CSCI 2350: Social & Economic Networks Are all the links in a - - PDF document

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CSCI 2350: Social & Economic Networks

Are all the links in a network the same? What is the effect of different types of links? “The strength of weak ties”

Mohammad T . Irfan

Networks effects

u Measles outbreak (CA, December’14—Feb’15)

Image: Fox 40

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Granovetter’s study (1960s)

u Acquaintances, not friends, hold critical

information about job opportunities

The strength of weak ties

u Connect local/interpersonal properties to

global/structural properties

u Mathematically prove this local to global

connection

u Show that the “critical” ties are actually

weak ties

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Triadic closure

A B C B and C are very likely to become friends

Local bridge

u An edge whose

endpoints do not have a common friend

ó An edge which is not a

side of a triangle

ó An edge whose deletion

causes the distance between its endpoints to be > 2

Under some condition, every local bridge must be a weak tie

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Big data (real-world) and the strength of weak ties

Weak ties in Facebook (Marlow et al., 2009)

Colleagues High school friends

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Weak ties in Facebook (cont...)

Weak ties (passive network) in FB facilitate newsfeed

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Twitter (Huberman et al., 2009)

Are weak ties really powerful?

Gladwell: “weak ties seldom lead to high-risk activism”

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Counter-argument Counter-argument

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Counter-argument

Community detection in social networks using local bridges

Section 3.6 (Advanced)

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Coauthorship network (Newman- Girvan, 2004)

Idea

u Delete local bridges one after another u Get connected components

u close-knit communities

u Divisive graph partitioning

(as opposed to agglomerative)

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But… which local bridge to delete first? Also… what if there’s no local bridge?

Need some form of “betweenness” measure for the edges!

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University ranking (unrelated to current topic)

http://www.wired.com/2015/02/academic-hiring-uphill-battle

Eigenvector centrality

Solution: Girvan-Newman algorithm (2002)

u Calculate the betweenness of each edge u Successively delete the edge(s) with the highest

betweenness (and recalculate betweenness)

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Betweenness of an edge

u Every node is sending 1 gallon of water to

every other node

u Water will only flow through the shortest

paths

u Equally distributed among multiple shortest paths

u Betweenness of an edge

= Quantity of water flowing through it

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How to compute the betweenness of an edge?

A B C D E F G H

Algorithm

u A. For each node X do the following:

1.

Do BFS starting with X

2.

Calculate the # of S.P . from X to every other node

3.

Calculate the quantity of water flow through each edge u B. Betweenness of an edge

= sum of all water flow (over all the BFS)

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BFS starting at node A

A B C D E F G H Remember, BFS needs to be done starting at each node!

# of S.P . from A to every node

A B C D E F G H This is the # of S.P . from A to G, not distance from A to G 1 1 1 2 1 3 3 Formula: # of S.P . from A to E = Sum of the # of S.P . from A to each friend of E in the previous level

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Calculate water flow on each edge

A B C D E F G H 1 1 1 2 1 3 3 Node A has sent 1 gallon of water to every other node. Water flows only on S.P .– splits evenly on multiple S.P .

2/3 gal. 1/3 gal.

Why?

Calculate water flow on each edge

A B C D E F G H 1 1 1 2 1 3 3 Node A has sent 1 gallon of water to every other node. Water flows only on S.P .– splits evenly on multiple S.P .

2/3 gal. 1/3 gal.

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Calculate water flow on each edge

A B C D E F G H 1 1 1 2 1 3 3 Node A has sent 1 gallon of water to every other node. Water flows only on S.P .– splits evenly on multiple S.P .

2/3 gal. 1/3 gal. 2/3 gal. 1/3 gal. 7/6

• gal. 7/6

gal. 5/3 gal. 13/6 gal. 23/6 gal. 1 gal. Why 7/6 gal. from D to E?

• Ans. E consumes 1 gallon and

passes 2/3 + 2/3 = 4/3 gallons below it. So, E needs a supply

• f 1 + 4/3 = 7/3 gallons, which

is split evenly into two S.P . from A to E.

We are not done yet! For each edge, we need to sum up the water flow from each BFS.

Karate club (Zachary, 1977)