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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