Social Networks 101 P ROF. J ASON H ARTLINE AND P ROF. N ICOLE I MMORLICA

Lecture Six: The mathematics of decentralized search

Small world phenomenon Milgram’s experiment (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 Problem. How can I get this message from me to the far-away target? Solution. Pass message to a friend. closer (sub)

Time for

Scales of resolution Each new scale doubles distance from the center.

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

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

Trace of route

Analysis old dist. 1242j2j 2 j+1 new dist.

Distance is cut in half every step!

Analysis 1.Original distance is ? 2.Distance is cut in half every step (at least). 3.Number of steps is ? at most n. at most log n.

And in real life …

Strength of weak ties 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?

Job search Granovetter: Most people learn about jobs through personal friends, who are mere acquaintances!

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 Which is more likely?

How will this network evolve?

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

Explaining triadic closure 1.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 closure 2.Incentive. If your best friend hates your girlfriend, it stresses both relationships.

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.

Does this happen in real graphs?

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 Clustering coefficient = 0.14 Density of edges =

Bridges 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

Weak Versus Strong Ties

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”: v u Suppose edge v-u is a local bridge and it is a strong tie w Then u-w must exist because of Strong Triadic Closure But then v-u is not a bridge

Conclusion Local bridges are necessarily weak ties. Structural explanation as to why job information comes from acquaintances.

Next time Structural holes and balance