1. Small World Networks Somsubhra Sharangi Computing Science, Simon Fraser University

2. Agenda • 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

3. Concepts: Modeling Large Networks There exist short chains of acquaintances linking together arbitrary pairs of strangers. Milgram’s Experiment: • A package to be transported from • source to destination • Has to be transported only through • already known persons 5.5 hops on average 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

4. Concepts: Modeling Large Networks pk = k-α

5. 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

6. 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

7. 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 contacts of a node u with p = 1 and q = 2. v and w are the two long-range contacts. A two-dimensional grid network with n = 6, p = 1, and q = 0

8. Resilience in Small World Networks

9. 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

10. More Applications • Internet Topology Generators • BA Model Inadequate • Random re-wiring model • Weibull Distribution • Simulating Synthetic Topologies • End System Multicast Scaling • -by Jin & Bestavros[7] • 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 • Contact Based Query in Wireless Networks • Mobile Ad-Hoc Networks with limited Infrastructure

11. 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.