Small Worlds
Presented by
Geetha Akula
For the Faculty of Department of Computer Science, CALSTATE LA.
On 8th June 07
Structure of the Thesis
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Introduction
The Small World Phenomenon
Applications to Routing
Modeling Internet
Social Networks
Bibliography
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”
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.
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.
Navigability Model by Kleinberg
 Another interesting aspect of Milgram's experiment is
why people are able to find short paths
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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
(inverse αth-power distribution)
 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
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
Barabasi-Albert Model
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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.
Applications to Computer Networks
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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
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)).
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
 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.
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).
Many real networks are small-world networks
Albert and Barabasi. REVIEWS OF MODERN PHYSICS, 74 2002 48-97
Map of Internet
Internet Mapping Project: http://research.lumeta.com/ches/map/gallery/index.html
The Sept 11 Hijackers and their Associates
Syphilis transmission in Georgia
Corporate Partnerships
Thank you