Information Survival Threshold in Sensor and P2P Networks

1. Information Survival Threshold inSensor and P2P Networks Deepayan Chakrabarti Yahoo (&CMU) Jure Leskovec CMU Christos Faloutsos CMU Samuel Madden MIT Carlos Guestrin CMU Michalis Faloutsos UCR D. Chakrabarti et al

2. Will a Datum Spread/Survive in a Network? • Nodes transmit the datum • Sensors go down (and lose memory) • Sensors come up with some probability • Will the datum survive or become extinct? • What are the conditions that define this? D. Chakrabarti et al

3. Applications • Sensor net; message -> a query (‘keep reporting average temperature’) • Corporate memory: people in a company; message -> verbal rules • Peer-to-peer community and shared storage of information • Virus propagation, for humans or machines D. Chakrabarti et al

4. Our Contributions • We formulate the problem in a general way • We develop an approximate analytic solution • We associate the transition threshold to • the eigenvalue of the “adjacency” matrix • We validate the accuracy of the model • In many realistic topologies and scenarios D. Chakrabarti et al

5. Visual Depiction of Major Result #carriers (lin) #carriers (log) #carriers (log) There is a clear transition phase phenomenon • We identify a threshold • below it, message dies off; • above it, some (not all) nodes will carry the message time (log) time (lin) time (lin) D. Chakrabarti et al

6. Outline • Problem definition - Motivation • Proposed solution • Experiments • Discussion - Conclusions D. Chakrabarti et al

7. Our Quite General Model • Consider a network of nodes • A node can be in three states • have, not-have, down • A node ui transmits message • which reach neighbors with prob. bij  • Nodes go down (lose datum) with failure prob. di • Nodes come up with resurrection prob. gi • Consider small time steps: where changes happen D. Chakrabarti et al

8. Dynamical System:State Transitions for a Node prob. that node has datum prob. that node does not have it prob. that node is down D. Chakrabarti et al

9. Eq. 1: Having the datum • A node can have datum at time t if: • Had it at (t-1) and does not die • Did not have it, but gets it from a neighbor D. Chakrabarti et al

10. Eq.2: Not having the datum • A node will not have the datum at time t if: • Did not have it at t-1 and did not get it from neighbors • Was down before and just woke up D. Chakrabarti et al

11. System Matrix fo Dynamic System • After some manipulation of the DS equations • Consider the system matrix: D. Chakrabarti et al

12. Main Result: Survivability Threshold We prove the following: Theorem 1: (condition for fast extinction): if the survivability score s=| l1,S | obeys s < 1 the datum will be extinct fast. Dfn: When s < 1, system is “below threshold” D. Chakrabarti et al

13. We can also prove more stuff: D. Chakrabarti et al

14. Generality of the Model • Our model subsumes the SIS epidemic model • SIS: Susceptible Infected Susceptible • Only two states: does not have a down state D. Chakrabarti et al

15. Outline • Problem definition - Motivation • Proposed solution • Experiments • Discussion - Conclusions D. Chakrabarti et al

16. Overview of Simulations • The dynamical system is very accurate • close to simulations • The threshold is accurate • datum dies out very quickly below the threshold, • it “survives” above the threshold. D. Chakrabarti et al

17. Network Topologies Intel MIT D. Chakrabarti et al

18. Distribution of link qualities Intel MIT D. Chakrabarti et al

19. Scenario description D. Chakrabarti et al

20. Major Result Revisited - 1 • Grid network • # carriers, vs #epochs (timesteps) - lin-lin #carriers (lin) ‘survival’ fast extinction time (lin) D. Chakrabarti et al

21. Major Result Revisited - 2 • # carriers, vs #epochs (simulation) - log-lin #carriers (log) below threshold: exponential drop time (lin) D. Chakrabarti et al

22. Major Result Revisited - 3 • # carriers, vs #epochs (simulation) - log-log #carriers (log) at threshold: power-law drop time (log) D. Chakrabarti et al

23. Simulation and dynamical system • Our dynamical system and threshold are accurate • in many different topologies Grid Gnutella Intel MIT D. Chakrabarti et al

24. Survivability vs. retransmission probability • The dashed vertical line is our threshold (s = 1) • The datum dies out below our threshold, • …but survives above it. Grid Intel D. Chakrabarti et al

25. Survivability vs. resurrection probability • The dashed vertical line marks our threshold. • Again, our threshold is very accurate. Gnutella MIT D. Chakrabarti et al

26. Insensitive to Initial State Gnutella network D. Chakrabarti et al

27. Conclusions • Model propagation in a very general way • Directed edges, lossy links, and node reboots • Develop an approximate Dynamical System • Provide a closed formula for threshold • Validate dynamic system and threshold • Extensive simulations on many diff. topologies D. Chakrabarti et al