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Social Networks 101. Prof. Jason Hartline and Prof. Nicole Immorlica. Lecture Six : The mathematics of decentralized search. Small world phenomenon. Milgram’s experiment (1960s ).

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social networks 101

Social Networks 101

Prof. Jason Hartline and Prof. Nicole Immorlica


Lecture Six:

The mathematics of

decentralized search

small world phenomenon
Small world phenomenon

Milgram’sexperiment (1960s).

Ask someone to pass a letter to another person via friends knowing only the name, address, and occupation of the target.

how to route
How to route

Problem. How can I get this message

from me to the far-away target?

Solution. Pass message to a friend.




Time for

Math Corner

scales of resolution
Scales of resolution

Each new scale doubles distance from the center.

long range links
Long-range links

Suppose each person has a long-range friend in each scale of resolution.

how to route1
How to route

Algorithm. Pass the message to your farthest friend that is to the left of the target.


new dist.






old dist.

  • Original distance is ?
  • Distance is cut in half every step (at least).
  • Number of steps is ?

at most n.

at most log n.

strength of weak ties
Strength of weak ties

Long-range links are often casual acquaintances,

… but are very important for search and other network phenomena

job search
Job search

Granovetter: Most people learn about jobs through personal friends,

who are mere acquaintances!

weak ties
Weak ties

Idea. Weak ties are likely to link distant parts of the network and so are particularly well-suited to information flow.

social network structure
Social network structure

Which is more likely?


Triadic Closure: If two nodes have common neighbor, there is an increased likelihood that an edge between them forms.

explaining triadic closure
Explaining triadic closure
  • Opportunity. If you spend a lot of time with your best friend and your girlfriend, there is an increased chance they will meet.
explaining triadic closure1
Explaining triadic closure

2. Incentive. If your best friend hates your girlfriend, it stresses both relationships.

explaining triadic closure2
Explaining triadic closure

3.Homophily. If you have things in common with both your best friend and your girlfriend, they have things in common too.


Definition: The clustering coefficient of a node v is the fraction of pairs of v’s friends that are connected to each other by edges.

Clustering Coefficient = 1/2

The higher the clustering coefficient of a node, the more strongly triadic closure is acting on it

collaboration graph
Collaboration graph

Clustering coefficient = 0.14

Density of edges = 0.000008



An edge is a bridge if deleting it would cause its endpoints to lie in different components


Local bridges

An edge is a local bridge if its endpoints have no common friends


Definition: Node v satisfies the Strong Triadic Closure if, for any two nodes u and w to which it has strong ties, there is an edge between u and w (which can be either weak or strong)

This graph satisfies the strong triadic closure


Claim: If node v satisfies the Strong Triadic Closure and is involved in at least two strong ties, then any local bridge it is involved in must be a weak tie

Argument “by contradiction”:

Suppose edge v-u is a local bridge and it is a strong tie


Then u-w must exist because of Strong Triadic Closure



But then v-u is not a bridge


Local bridges are necessarily weak ties.

Structural explanation as to why job information comes from acquaintances.

next time
Next time

Structural holes and balance