Small World Social Networks

With slides from Jon Kleinberg, David Liben-Nowell, and Daniel Bilar

What is a social network?

A set of relationships between entities

Social Network Analysis

[Social network analysis] is grounded in the observation that social actors are interdependent and that the links among them have important consequences for every individual [and for all of the individuals together]. ...

[Relationships] provide individuals with opportunities and, at the same time, potential constraints on their behavior. ...

Social network analysis involves theorizing, model building and empirical research focused on uncovering the patterning of links among actors. It is concerned also with uncovering the antecedents and consequences of recurrent patterns. [Linton C. Freeman, UC-Irvine]

Representing a Social Network

a set V of n nodes or vertices,usually denoted {v1, …, vn}

a set E of m edges between nodes,usually denoted {ei,j}

edge e8,3

node v2

Examples of Social Networks

Nodes are high-school students. Boys are red, Girls are blue…What is the meaning of a bidirectional edge?

Meaning of Bidirected Edges

“I date(d) you.”

Paths

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 a directed edge ei,i+1 between them.

Path (v1,v2,v8,v3,v7)

Paths

“I date(d) someone who date(d) someone who date(d) you.”

Examples of Social Networks

Nodes are high-school students. Boys are red, Girls are blue…What is the meaning of a bidirectional edge?

Path length

4

5

6

Definition: The length of a path is the number of edges it contains.

Path (v1,v2,v8,v3,v7)has length 4.

Distance

5

6

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.

Famous distances

nodes = {actors}edges = if two actors star in same film

Kevin Bacon number = distance between actor and

Bacon

Kevin Bacon number

The Kevin Bacon Game

Invented by Albright College students in 1994: Craig Fass, Brian Turtle, Mike Ginelly

Goal: Connect any actor to Kevin Bacon, by linking actors who have acted in the same movie.

Oracle of Bacon website uses Internet Movie Database (IMDB.com) to find shortest link between any two actors:

http://oracleofbacon.org/

Title

Data

Famous distances

Math PhD genealogies

Famous distances

nodes = {mathematicians}edges = if 2 mathematicians co-author a

paper

Erdős number = distance between mathematican and Erdos

Paul Erdős number

Erdős NumbersErdős wrote 1500+ papers with 507 co-authors. Number of links required to connect scholars to Erdős, via co- authorship of papersWhat type of graph do you expect?Jerry Grossman (Oakland Univ.) website allows mathematicians to compute their Erdős numbers: http://www.oakland.edu/enp/

Connecting path lengths, among mathematicians only:

avg = 4.65 max = 13

Famous distances

Erdős number of …

Famous distances

Erdős number of …

Fan Chung F.T. Leighton P.T. MetaxasErdos

Famous distances

Erdos number of …

if you publish with me!

Diameter

5

6

Definition: The diameter of a graph is the maximum shortest-path distance between any two nodes.

The diameter is 3.

Six degrees of separation

The diameter of a social network is typically small.

Milgram: Six Degrees of Separation

296 People in Omaha, NE, were given a letter,asked to try to reach a stockbroker in Sharon, MA,via personal acquaintances20% reached target

average number of “hops” in the completed chains = 6.5Why are chains so short?

“Random Graphs have small diameter”Do they?

Why are Chains so Short?

Maybe exponential growth of acquaintances…@d=1: Most people know at least 100 others@d=2: Through their friends: 10K@d=3: Through their friends’ friends: 1M@d=4: Through their friends’ friends’ friends: 100M@d=k: 10^k

Not so fast…

Your friends mostly know each other…In high school self-reported friendships, clusters based on race (left-right)and age (top-bottom)

Homophily: Your friends are similar to you!Pr [two of your friends are friends] is high

Social networks have vertices with high clustering coefficient(how much its neighborhood

resembles a clique)

So, exponential growth does not explain itWe want a model with small diameter and large clustering coefficient

Watts/Strogatz: Rewire Ring Lattice

Proposed a model (ring lattice)with small diameter and large clustering coefficient:Put people on circleconnect each to x closest neighbors;with prob. p, rewire each connection randomlyResult: Yes, short chains exist for p>0.1!

p =0.0 p =0.1 p =1.0

Ok, short chains exist, but…

Will people be able to find the short chains?Milgram showed that people were able to find them.

Kleinberg [2000]: No search strategy in a Watts/Strogatz network,based only on local information,can find short chains…

Kleinberg’s Rewire Grid

Now you can find short paths!

The effect of distance

Searching with local information gets more efficient as increases up to 2, then gets worse again!In fact, it finds short paths in logarithmic time!Theory and practice agree!

Translated into English?

“Distance scales”Count friends within log distances:1 - 1010 - 100100 - 10001000 - 10000 …When = 2,nodes have the same volume of links to each distance scale

The Power of Long Distance Relations

Probability of friendship is falling offlike the square of the distance!Geographic location isa primary reason for selecting next person in chainWe have eventually understood Milgram’s experiment

But does this explains what happens on the internet?

(It depends on how you define distance:see Liben-Nowell’s paper)