1. Siddhartha Gunda Sorabh Hamirwasia

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

3.  What is small world network model ?  Watts-Strogatz vs Kleinberg’s Model.  BFS vs Decentralized search.

5. 2D latticeKleinberg’s Model

6.  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. 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.  Epidemic Models.  Branching Model.  SIS Model  SIR Model  SIRS Model  SIRS over SIR

10.  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 T I expires. If yes change the state to recovery.  If state = Recovered Check if T R expires. If yes change the state to susceptible.  Step 3 – Store number of infected nodes.  Step 4 – Repeat above steps for N cycles.

11. 1000 Cycles

15.  ‘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.  [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!