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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Prof. Jason Hartline and Prof. Nicole Immorlica
The mathematics of
Ask someone to pass a letter to another person via friends knowing only the name, address, and occupation of the target.
Problem. How can I get this message
from me to the far-away target?
Solution. Pass message to a friend.
Each new scale doubles distance from the center.
Suppose each person has a long-range friend in each scale of resolution.
Algorithm. Pass the message to your farthest friend that is to the left of the target.
at most n.
at most log n.
Long-range links are often casual acquaintances,
… but are very important for search and other network phenomena
Where do the best job leads come from: your close friends or your acquaintances?
Granovetter: Most people learn about jobs through personal friends,
who are mere acquaintances!
Idea. Weak ties are likely to link distant parts of the network and so are particularly well-suited to information flow.
Which is more likely?
Triadic Closure: If two nodes have common neighbor, there is an increased likelihood that an edge between them forms.
2. Incentive. If your best friend hates your girlfriend, it stresses both relationships.
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
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
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.
Structural holes and balance