Decentralized Search and Epidemics in Small World Network

Siddhartha GundaSorabh Hamirwasia

Generating small world network model. Optimal network property for decentralized

search. Variation in epidemic dynamics with

structure of network.

Introduction

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

Background

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 =

Generating Kleinberg’s Model

Generating Kleinberg’s Model

2D lattice Kleinberg’s Model

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.

Decentralized Algorithm

ResultsParameters: 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.

Epidemic Models. Branching Model. SIS Model SIR Model SIRS Model

SIRS over SIR

Background

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 =

Epidemic Model

Step 1 – Generate Kleinberg’s graph. Step 2 – Simulate SIRS algorithm.

If state = SusceptibleCheck if node can get infection.If yes change the state to infected.

If state = InfectedCheck if TI expires.If yes change the state to recovery.

If state = RecoveredCheck 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.

Epidemic Model

Results Model 1

1000 Cycles

Results Model 1

1000 Cycles

Results Model 2

1000 Cycles

Results Model 2

1000 Cycles

‘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.

Observations:

[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.

References

Questions ?

Thank You!