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.