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Small World Six Degrees of Separation-

Contents. IntroductionExamples of Six Degrees of SeparationJustification of Six DegreesScale Free NetworkConclusion. . . Introduction. Do you have an experiment such that new acquaintance for you is your friend's friend?Have you ever said

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Small World Six Degrees of Separation-

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    1. Small World –Six Degrees of Separation- Teruhiko Yoneyama

    3. Introduction Do you have an experiment such that new acquaintance for you is your friend’s friend? Have you ever said “What’s a Small World!!” ?

    4. Introduction Question: For given any two persons in the worlds, how many intermediate people are needed to connect the two person?

    5. Example 1 Stanley Milgram’s Experiment; He sent mails to random people in Kansas and Nebraska, and asked them to readdress the mail to their acquaintance who may know the “target” person in Boston.

    7. Average Number of Intermediate people is 5.5

    8. Example 2 Erdos Number: Paul Erdos is very famous mathematician who published 1500 papers. Many Researchers are proud of being his collaborator. A person who writes a paper with him has Erdos Number of 1. A person who writes a paper with a person whose Erdos Number is 1 has Erdos Number of 2. And so on.

    11. Justification of Six Degrees In both examples, the number of degree of separation is less than 6. Is this value reasonable? Suppose the total population in this world is 6.5 billion and each person have 50 acquaintances.

    12. What is the degree number? Degree 1 A person links 50 people Degree 2 = 250 people : Degree 5 = 0.31 billion people Degree 6 = 15.6 billion people Six degrees are enough for 6.5 billion.

    13. Why is the degree so small? Suppose each node has averagely k links in the network. That is, there are k nodes which are reached with 1 step from a typical node. There are nodes with 2 steps. There are nodes with 3 steps. There are nodes with d steps.

    14. Why is the degree so small? Each node has averagely k links. There are nodes with d steps. If k is big, the number of reachable nodes becomes very large, even if d is small.

    15. Average Distance Let N be the number of nodes in network. is not more than N. Suppose , then we obtain the formula for average distance, d by

    16. If degree is six… How many people should ONE person know so that all people in the world completely connect? With d=6 and =6.5 billion, Then . and . Therefore 44 people are enough for the number of one person’s acquaintance.

    17. Random vs Scale Free Network So far, we considered this world as Random network. However, we know; -Some people have more chance to meet with new acquaintance than other normal people do. -Some portal sites, such as Yahoo! and MSN, is linking with more sites than other normal sites are.

    18. Random vs Scale Free Network Characteristics of Scale Free Network -Richer gets richer then, -Hub node appears

    19. Random vs Scale Free Network

    20. Scale Free Network Developing Scale Free Network New node precedes to select the node which has more nodes compared with other nodes.

    21. Scale Free Network

    22. Map of Internet

    23. FSN makes the degree be smaller Scale Free Network makes the degree of distance of nodes be smaller since one person have more chance to connect with others through hub nodes. Therefore, this world becomes more smaller.

    24. Bad effect of Hub node One example is epidemic of AIDS. If there is one person who has frequent sexual intercourse with many people, and if the person is infected by HIV, then the many people gets risk through the person. Also such person has usually higher risk to be infected because of large number of link. Computer virus is also this case. A significant site has higher risk to be invaded and has more possibility to scatter the virus to other sites. In other words, hub node has more influence to other node and more influence from other nodes.

    25. Conclusion Increasing population, N, doesn’t matter for the degree of separation, d, because of logarithm of N. However the number of one person’s average acquaintance, k, is an important factor. Progress of technologies, such as transportation and Internet, will make our world be smaller.

    26. References -Stanley Milgram, 1977, The Individual in a Social World -Mark Buchanan, 2002, Nexus: Small World and the Groundbreaking Science of Networks -Albert L. Barabasi, 2002, LINKED: The New Science of Networks -Erdos Number Project, http://www.oakland.edu/enp/index.html -Internet Mapping Project, http://research.lumeta.com/ches/map/gallery/index.html

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