1. Scalable and Fully Distributed Localization With Mere Connectivity

2. Localization with Mere Connectivity Localization is imperative to a variety of applications in wireless sensor networks. Considering the cost of extra equipment, localization from mere connectivity is ideal for large-scale sensor networks.

3. Previous Methods Previous mere connectivity based localization methods: ◦ Multi-Dimensional Scaling (MDS) based  low scalability  centralized ◦ Neural Network based  numerically unstable ◦ Graph Rigidity Theory based  low localization accuracy

4. Our Proposed Method We model a planar sensor network as a discrete surface with its metric represented as edge lengths (approximated by hop counts) and curvatures at each node as the angle deficits. Due to the approximation error, the surface is curved, not flat in plane. We compute the provably optimal flat metric which introduces the least distortion from estimated metric to isometrically embed the surface to plane.

5. Contributions of Our Proposed Method Fully distributed ◦ All the involved computations for each node only require information from its direct neighbors Scalable ◦ The computational cost and communication cost are both linear to the size of the network ◦ Limited error propagation Theoretically sound ◦ Provably optimal Numerically stable ◦ Free of the choice of initial values and local minima with theoretical guarantee

6. Talk Overview Theory of flat metric Distributed algorithm on sensor network Discussions ◦ Costs ◦ Error propagation Experiments and Comparison

7. Talk Overview Theory of flat metric Distributed algorithm on sensor network Discussions ◦ Costs ◦ Error propagation Experiments and Comparison

8. Discrete Metric and Discrete Gaussian Curvature Discrete metric: edge length of the triangulation Discrete Gaussian curvature: induced by metric, measured as angle deficit

9. Circle Packing Metric

10. Flat Metric Flat metric: a set of edge lengths which induce zero Gaussian curvature for all inner vertices such that the triangulation can be isometrically embedded into plane. Infinite number of flat metrics for a given connectivity triangulation.

11. Optimal Flat Metric

12. Tool to Compute Optimal Flat Metric Discrete Ricci flow ◦ [Hamilton 1982]: Ricci flow on closed surfaces of non-positive Euler characteristic ◦ [Chow 1991]: Ricci flow on closed surfaces of positive Euler characteristic ◦ [Chow and Luo 2003]: Discrete Ricci flow including existence of solutions, criteria, and convergence. ◦ [Jin et al. 2008]: a unified framework of computational algorithms for discrete Ricci flow

13. Tool to Compute Optimal Flat Metric

14. Talk Overview Theory of flat metric Distributed algorithm on sensor network Discussions ◦ Costs ◦ Error propagation Experiments and Comparison

15. Distributed Algorithm Preprocessing: build a triangulation from network graph ◦ The dual of the landmark-based Voronoi diagram

16. Distributed Algorithm Computing optimal flat metric (edge length) ◦ All edge lengths initially = unit  The curvature of each vertex is not zero  The initial triangulation surface can’t be embedded in plane. ◦ Find the flat metric, such that  The triangulation surface can be isometrically embedded in plan  The introduced localization error is minimal

17. Distributed Algorithm

18. Distributed Algorithm Isometric embedding ◦ Start embedding from one vertex ◦ Continuously propagate to the whole triangulation

19. Talk Overview Theory of flat metric Distributed algorithm on sensor network Discussions ◦ Costs ◦ Error propagation Experiments and Comparison

20. Costs Preprocessing step - building triangulation ◦ Time complexity: ◦ Communication cost: Step 1 – computing flat metric ◦ Time complexity: ◦ Communication cost: Step II – isometric embedding ◦ Time complexity: ◦ Communication cost:

21. Convergence Rate of Step I Number of iterations =

22. Error Propagation If error is introduced to the estimated or measured metric in a small area, error will propagate to the entire network for general localization methods. The impact to our computing optimal flat metric based method can be modeled as a discrete Green function:

23. Testing of Error Propagation One scenario: a much longer distance measurement around one vertex due to slower response of the node to its neighboring nodes’ signals.

24. Talk Overview Theory of flat metric Distributed algorithm on sensor network Discussions ◦ Costs ◦ Error propagation Experiments and Comparison

25. Variant Density

26. Experiments and Comparison Different transmission models Trans. modelsUDGQuasi-UDGLog-NormProbability Localization error0.250.340.420.43

27. Experiments and Comparison Non-uniform distribution Density ranging from 11.3 to 18.7 Localization error0.46

28. Experiments and Comparison Comparison with other methods on networks with different topologies NetworksFlat MetricMDS-MAPMDS-MAP(P)C-CCAD-CCA 0.292.520.89 2.100.88 0.320.560.680.710.69 0.480.620.750.72 0.64 0.551.180.610.780.70 0.631.270.991.170.8

29. Summary and Future Work Mere connectivity based localization method ◦ Theoretically guaranteed and numerically stable ◦ Fully distributed and highly scalable with linear computation time and communication cost and dramatically decreased error propagation Future work: localization for sensor nodes deployed on general 3D surface