1. Complex Systems Models
in the Social Sciences
(Lecture 2)
daniel martin katz
illinois institute of technology
chicago kent college of law
@computationaldanielmartinkatz.com computationallegalstudies.com

3. Introduction to
Network Analysis

4. Introduction to
Network Analysis
What is a Network?
What is a Social Network?
Mathematical Representation of the
Relationships Between Units such as
Actors, Institutions, Software, etc.
Special class of graph Involving
Particular Units and Connections

5. Introduction to
Network Analysis
Interdisciplinary Enterprise
Applied Math
(Graph Theory, Matrix Algebra, etc.)
Statistical Methods
Social Science
Physical and Biological Sciences
Computer Science

6. Social Science
For Images and Links to
Underlying projects:
http://jhfowler.ucsd.edu/
3D HiDef SCOTUS Movie
Co-Sponsorship in Congress
Spread of Obesity
Hiring and Placement of
Political Science PhD’s

7. Social Science
The 2004 Political Blogosphere
(Adamic & Glance)
High School Friendship
(Moody)
Roll Call Votes in Congress
(Mucha, et al)

8. Physical and
Biological
Sciences
For Images and Links to
Underlying projects:
http://www.visualcomplexity.com/vc/

9. Computer Science
Mapping
of the
Code
Networks are ways
to represent
dependancies
between software

10. Computer Science
Internet is one of
the largest
known and most
important networks

11. Computer Science
Mapping
the
Iranian
Blogsphere
http://cyber.law.harvard.edu/publications/2008/Mapping_Irans_Online_Public

12. Primer on
Network
Terminology

13. Terminology & Examples
Institutions
Firms
States/Countries
Actors
NODES
Other

14. Example: Nodes in an actor-
based social Network
Alice
Bill
Carrie
David
Ellen
How Can We Represent The
Relevant Social Relationships?
Terminology & Examples

15. Edges
Alice
Bill
Carrie
David
Ellen
Arcs
Terminology & Examples

16. Edges
Alice Bill
Carrie
David
Ellen
Arcs
Terminology & Examples

17. Edges Alice Bill
Carrie
David
Ellen
Arcs
Terminology & Examples

18. Alice Bill
David
Carrie
Ellen
A Full Representation
of the Social Network
Terminology & Examples

19. Bill
David
Carrie
Ellen
Terminology & Examples
Alice
A Full Representation
of the Social Network
(With Node Weighting)

20. Bill
David
Carrie
Ellen
A Full Representation
of the Social Network
(With Node Weighting
and Edge Weighting)
Terminology & Examples
Alice

21. 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

22. 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 2
1 3
1 4
1 5
2 1
2 3
3 4
3 5
3 2
5 1
5 4
5 3
5 2
ROWS è COLUMNS
*How to Read the Edge List: (Person in Column 1 is friends with Person in Column 2)

23. 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

24. A Quick Example of a
Dynamic Network

25. United States Supreme Court
To Play Movie of the Early SCOTUS Jurisprudence:
http://vimeo.com/9427420

26. Some Other
Examples
of Networks

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

28. Corporate Boards
http://www.theyrule.net/

29. Transportation Networks
We might be interested in developing
transportation systems that are minimize
total travel time per passenger

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

31. Campaign Contributions
Networks
http://computationallegalstudies.com/tag/110th-congress/

32. Some Recent
Network Related
Publications
Special Issue:
Complex systems
and Networks
July 24, 2009
Special 90th anniversary
Issue:
May 7, 2007

33. History of
Network Science

34. 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?

35. The Königsberg Bridge Problem
Leonhard Euler
proved that this was
not possible
Is It Possible to
cross each bridge
each and only once?

36. 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 point
can have 0 or 2 nodes with an odd degree
Hamiltonian path: visit
each vertex exactly once

37. Modern
Network Science

38. 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

39. 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?

40. Stanley Milgram’s
Other Experiment
NE
MA

41. 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”

42. 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/

43. Six Degrees of Kevin Bacon

44. Visualization Source: Duncan J. Watts, Six Degrees
Six Degrees of Kevin Bacon

45. 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

46. 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

47. 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