Decentralized Search and Epidemics in Small World Network
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Transcript of Decentralized Search and Epidemics in Small World Network
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!