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
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- The small world phenomenon
Please hold applause until the end of the presentation.
Introduction and Conclusion
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.
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 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
Duncan Watts - social scientist at Columbia University
In 1998, the two developed a more refined model to represent the Small-World Network.
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.
Supported the idea that the Small-World Phenomenon is pervasive in a wide range of networks in nature and technology.
Interest has spread to many areas of study including:
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.
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.
For p > 0 the node u has a directed edge on every other node within lattice distance p.
p = 1 and q = 2
For and a directed edge is made using independent random trials
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 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.
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 is
*We get (2n-2) as an upper limit because we are dealing with a finite lattice structure and the furthest point from the message holder is
(n-1) + (n-1) = (2n-2).*
Now, to find the lower bound, we simply put ln (n) back into the original equation:
For j > 0, phase j is defined as
For j > 0, Ball j is defined as
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.
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
...By the law of total probability
We’ll let you decide for yourself and come up with your own model.
Thanks for coming!
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!