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Social Networks How important?
How the people you are connected to can influence you
Study of influence In this lecture
Influential models Two famous social networking sites
Facebook Early timeline
Jan. ‘04: “thefacebook” for Harvard students Mar. ‘04: Expanded to other schools ‘05: changed name to “Facebook” Sept. ‘06: Opened to all 13 and older
Largest social networking site today ’08: Hit 100M users worldwide ’12: Hit 1B users worldwide
What is a “link” between FB users?
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Twitter Timeline and growth
Mar. ’06: Created and launched in July. ’06 In just six years, more than 500M users Mar. ’13: More than 400M tweets each day
Main features One-way “following” relationships
Spike in usage during major events Summer ’11 US east-coast earthquake 2013 largest “tweeted”: Boston Marathon Bombings
(27.8M), NBA Finals (26.7M), Super Bowl (24.1M)
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Who is “important?” How to analyze “opinions”? How to measure influential power?
(analysis) Companies charting influential power on
Twitter: # followers, # retweets … Companies attempting to put together FB
graph How to leverage this to influence?
(synthesis) Marketing campaigns “buy off” influential
people and seed them with products
In this lecture, we will focus on how to quantify a node’s importance?
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Measuring node important Social graph
Nodes: people Links: relationship between nodes
Not clear – assume nodes know eachother
Links can be Bidirectional (Facebook) Unidirectional (Twitter)
Which node is the most important? Three different approaches
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Degree centrality Centrality: measure of node important
The number of connected nodes Degree = 3 (Dana, Evan, Cara) Degree = 2 (Anna, Frank) Degree = 1 (Ben)
Reasonable? Cara (causes network partition)
versus (Dana or Evan) Anna versus Frank
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Closeness centrality Degree: not count distance between nodes in the graph
Distance = the number of links traversed over the path Path: set of links connecting 2 nodes (referred as the nodes
visited) Example
A path from Ben to Frank could be (B,A,C,E,D,F)
The distance of this path is 5 (5 links).
Shortest path between 2 nodes Example
The shortest path from Ben to Frank is (B,A,C,D,F) or (B,A,C,E,F) The distance of this path is 4 (4 links).
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Closeness centrality (2) Closeness centrality for node:
Find shortest path lengths to others Take average of these Closeness = 1/average shortest path length
Cara:
Dana
Others
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Closeness centrality (3) Ranking:
Clearly more reasonable than degree centrality Cara and Anna promoted
Reasonable? Dana and Evan don’t hold graph together
Why should Anna be less important?
Cara should be even more “central” than Dana and Evan She is the most vital to connectivity
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Betweenness centrality How to bring “connectivity” into important
Message-passing examples Anna to Frank Ben to Dana
Betweenness centrality for node: Find shortest path(s) for each pair Award other nodes “points” for being on shortest
path(s) Points awarded to node for a pair is fraction of shortest
paths between pair the node is on Add up number of points for each node
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Betweenness centrality (2) Cara:
For each pair, consider two questions: How many shortest paths are there between the pair of
people? How many of these shortest paths contain Cara?
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Betweenness centrality (3) Dana:
Others
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Betweenness centrality (4) Ranking:
Now, Cara is by far most important
Now, Anna is more important than Dana and Evan
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Summary of different centrality measures
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Contagion How do social relationships influence adoption of
products?
Star network state-0: “N”, have not adopted state-1: “Y”, have adopted Enough social influence for center node to flip (adopt
product)?
Flipping threshold Fraction of neighbors that must have flipped for node to flip Hard to estimate: depends on product and person Assume we know it, and that it’s the same for each node
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Contagion process Each iteration, go through person by person to
see if thresholds have been met
Threshold: 50%
Time 1
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Contagion process (2) Time 2
Time 3
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Cluster density Cluster: any group of nodes that have
connections among themselves
Cluster density For each node, find fraction of neighbors inside the
cluster Smallest of these fractions is the density
Need at least the threshold outside of the cluster in order to penetrate it Only 40% outside in this case!
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Marketing strategies How to increase number who adopt?
Lower threshold Break clusters: cut social ties within Seed node in a cluster
Seeding is most plausible Which to seed in order to guarantee … Maximum number will flip? Minimum time to reach new equilibrium?
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Marketing strategies (2) Seeding one node
Should be most important (e.g., celebrities) What is most important node?
Using centrality measures May be under budget constraint
Seeding multiple nodes Don’t want just two most central Must consider combined influence
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Marketing strategies (3) Consider original social graph One node: seed Cara Two nodes
Seed Cara and Frank or Anna and Dana (don’t need Cara!)
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Summary Social networking sites
Facebook and Twitter Charting influential power Marketing campaigns
Measuring influence (analysis) Centrality measures
Influencing adoption (synthesis) Contagion
Reference Brinton, Christopher; Chiang, Mung (2013-06-
10). Networks Illustrated: 8 Principles Without Calculus (Kindle Locations 1119-1123). Edwiser Scholastic Press. Kindle Edition.
https://en.wikipedia.org/wiki/Social_network_analysis
https://en.wikipedia.org/wiki/Social_network
