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Small Worlds Presented by Geetha Akula For the Faculty of Department of Computer Science, CALSTATE LA. On 8 th June 07

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Structure of the Thesis Introduction The Small World Phenomenon Applications to Routing Modeling Internet Social Networks Bibliography

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The Small World Phenomena Stanely Milgram’ s work on the small world is responsible for the standard believe that “everyone is connected by a chain of about six steps” Their experiment “Send a packet from sets of randomly selected people to a stock broker in Boston”

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Graphs Regular Graphs –High characteristic path length –High degree of clustering Random Graphs –Low characteristic path length –Low degree of clustering Graphs of real life networks lie in between these two extremes. Small World Graph Most Large Scale Sparse Networks are found to be of the small world type e.g. ‘Internet’, ‘Electronic Circuits’, ‘Neurons’, ‘Human beings’ (Friendship Networks) ‘Six Degrees of Separation’ (Strangers -- Sociological Concept) Mathematically: In between ‘Regular Networks’ and ‘Random Networks’ A small world graph is any graph with a relatively small characteristic path length and a relatively large Clustering coefficient.

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Small World models Watts and Strogatz (1998) –Very small number of long range contacts needed to decrease path lengths without much reduction in cliquishness. –Long range contact picked uniformly at random (u.a.r) –Small world networks in 3 different areas esp. spread of infectious disease. Probabilistic reach. No specific destinations. Doesn’t require knowledge of paths and no active path selection.

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Another interesting aspect of Milgram's experiment is why people are able to find short paths Navigability Model by Kleinberg Let the routing algorithm take place on the following network model –Start with a d-dimensional grid –Add random edges between vertices v and w with a probability of Theorem: The routing algorithm will find ‘short‘ paths, if and only if α = d –‘short‘ means paths with a length of O(log n) from any given source to any given target vertex (inverse αth-power distribution) The idea behind the greedy alg. is that for any α < d there are too little random edges to make the paths short For α > d there are too many random edges, and hence too many choices to which the message could be passed on The message will make a (long) random walk through the network

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Barabasi-Albert Model Preferential attachment defines the probability for a vertex to get an edge to the new vertex 1.network has to be expanding, growing. This precondition of growth is very important as the idea of emergence comes with it. It is constantly evolving and adapting. 2.The second is the condition of preferential attachment that is, nodes (webpages) will wish to link themselves to hubs (websites) with the most connections.

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Applications to Computer Networks P2P overlay networks Distributed hashing protocols Security systems in mobile ad hoc networks Hybrid sensor networks Referral systems Links between webpages. Freenet. The Internet. Large Scale Ad-hoc Multicast

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Applications: Hybrid Sensor Networks Sharma & Mazumdar (2005) – –Adding of a few shortcut wires between wireless sensors. –Reduced energy dissipation per node as well as non-uniformity in expenditure. –Deterministic as well as probabilistic placement of wires. –Few wires unlike 1 long range contact per node in Kleinberg’s model. One in a cell / group of cells of sensors is wired. –Very good performance in static sink node case with addition of Θ(nl(n)/log n) wires, average hop count reduced to Θ(1/√l(n)) and EDS to Θ(1/l(n)). –In dynamic case, with greedy routing, hop count cant be reduced below Ω(1/l(n)).

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Links Between Webpages A study looked at homepages and mailing lists at Stanford and MIT. Looked at the contents, out-links, and in-links. Tried to determine association network from the webpage links. Assumptions of the study: –Links are bidirectional. –Easy to weed out links where users don’t know each other. L = 0.35 + 2.06 log N

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Findings: –Average 2.5 links per person. –This leads to 1265 users (58%) connected at Stanford. 9.2 hops average path. –It was 1281 users (85.6%) connected at MIT. 6.4 hops average path. –High clustering coefficient of 0.22 and 0.21 greater than that of random networks. Conclusion – we have a small world network.

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The Internet A study found that at the site level, the Internet has a small characteristic path length, and a large clustering coefficient orders larger than that of a random network. Can exploit this property to build a smarter search engine. –Look for documents corresponding to search string. –Identify strongly connected component, find largest. –Calculate score (path length, clustering coefficient).

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Albert and Barabasi. REVIEWS OF MODERN PHYSICS, 74 2002 48-97 Many real networks are small-world networks

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Map of Internet Internet Mapping Project: http://research.lumeta.com/ches/map/gallery/index.html

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The Sept 11 Hijackers and their Associates

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Syphilis transmission in Georgia

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Corporate Partnerships

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Thank you

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