It’s a Small World After All

Kim Dressel

- The small world phenomenon

Please hold applause until the end of the presentation.

Angie Heimkes

Eric Larson

Kyle Pinion

Jason Rebhahn

Kyle PinionIntroduction and Conclusion

Jason RebhahnThe 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 LarsonDefinitions 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 DresselProof 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

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Stanley Milgram - social psychologist at Harvard University

• 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 Small-World Network.

Watts-Strogatz ModelBased 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 ModelSupported the idea that the Small-World 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 ModelInterest 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

Lattice Drawing • n x n grid

• nodes represent individuals in a social network

•

Lattice Distance

Lattices

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.

Long-range - For and a directed edge is made using independent random trials

p = 1 and q = 2

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

p = 1 and q = 2

To obtain a probability distribution, this is divided by an appropriate normalizing constant.

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

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 isgiven by:

Kleinberg’s Theorem

To find the upper bound, we have:

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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

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Mathematical Background

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Geometric Series: Each term in the series is obtained from the preceding one by multiplying it by a common ratio.

Probability: It is used to mean the chance that a particular event will occur expressed on a linear scale 0 to 1.

= a/(1-r)

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

Mathematical Background

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?

• 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?

What if...

We’ll let you decide for yourself and come up with your own model.

Maybe the world is not as small as we think.

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 World- Wide Web” Nature. 401, 130 (1999)

Thanks again Steve!