Post on 02-Jan-2016
description
Small World Networks
Somsubhra Sharangi Computing Science, Simon Fraser University
• Definition & Some Terminology– Random Graphs– Scale Free Graphs
• Some Properties– Navigation– Resilience
• Some Relevant Applications– P2P Overlay Construction– Internet Topology Modeling– Wireless and Mobile Networks
Agenda
Milgram’s Experiment:
There exist short chains of acquaintances linking together arbitrary pairs of strangers.
Random Graph:n nodes or vertices, where each possible edge between two vertices is present with independent probability p
average degree of a vertex is
Mean Shortest Path: the average length of shortest paths over all pairs of vertices
• A package to be transported from source to destination• Has to be transported only through already known persons
Concepts: Modeling Large Networks
5.5 hops on average
Concepts: Modeling Large Networks
pk = k-α
Concepts: Modeling Large Networks
Scale Free Graph: Degree distribution of the vertices follows the power law
Clustering coefficient: the average probability that two neighbors of a given vertex are also neighbors of one another.
Small World Networks:• Small mean shortest path
• High clustering coefficient
• L ~ Lrand but C >> Crand
Network n k L Lrand C Crand
WWW pages 153127 35.21 3.1 3.35 0.1078 0.00023
Internet AS 6209 4.11 3.76 6.18 .3 .001
Math co-authors 70975 3.9 9.5 8.2 .59 5.4x10-5
Power Grid 4941 2.67 18.7 12.4 0.08 0.005
E-coli reaction 315 28.3 2.62 1.98 .59 0.09
Genesis of Small World Networks
• Barabasi-Albert Model for Scale Free Networks
• Growth• Preferential Attachment
• BA biased towards history
• Weighted Preferential Attachment
• Random Re-Wire Model
Navigation in Small World Networks
Why should arbitrary pairs of strangers be able to find short chains of acquaintances that link them together?
Kleinberg’s result: Prd ≈ d(x,y)-α
A two-dimensional grid network with n = 6, p = 1, and q = 0
contacts of a node u with p = 1 and q = 2. v and w are the two long-range contacts.
Resilience in Small World Networks
Application : P2P Overlay Construction
• Head Node• Inner Node
• Long Link• Cluster Link
• Works on Top of P2P network layer• Joining Network
• Position determined by network layer• Determines whether to act as Head Node• If Head Node, create random links biased towards far away Head Nodes.
• Leaving Network• Normal restructuring of topology• New Head Node finds new long links
• Object Lookup• Search in local cluster• Determine long link and remote Head Node• Search in cluster of remote Head Node
• Resilience against flash crowd traffic
More Applications
• Hybrid Wireless Sensor Networks • Energy Dissipation proportional to number of hops in routing.• Divide the sensor space into cells• Place wire in each cell and flood• Greedy Geographic routing
• Internet Topology Generators• BA Model Inadequate• Random re-wiring model• Weibull Distribution
• Simulating Synthetic Topologies• End System Multicast Scaling
-by Jin & Bestavros[7]
• Contact Based Query in Wireless Networks• Mobile Ad-Hoc Networks with limited Infrastructure
References:
[1] M. E. J. Newman, Random Graphs as Models of Networks, Handbook of Graphs & Networks, Berlin,2003.[2] Albert-László Barabási and Eric Bonabeau, Scale Free Networks, Scientific American 288, pp 60-67, 2003.[3]J. Kleinberg. Navigation in a small-world. Nature, 406, 2000.[4] J. Kleinberg. The small-world phenomenon: an algorithmic perspective. Cornell Computer Science Technical Report 99-1776,2000.[5]KYK Hui, JCSLiu and DKYau, Small World Overlay p2p Networks.IWQoS 2004, pp 201-210.[6]Gaurav Sharma, Ravi Majumdar, Hybrid Sensor Networks: A Small World,MobiHoc 2005[7]Shudong Jin and Azer Bestavros, Small world Internet Topologies, Boston University BUCS-TR-2002-004.[8]Ahmed Helmy, Small Worlds in Wireless Networks, Communication Letters IEEE,vol-7, issue-10, Oct-2003, pp 490-492.