Small Worlds

1
Small Worlds

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

2
Structure of the Thesis

• Introduction
• The Small World Phenomenon
• Applications to Routing
• Modeling Internet
• Social Networks
• Bibliography

3
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

4
Graphs
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

• 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.

A small world graph is any graph with a
relatively small characteristic path length and a
relatively large Clustering coefficient.
5
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.
• Doesnt require knowledge of paths and no active
  path selection.

6
Navigability Model by Kleinberg

• Another interesting aspect of Milgram's
  experiment is why people are able to find short
  paths

• 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 a d
• short means paths with a length of O(log n)
  from any given source to any given target vertex

(inverse ath-power distribution)
The idea behind the greedy alg. is that for any
a lt d there are too little random edges to make
the paths short For a gt 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
7
Barabasi-Albert Model

• Preferential attachment defines the probability
  for a vertex to get an edge to the new vertex

• 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.
• The second is the condition of preferential
  attachment
• that is, nodes (webpages) will wish to link
  themselves to hubs (websites) with the most
  connections.

8
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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ApplicationsHybrid 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
  Kleinbergs model. One in a cell / group of cells
  of sensors is wired.
• Very good performance in static sink node case
• with addition of T(nl(n)/log n) wires, average
  hop count reduced to T(1/vl(n)) and EDS to
  T(1/l(n)).
• In dynamic case, with greedy routing, hop count
  cant be reduced below ?(1/l(n)).

10
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 dont 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.

12
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).

13
Many real networks are small-world networks
Albert and Barabasi. REVIEWS OF MODERN PHYSICS,
74 2002 48-97
14
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
16
Syphilis transmission in Georgia
17
Corporate Partnerships
18
Thank you