1. Geographic Gossip: Efficient Aggregations for Sensor Networks Author: Alex Dimakis, Anand Sarwate, Martin Wainwright University: UC Berkeley Venue: IPSN 2006 Presentatior: Yunhuai LIU

2. Outline Introduction Proposed algorithm Analysis Conclusion

3. Distributed Aggregation Every node has a measurement (e.g sensing temperature) Every node wants to access the global average Want a truly distributed, localized and robust algorithm to compute the average. 2 2 3 5 12 Goal: every node gets (2+2+3+5+12)/5=4.8 with the minimized energy cost

4. Gossip Algorithms for Aggregations Start with initial measurement as an estimate for the average and update Each node interacts with a random neighbor and both compute pairwise average (one update) Converges to true average Useful building block for more complex problems 2 2 3 5 2.5 12 3.75 7.87

5. How Many Interactions ? ε-averaging time T ave (n, ε): First time where x(k) is ε-close to the normalized true average with probability greater than 1-ε. Averaging time connected with mixing time Cost: Number of radio transmissions (fixed Tr radius) The physical meaning of T ave (n, ε): after T ave rounds, with very low probability the difference between the obtained value and the true value is high.

6. Cost of Standard Gossip Depends on graph and the transition probabilities:  Complete graph: T mix =Θ(1)  T ave =Θ(n log(n))  Small World/Expander: T mix =Θ(log(n))  T ave =Θ(n log(n))  Random Geometric Graph: T ave =Θ(n 2 )

7. Outline Introduction Proposed algorithm Analysis Conclusion

8. In standard gossip algorithms useful information performs random walks, which diffuses slowly Basic idea: add a random direction to gossip in order to diffuse faster. Assume:  Given location information  round-based, i.e., there is a virtual, global clock tick so that in each tick just one node active and conduct pairwise update.  The transmission range r follows: Basic Idea of Geographic Gossip

9. Random Target Routing Node picks a random location (=“target”) Greedy routing towards the target Perform pairwise update between the source and destination nodes

10. Geographic Gossip Nodes use random routing to gossip with nodes far away in the network Each interaction costs But faster mixing (convergence) 2 3 2. 5

11. Geographic Gossip When  Expected averaging cost of  w.h.p. the averaging cost is bounded by  The expected averaging cost of

12. Outline Introduction Proposed algorithm Analysis Conclusion

13. Some Interesting Things How we uniformly randomly select a target node to perform pairwise updates  Rejection sampling technique When the application field is partitioned by squares with length of with high probability each square contains at least one node.  It equals the probability that nlogn balls are thrown randomly to cover n bins Averaging algorithms based on pairwise updates have the convergence rate determined by the second largest eigenvalue λ 2 (W) of the matrix

14. Outline Introduction Proposed algorithm Analysis Conclusion

15. Conclusions Geographic gossip saves a factor of n 1/2 in energy required for aggregation. For realistic graph topologies Achieves with location information. Only localized distributed operations. distributed, localized, robust. Can be combined with other related consensus algorithms

16. Why I Choose This Paper Learn more about the random walk based algorithms  How others give analysis How to apply mathematics in research studies  Eigenvalue in convergence for optimization problems How others study the issue  Start from real applications: aggregations in WSNs  Narrow down the research domain to a specific problem: get the global average for each node with minimized energy cost  Proposal an algorithm to tackle the problem  Show the effectiveness of the algorithm as deep as possible: depth >> breadth (this is your contribution)  Do not involve too many things; make reasonable assumptions as many as possible to walk around those un-related things

17. Thanks Qeuestion and Answer