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Decentralized Search and Epidemics in Small World Network

Decentralized Search and Epidemics in Small World Network. Siddhartha Gunda Sorabh Hamirwasia. Introduction. Generating small world network model. Optimal network property for decentralized search. Variation in epidemic dynamics with structure of network. Background.

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Decentralized Search and Epidemics in Small World Network

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  1. Decentralized Search and Epidemics in Small World Network Siddhartha Gunda SorabhHamirwasia

  2. Introduction • Generating small world network model. • Optimal network property for decentralized search. • Variation in epidemic dynamics with structure of network.

  3. Background • What is small world network model ? • Watts-Strogatzvs Kleinberg’s Model. • BFS vs Decentralized search.

  4. Generating Kleinberg’s Model • Form a 2D lattice. • Manhattan distance between nodes. d[u,v]= |ux – vx| + |uy – vy| • Generate long edge using “inverse rth-power distribution”: • p α • p=

  5. Generating Kleinberg’s Model 2D lattice Kleinberg’s Model

  6. Decentralized Algorithm • Step 1- Select source and target node randomly. • Step 2 – Send message using decentralized search. • At each node find neighbor nearest to the target. • Pass message to the neighbor found above. • Repeat till message reaches target node. • Compute hops required. • Step 3 - Repeat Step1and Step2 for N cycles. • Step 4 - Calculate average number of hops.

  7. Results Parameters: • Lattice dimension = 2, Number of Nodes/dimension = 100, Number of iterations = 10000 • For same value of r, decrease in q results in increase in average path length. • For different values of r, optimal average length is found at r = 2.

  8. Background • Epidemic Models. • Branching Model. • SIS Model • SIR Model • SIRS Model • SIRS over SIR

  9. Epidemic Model • Valid states of a node - {Infected, Susceptible, Recovered} • TI cycles – Infection Period. • TR cycles – Recovery Period. • Ni – Initial count of infected nodes. • q – Probability of contagion. • Model 1: Probability of getting infected pi = • Model 2: Probability of getting infected pi =

  10. Epidemic Model • Step 1 – Generate Kleinberg’s graph. • Step 2 – Simulate SIRS algorithm. • If state = Susceptible Check if node can get infection. If yes change the state to infected. • If state = Infected Check if TI expires. If yes change the state to recovery. • If state = Recovered Check if TR expires. If yes change the state to susceptible. • Step 3 – Store number of infected nodes. • Step 4 – Repeat above steps for N cycles.

  11. Results Model 1 1000 Cycles

  12. Results Model 1 1000 Cycles

  13. Results Model 2 1000 Cycles

  14. Results Model 2 1000 Cycles

  15. Observations: • ‘r=0’ means uniform probability. Behavior same as Watts-Strogatz Model. • For constant “q”, Decrease in “r” results in increase in “p” for same distance. Hence high synchronization. • For constant “r”, Decrease in “q” results in decrease in “p”. Hence low synchronization.

  16. References • [1] Jon Kleinberg. The small-world phenomenon: an algorithmic perspective. In Proc.32nd ACM Symposium on Theory of Computing, pages 163–170, 2000. • [2] Marcelo Kuperman and Guillermo Abramson. Small world effect in an epidemiological model. Physical Review Letters, 86(13):2909–2912, March 2001.

  17. Questions ?

  18. Thank You!

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