other nodes that use each edge
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32. How to select the number of clusters?
Define: Modularity 𝑸
A measure of how well a network is partitioned into communities
Given a partitioning of the network into groups 𝒔 𝑺:
Q ∑s S [ (# edges within group s) – (expected # edges within group s) ]
Null / Erdos-Renyi model
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33. Modularity values take range [−1,1]
0.3-0.7<Q means significant community structure
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35. Community Detection – Agglomerative
Approach
▪Modularity optimization is NP Hard
▪Greedy Heuristic - Trivial clustering with each node in its own cluster
▪Repeat:
▪ Merge the two clusters that will increase the modularity by the largest amount
▪ Stop when all merges would reduce the modularity
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