Complex Systems Models in the Social Sciences

(Lecture 2)

Daniel Martin KatzMichigan State University

College of Law

Introduction to Network Analysis

Introduction to Network Analysis

What is a Network?

What is a Social Network?

Mathematical Representation of theRelationships Between Units such asActors, Institutions, Software, etc.

Special class of graph Involving Particular Units and Connections

Introduction to Network Analysis

Interdisciplinary Enterprise

Applied Math(Graph Theory, Matrix Algebra, etc.)

Statistical Methods

Social Science

Physical and Biological Sciences

Computer Science

Social Science

For Images and Links to Underlying projects:

http://jhfowler.ucsd.edu/

3D HiDef SCOTUS Movie

Co-Sponsorship in CongressSpread of Obesity

Hiring and Placement of Political Science PhD’s

Social Science

The 2004 Political Blogosphere (Adamic & Glance)

High School Friendship(Moody)

Roll Call Votes in Congress(Mucha, et al)

Physical and Biological Sciences

For Images and Links to Underlying projects:

http://www.visualcomplexity.com/vc/

Computer Science

Mapping

of the

Code

Networks are waysto represent dependanciesbetween software

Computer Science

Internet is one ofthe largest

known and most important networks

Computer Science

Mappingthe

Iranian Blogsphere

http://cyber.law.harvard.edu/publications/2008/Mapping_Irans_Online_Public

Primer on Network

Terminology

Terminology & Examples

Institutions

Firms

States/Countries

Actors

NODES

Other

Example: Nodes in an actor- based social Network

Alice

Bill

Carrie

David

Ellen

How Can We Represent The Relevant Social Relationships?

Terminology & Examples

Edges

Alice

Bill

Carrie

David

Ellen

Arcs

Terminology & Examples

Edges

Alice Bill

Carrie

David

Ellen

Arcs

Terminology & Examples

Edges Alice BillCarrie

David

Ellen

Arcs

Terminology & Examples

Alice Bill

David

Carrie

Ellen

A Full Representation of the Social Network

Terminology & Examples

Bill

David

Carrie

Ellen

Terminology & Examples

Alice

A Full Representation of the Social Network(With Node Weighting)

Bill

David

Carrie

Ellen

A Full Representation of the Social Network(With Node Weighting and Edge Weighting)

Terminology & Examples

Alice

A Survey Based Example

“Which of the above individuals do you consider a close friend?”

Image We Surveyed 5 Actors:

(1) Daniel, (2) Jennifer, (3) Josh, (4) Bill, (5) Larry

From an EdgeList to Matrix

1 2 3 4 5 --------------------------- Daniel (1) 0 1 1 1 1 Jennifer (2) 1 0 1 0 0 Josh (3) 0 1 0 1 1 Bill (4) 0 0 0 0 0 Larry (5) 1 1 1 1 0

*Directed Connections (Arcs) 13

1 21 31 41 52 12 33 43 53 25 15 45 35 2

ROWS è COLUMNS

*How to Read the Edge List: (Person in Column 1 is friends with Person in Column 2)

1 2 3 4 5 --------------------------- Daniel (1) 0 1 1 1 1 Jennifer (2) 1 0 1 0 0 Josh (3) 0 1 0 1 1 Bill (4) 0 0 0 0 0 Larry (5) 1 1 1 1 0

From a Survey to a Network

A Quick Example of a Dynamic Network

United States Supreme Court

To Play Movie of the Early SCOTUS Jurisprudence: http://vimeo.com/9427420

Some Other Examples

of Networks

Consumer Data

Knowing Consumer Co-Purchases can help ensure that “Loss Leader” Discounts can be recouped with other purchases

Corporate Boards

http://www.theyrule.net/

Transportation Networks

We might be interested in developing transportation systems that are minimize

total travel time per passenger

Power Grids

We might be interested in developing Power Systems that are Globally Robust to Local Failure

Campaign Contributions Networks

http://computationallegalstudies.com/tag/110th-congress/

Some Recent Network Related

Publications

Special Issue: Complex systems

and NetworksJuly 24, 2009

Special 90th anniversary Issue:

May 7, 2007

History ofNetwork Science

The Origin of Network Science is Graph Theory

The Königsberg Bridge Problem the first theorem in graph theory

Is It Possible to cross each bridge each and only once?

The Königsberg Bridge Problem

Leonhard Euler proved that this was not possible

Is It Possible to cross each bridge each and only once?

Eulerian and Hamiltonian Paths

Eulerian path: traverse each edge exactly once

If starting point and end point are the same: only possible if no nodes have an odd degree

each path must visit and leave each shore

If don’t need to return to starting pointcan have 0 or 2 nodes with an odd degree

Hamiltonian path: visiteach vertex exactly once

ModernNetwork Science

Moreno, Heider, et. al. and the Early Scholarship

Focused Upon Determining the Manner in Which Society was Organized

Developed early techniques to represent the social world Sociogram/ Sociograph

Obviously did not have access to modern computing power

Stanley Milgram’s Other Experiment

Milgram was interested in the structure of society

Including the social distance between individuals

While the term “six degrees” is often attributed to milgram it can be traced to ideas from hungarian author Frigyes Karinthy

What is the average distance between two individuals in society?

Stanley Milgram’s Other Experiment

NE

MA

Six Degrees of Separation?

NE

MA

Target person worked in Boston as a stockbroker

296 senders from Boston and Omaha.

20% of senders reached target.

Average chain length = 6.5.

And So the term ... “Six degrees of Separation”

Six Degrees

Six Degrees is a claim that “average path length” between two individuals in society is ~ 6

The idea of ‘Six Degrees’ Popularized through plays/movies and the kevin bacon game

http://oracleofbacon.org/

Six Degrees of Kevin Bacon

Visualization Source: Duncan J. Watts, Six Degrees

Six Degrees of Kevin Bacon

But What is Wrong with Milgram’s Logic?

150(150) = 22,500

150 3 = 3,375,000

150 4 = 506,250,000

150 5= 75,937,500,000

The Strength of ‘Weak’ Ties

Does Milgram get it right? (Mark Granovetter)

Visualization Source: Early Friendster – MIT Network

www.visualcomplexity.com

Strong and Weak Ties (Clustered

v. Spanning)

Clustering ---- My Friends’ Friends are also likely to be friends

So Was Milgram Correct?

Small Worlds (i.e. Six Degrees) was a theoretical and an empirical Claim

The Theoretical Account Was Incorrect

The Empirical Claim was still intact

Query as to how could real social networks display both small worlds and clustering?

At the Same time, the Strength of Weak Ties was also an Theoretical and Empirical proposition

We Will Continue Here Next Time