It’s a Small World After All
- The small world phenomenon
Please hold applause until the end of the presentation.
Kim Dressel
Angie Heimkes
Eric Larson
Kyle Pinion
Jason Rebhahn
Kyle Pinion
Introduction and Conclusion
Jason Rebhahn
The research of the small world phenomenon by Stanley Milgram, Steven
Strogatz, and Duncan Watts. Examples how this phenomenon can be applied to
realistic situations, including the world wide web.
Eric Larson
Definitions and terms involved in the mathematics behind the small world
phenomenon. Introduction to lattice representations, short and long range
contacts, metrics, and phase j.
Angie Heimkes & Kim Dressel
Proof of the main theorem behind Jon Kleinberg’s model of the Small-World
Network.
The Small-World Phenomenon
“The idea that even in a planet with
billions of people, everyone is
connected in a tight network.”
Also known as the Six Degrees of
Separation
Stanley Milgram - social psychologist
at Harvard University
kissm ywhite  clown
• First to study the Small-World Phenomenon
• 1967 - Performed chain letter experiment
from the Midwest to Boston
• Averaged 6 transitions of the letter
• Sparked a wide interest in the study of the
Small-World Phenomenon
Steven Strogatz - mathematician
at Cornell University
Duncan Watts - social scientist at
Columbia University
In 1998, the two developed a more
refined model to represent the SmallWorld Network.
Watts-Strogatz Model
Based on Regular and Random Networks
Regular Network: A given point is only
directly linked to its four nearest
neighbors.
Random Network: Each point has a
connection to a more distant point.
Small-World Network: A given point has
four local connections plus a distant
connection.
Watts-Strogatz Model
Supported the idea that the SmallWorld Phenomenon is pervasive in a
wide range of networks in nature and
technology.
• Six Degrees of Kevin Bacon
• Neural networks in elegan worms
• Power grid of the United States
Small-World Model
Interest has spread to many areas of
study including:
• Economics
• Physics
• Neurophysiology
• Biochemistry
World-Wide Web
Estimated size of 800 million documents.
The Northern Light search engine covers
the largest amount at 38% of the web.
Since the Small-World Network applies
very well to the WWW, search engines
could make use of it to make more efficient
searches over a larger amount of the web.
Jon Kleinberg - Professor at
Cornell University
Developed the Clever Algorithm for
searching the web more efficiently.
Determined the Watts-Strogatz Model
was insufficient to explain the
algorithmic concepts of Milgram’s
Small-World Phenomenon.
Definitions and Terms
Lattices
Lattice Drawing
• n x n grid
• nodes represent individuals in
a social network
•
Lattice Distance
Short and Long-range
Contacts
Short-range -
For p > 0 the node u
has a directed edge
on every other node
within lattice
distance p.
p = 1 and q = 2
Long-range -
For
and
a directed edge is made
using independent random trials
The Decentralized
algorithm A
• Determine the long-range contact(s)
• Transfer the message to the node closest to the target node
Inverse rth-power
distribution
The ith directed edge from u has endpoint
v with probability proportional to
[D(u,v)]-r
To obtain a probability distribution, this is
divided by an appropriate normalizing
constant.
p = 1 and q = 2
Performance Metric
Performance in this system is measured by the
average number of steps it takes to get from the
source to the target. This can be defined
mathematically as the Expectation of X.
Here we go….
Kleinberg’s Theorem
The theorem behind the model states that there
is a decentralized algorithm A and a constant c,
independent of n, so that when r = 2 and
p = q = 1, the expected delivery time of A is at
most
Kleinberg’s Theorem
First of all, we will find the upper and lower
bounds for the probability that u chooses v as
its long-range contact. The probability that u
chooses v as a long-range contact is
given by:
To find the upper bound, we
have:
*We get (2n-2) as an upper limit
 4  ln(2n  2) because we are dealing with a
dx
j6
v4)ln(
n
)
)4 (4 j )( jfinite
lattice structure and the
 d(u4,
x
 ln(n)
furthest point from the message
holder is
(n-1) + (n-1) = (2n-2).*
2n2
2 n  22 n  2
 2 1
v u
j 1
j 1 j 1
2
Now, to find the lower bound, we
simply put ln (n) back into the
original equation:
1
1

ln(n)  d (u, v) 2
v u
2
d (u, v)
ln(n)

d (u, v) 2
 d (u, v)2
v u
more definitions...
Phase j
For j > 0, phase j is defined as
In this picture,
• Red is the target
• Green is phase j = 0
• Yellow is phase j = 1
• Blue is phase j = 2
• Black is the start of phase j = 3
Ball j
For j > 0, Ball j is defined as
In this picture,
• Red is the target
• Green is B0
• Yellow and green make B1
• Blue, yellow, and green make B2
• Black, blue, yellow, and green
make the start of B3
And now - the rest of the proof
Mathematical Background

a  ar  ar 2  ar3  .....   ar( n1) foralla o
n 1
a  ar  ar
If | r | 1, it ' s sum is 
a  ar  ar 2  ar3  .....   ar( n1) foralla o
n 1
a  ar  ar
If | r | 1, it ' s sum is
Geometric Series: Each term in the series is
obtained from the preceding one by multiplying
it by a common ratio.
= a/(1-r)
Probability: It is used to mean the chance that a
particular event will occur expressed on a linear
scale 0 to 1.
Mathematical Background
Discrete Random Variable: Assumes each of
its values with a certain probability. Must be
between 1 and 0 with the sum of 1
Logarithms: log n denotes the logarithm base 2,
while ln n denotes the natural logarithm, base e
Number of Nodes in Bj
Probability that a node will be in Bj
Proof of expectation
...By the law of total probability
Proof of expectation (continued)
It’s okay, take a deep breath.
So is it really a small world after all?
What if...
• there were different values for p and q?
• different formulas were used for probability and the
decentralized algorithms?
• the definitions for long and short range contacts
changed?
• metric spaces allowed diagonal movement, not just up
& down?
Maybe the world is not as small as we think.
We’ll let you decide for yourself and come up with your own model.
Wrapping it up….
Thanks for coming!
Kim Dressel
Angie Heimkes
Eric Larson
Kyle Pinion
Jason Rebhahn
REFERENCES
1. L. Adamic, “The Small World Web” , manuscript available at
http://www.parc.xerox.com/istl/groups/iea/www/smallworld.html
2. Sandra Blakeslee, “Mathematics Prove That It’s a Small World”
3. Dr. Steve Deckelman, His Extensive Mathematical Knowledge
4. Jon Kleinberg, “The Small-World Phenomenon: An Algorithmic Perspective”
5. Stanley Milgram, “The Small World Problem” Psychology Today 1, 61 (1967)
5. Beth Salnier, “Small World”
6. Reka Albert, Hawoong Jeong, Albert-Laszlo Barabasi, “Diameter of the WorldWide Web” Nature. 401, 130 (1999)
Thanks again Steve!