![Page 1: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/1.jpg)
Decentralized Search and Epidemics in Small World Network
Siddhartha GundaSorabh Hamirwasia
![Page 2: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/2.jpg)
Generating small world network model. Optimal network property for decentralized
search. Variation in epidemic dynamics with
structure of network.
Introduction
![Page 3: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/3.jpg)
What is small world network model ? Watts-Strogatz vs Kleinberg’s Model. BFS vs Decentralized search.
Background
![Page 4: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/4.jpg)
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
![Page 5: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/5.jpg)
Generating Kleinberg’s Model
2D lattice Kleinberg’s Model
![Page 6: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/6.jpg)
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
![Page 7: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/7.jpg)
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.
![Page 8: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/8.jpg)
Epidemic Models. Branching Model. SIS Model SIR Model SIRS Model
SIRS over SIR
Background
![Page 9: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/9.jpg)
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
![Page 10: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/10.jpg)
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
![Page 11: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/11.jpg)
Results Model 1
1000 Cycles
![Page 12: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/12.jpg)
Results Model 1
1000 Cycles
![Page 13: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/13.jpg)
Results Model 2
1000 Cycles
![Page 14: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/14.jpg)
Results Model 2
1000 Cycles
![Page 15: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/15.jpg)
‘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:
![Page 16: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/16.jpg)
[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
![Page 17: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/17.jpg)
Questions ?
![Page 18: Decentralized Search and Epidemics in Small World Network](https://reader036.vdocuments.site/reader036/viewer/2022062315/5681610a550346895dd056ae/html5/thumbnails/18.jpg)
Thank You!