small worlds

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

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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. Structure of the Thesis. Introduction The Small World Phenomenon Applications to Routing Modeling Internet Social Networks Bibliography. The Small World Phenomena. - PowerPoint PPT Presentation

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Page 1: Small Worlds

Small WorldsSmall WorldsPresented byPresented byGeetha AkulaGeetha Akula

For the Faculty of Department of Computer Science, For the Faculty of Department of Computer Science, CALSTATE LA.CALSTATE LA.

On 8On 8thth June 07 June 07

Page 2: Small Worlds

Structure of the Thesis IntroductionIntroduction The Small World PhenomenonThe Small World Phenomenon Applications to RoutingApplications to Routing Modeling InternetModeling Internet Social NetworksSocial Networks BibliographyBibliography

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The Small World PhenomenaThe Small World Phenomena

Stanely Milgram’ s work on the small world is Stanely Milgram’ s work on the small world is responsible for the standard believe that “everyone is responsible for the standard believe that “everyone is connected by a chain of about six steps”connected by a chain of about six steps”

Their experiment “Send a packet from sets of Their experiment “Send a packet from sets of randomly selected people to a stock broker in Boston”randomly selected people to a stock broker in Boston”

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GraphsGraphs Regular GraphsRegular Graphs

– High characteristic path High characteristic path lengthlength

– High degree of clusteringHigh degree of clustering

Random GraphsRandom Graphs– Low characteristic path Low characteristic path

lengthlength– Low degree of clusteringLow degree of clustering

Graphs of real life networks Graphs of real life networks lie in between these two lie in between these two extremes.extremes.

Small World GraphSmall World Graph Most Large Scale Sparse Most Large Scale Sparse

Networks are found to be of the Networks are found to be of the small world type e.g. ‘Internet’, small world type e.g. ‘Internet’, ‘Electronic Circuits’, ‘Neurons’, ‘Electronic Circuits’, ‘Neurons’, ‘Human beings’ (Friendship ‘Human beings’ (Friendship Networks) Networks)

‘‘Six Degrees of Separation’ Six Degrees of Separation’ (Strangers -- Sociological (Strangers -- Sociological Concept) Concept)

MathematicallyMathematically:: In between In between ‘Regular Networks’ and ‘Regular Networks’ and ‘Random Networks’‘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 modelsSmall World models Watts and Strogatz (1998)Watts and Strogatz (1998)

– Very small number of long range contacts needed to decrease path Very small number of long range contacts needed to decrease path lengths without much reduction in cliquishness.lengths without much reduction in cliquishness.

– Long range contact picked uniformly at random (u.a.r)Long range contact picked uniformly at random (u.a.r)– Small world networks in 3 different areas esp. Small world networks in 3 different areas esp. spread of infectious spread of infectious

disease.disease. Probabilistic reach. No specific destinations.Probabilistic reach. No specific destinations. Doesn’t require knowledge of paths and no active path selection.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 shortFor 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

Page 7: Small Worlds

Barabasi-Albert Model 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.

Page 8: Small Worlds

Applications to Computer NetworksApplications to Computer Networks

P2P overlay networksP2P overlay networks Distributed hashing protocolsDistributed hashing protocols Security systems in mobile ad hoc networksSecurity systems in mobile ad hoc networks Hybrid sensor networksHybrid sensor networks Referral systemsReferral systems Links between webpages.Links between webpages. Freenet.Freenet. The Internet.The Internet. Large Scale Ad-hoc MulticastLarge Scale Ad-hoc Multicast

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Applications:Applications:Hybrid Sensor NetworksHybrid Sensor Networks

Sharma & Mazumdar (2005) – Sharma & Mazumdar (2005) – – Adding of a few shortcut wires between wireless sensors. Adding of a few shortcut wires between wireless sensors. – Reduced energy dissipation per node as well as non-uniformity Reduced energy dissipation per node as well as non-uniformity

in expenditure.in expenditure.– Deterministic as well as probabilistic placement of wires. Deterministic as well as probabilistic placement of wires. – Few wires unlike 1 long range contact per node in Kleinberg’s Few wires unlike 1 long range contact per node in Kleinberg’s

model. One in a cell / group of cells of sensors is wired.model. One in a cell / group of cells of sensors is wired.– Very good performance in static sink node case Very good performance in static sink node case

with addition of Θ(nl(n)/log n) wires, average hop count reduced to 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)).Θ(1/√l(n)) and EDS to Θ(1/l(n)).

– In dynamic case, with greedy routing, hop count cant be reduced In dynamic case, with greedy routing, hop count cant be reduced below Ω(1/l(n)).below Ω(1/l(n)).

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Links Between WebpagesLinks Between Webpages

A study looked at homepages and mailing lists A study looked at homepages and mailing lists at Stanford and MIT.at Stanford and MIT.

Looked at the contents, out-links, and in-links.Looked at the contents, out-links, and in-links. Tried to determine association network from the Tried to determine association network from the

webpage links.webpage links. Assumptions of the study:Assumptions of the study:

– Links are bidirectional.Links are bidirectional.– Easy to weed out links where users don’t know Easy to weed out links where users don’t know

each other.each other.L = 0.35 + 2.06 log N

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Findings:Findings:– Average 2.5 links per person.Average 2.5 links per person.– This leads to 1265 users (58%) connected This leads to 1265 users (58%) connected

at Stanford. 9.2 hops average path.at Stanford. 9.2 hops average path.– It was 1281 users (85.6%) connected at It was 1281 users (85.6%) connected at

MIT. 6.4 hops average path.MIT. 6.4 hops average path.– High clustering coefficient of 0.22 and 0.21 High clustering coefficient of 0.22 and 0.21

greater than that of random networks. greater than that of random networks.

Conclusion – we have a small world Conclusion – we have a small world network.network.

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The Internet The Internet

A study found that at the site level, the Internet A study found that at the site level, the Internet has a small characteristic path length, and a has a small characteristic path length, and a large clustering coefficient orders larger than large clustering coefficient orders larger than that of a random network.that of a random network.

Can exploit this property to build a smarter Can exploit this property to build a smarter search engine.search engine.– Look for documents corresponding to search string.Look for documents corresponding to search string.– Identify strongly connected component, find largest.Identify strongly connected component, find largest.– Calculate score (path length, clustering coefficient).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