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COLOR TESTCOLOR TESTCOLOR TESTCOLOR TEST
Social Networks:Structure and Impact
NICOLE IMMORLICA, NORTHWESTERN U.
Graph Representation
Nodes: =
Edges: =
New Testament
Visualization from ManyEyes
New Testament
Scientific Collaboration
Scientific Collaboration
World Wide Web
World Wide Web
Seattle
Honolulu
thesis:
networks are basic structure upon which society operates
spread of disease.
diffusion of ideas.
models of social networks
A theory must be tempered with reality.
– Jawaharlal Nehru, Indian Prime Minister
which is the best theory?
grid star cycle “tree”
“reality”:
1. power-law degree dist.Popular people are really
popular. Most people aren’t.
1
7
Degrees
2
8
3
4
5
6
Definition: The degree of a node vi is the number of nodes vj such that there’s an edge ei,j between them.
Degree v8 = 4.
Degree Distributions
0 1 2 3 4
0
1/4
1/2
3/4
5 6 7 8
1
Frequency
# of friend (degree)
A cycle?
Cycle deg. dist.
A star?
Star deg. dist.
Power-law Degree Dist.
0 1 2 3 4
0
1/3
1/2
5 6 7 8
1
Frequency
Degree
Power-law: P(∂) = c∂-α
Log-Log PlotsLog (Frequency)
Log (Degree)
Power-law: P(∂) = c∂-α
log (P(∂)) = log (c∂-α)= log (c) – α∙log (∂)
Straight line on a log-log plot!
Example: Web Graph In-Degree
Power law exponent: α = 2.09
Example: New York Facebook
Lognormal is better fit.
“reality”:
2. small diameterMost people know people who know people who know people who … know Obama.
Paths
1
2
8
3
7
4
5
6
Definition: A path is a sequence of nodes (v1, …, vk) such that for any adjacent pair vi and vi+1, there’s an edge ei,i+1 between them.
Path (v1,v2,v8,v3,v7)
Paths
“I know someone who knows someone who knows you.”
Path length
Definition: The length of a path is the number of edges it contains.
Path (v1,v2,v8,v3,v7)has length 4.
1
2
8
3
7
4
5
6
Distance
Definition: The distance between nodes vi and vj is the length of the shortest path connecting them.
The distance between v1 and v7 is 3.
1
2
8
3
7
4
5
6
Famous distances
nodes = {mathematicians}edges = if 2 mathematicians co-author a paper
Erdos number = distance between mathematican and Erdos
Paul Erdos number
Famous distances
Erdos number of …
http://www.oakland.edu/enp/
= 4
Diameter
Definition: The diameter of a graph is the maximum shortest-path distance between any two nodes.
The diameter is 3.
1
2
8
3
7
4
5
6
“longest shortest path”
The trace of a disease
1. Initially just one node is infected2. All nodes with an infected friend get infected
Day 0Day 1Day 2
The trace of a disease
# days ≤ diameter ≤ twice # days
Day 0Day 1Day 2
Because the trace defines the distance from the initially infected person to the last infected person.
Because there’s a path in the trace between any two people going through the initially infected person.
Six degrees of separation
The diameter of a social network is typically small.
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.
Short paths exist (and people can find them!).
Diameter
“longest shortest path”
grid star tree
√n 2 log n(for const. deg.)
Diameter
For the population of the US,
grid star tree
2,000 2 6
“reality”:
3. high clusteringMost people’s friends are
themselves friends.
Clustering Coefficient
“fraction of triangles bt. all connected triples”
ZERO …. > ZERO
grid star cycle “tree”
Why do we see these realities?
1. High clustering coefficient… triadic closure – tend to know your friend’s friends
2. Power-law degree distribution… popular people attract proportionally more friends
3. Low diameter… there is an element of chance to whom we meet
preferential attachment:
People are imitators. They make the choices their
friends make.
Preferential Attachment
1. People join network in order 1, 2, …, N2. When join, person t chooses friend by
a) With probability p, pick person t’ uniformly at random from 1, …, t-1
b) With probability (1-p), pick person t’ uniformly at random and link to person that t’ links too
Imitation
The rich get richer
2 b) With prob. (1-p), pick person t’ uniformly at random and link to person that t’ links too
1/43/4
The rich get richer
2 b) With prob. (1-p), pick person t’ uniformly at random and link to person that t’ links too
Equivalently,
2 b) With probability (1-p), pick a personproportional to in-degree and link to him
preferential attachment:
The degree distribution follows a power-law.
– Albert-L szl Barabasi and R ka Albert (1999): a� o̒� e�Emergence of Scaling in Random Networks, Science
can we use network structure to explain success of various technologies/fads/rumors?
impact of social networks
1. spread technology.impact of social networks
key: = Instant Msg A user= Instant Msg B user
f ( new ) > f ( old )
… if enough friends use !
1. relative quality: f ( new ) >> f ( old )
How can we convincepeople to use ?
1. relative quality, 2. compatibility:
How can we convincepeople to use ?
= user of both technologies
Lessons [IKMW’05].
1. Inferior incumbents can always survive invasion of slightly superior competitors by adopting limited levels of compatibility.
2. This happens through the formation of bi-lingual buffers.
2. generate revenue.impact of social networks
$10$0
$11
$8
Lesson [HIMM’11].
When selling a socially-enabled product in a social network, must subsidize influential nodes.
3. encourage cooperationimpact of social networks
nice guys have more friends
Lesson [ILR’10].
Conclusion
• Social networks have predictable structure– Power-law degree distribution
(preferential attachment model)– Low (logarithmic) diameter– High clustering coefficients
• Social networks impact many social processes– Spread disease/technology– Generate revenue– Sustain cooperation