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

2. 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

3. 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

4. Concepts: Modeling Large Networks p k = 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 ~ L rand but C >> C rand NetworknkLL rand CC rand WWW pages15312735.213.13.350.10780.00023 Internet AS62094.113.766.18.3.001 Math co-authors709753.99.58.2.595.4x10 -5 Power Grid49412.6718.712.40.080.005 E-coli reaction31528.32.621.98.590.09

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: Pr d ≈ 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.

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 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

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.