Connectivity Properties for Topology design in Sparse Wireless Multi-hop Networks

1. Connectivity Properties for Topology design in Sparse Wireless Multi-hop Networks Ph.D. Defense Srinath Perur Advisor: Sridhar Iyer IIT Bombay

2. Introduction

3. Multi-hop Wireless Networks (MWN) • Multi-hop Wireless Network • Decentralised • Infrastructure-less • Cooperative multi-hop routing • Examples: • Mobile ad hoc networks • Sensor networks • Mesh networks

4. Topology Design • Combination of network parameters for desired network graph • Ex: Transmission range, area of operation, number of nodes Topology design can be: • Deterministic • Ex: Mesh networks • Probabilistic • Ex: Sensor networks, MANETs

5. Connectivity Properties • Value associated with a network indicating extent to which nodes are connected • Connectivity: probability of nodes forming a single connected component • Size of largest connected component • Connectivity properties are often metrics for topology design • Ex: Transmission range required for a connected network

6. Sparse MWNs • A sparse MWN is one that is not connected with high probability • We assume < 0.95 • Examples: • Vehicular MWN at low traffic density • Sensor network after some nodes have died • Incrementally deployed MANET • 25/60 sets of network parameters used in MobiHoc papers were sparse

7. Sparse MWNs • Sparse networks can also occur by design • Trade-off connectivity for other network parameters in constrained scenarios • Ex: Delay tolerant networks • Networks tolerating 90% nodes in one connected component required significantly reduced transmission range [SB03]

8. Questions of Interest • Are currently used connectivity properties appropriate for topology design in sparse MWNs? • How can they be used? • What other connectivity properties can be used? • What trade-offs between network parameters can be made in sparse deployments? • What tools, such as models or simulators, would we require in order to accomplish these trade-offs while designing networks?

9. Organisation • Connectivity • Empirical characterisation for sparse region in finite domain • Reachability • Definition and properties • Applications • Characterisation • Simran - a topological simulator for MWNs • Spanner - a design tool for sparse MWNs • Edge effects in MWNs • Quantifying the edge effect • Applying it to use results for square area networks in rectangular networks

10. Network Model • We define a network as a tuple: <N, R, l, M> • N – number of nodes • R – uniform transmission range of nodes • l – side of square area of operation • M – mobility model and its parameters • Two nodes are connected • directly if they are within distance R of each other • if there is path between them in the network graph

11. Instances of a network

12. Connectivity for Sparse Networks

13. Connectivity • Defined as the probability that all nodes in the network form a single connected component • Many asymptotic results • Model connectivity as threshold function • Value of normalised range, r, where the network is connected • Ex: if r(n) decreases slower than the network is almost surely connected as

14. Connectivity • Assumption of threshold function does not hold for small N • We require a finite domain connectivity model valid for entire operating range

15. Connectivity • Existing work in the finite domain • Exact expression for one-dimensional network [DM02] • Empirical studies of k-connectivity [Kos04] • Tang and others [TFL03] • Empirical model of connectivity in two-dimensions for N between 3 and 125 and connectivity between 0.5 and 0.99 • We present a more general and accurate empirical model

16. Characterising Connectivity • We characterise C(N,r) in terms of • N - number of nodes • r - normalised transmission range for and • Nodes static and uniformly distributed • By exploring simulation data we found • Sigmoidal growth curve for C(N,r) vs. r • Asymmetric about point of inflection

17. Characterising Connectivity • We found the Gompertz model was the simplest to consistently fit C(N,r) vs. r • Three parmameter model • is the upper asymptote; and / gives the point of inflection • Since is 1, we write

18. Connectivity characterisation • 44 values of N between 2 and 500 • For each N, we conducted simulations to obtain r vs. C(N,r)values in the interval [0,1] • Simulations with Simran • 10000 runs for each N,r value • Simulations accurate to within 0.01 with 95% confidence

19. How many simulations? • The mean of n runs is known to have be within an error of where n is the number of samples and s is the standard deviation of the samples. • It can be shown that the largest value of s for connectivity experiments is 0.5 • It follows that by using n > 9604 we can ensure error within 0.01 with 95% confidence

20. Connectivity Characterisation • We obtained a table for each of the 44 values of N chosen

21. Connectivity Characterisation • We convert the Gompertz equation for C(N,r)to a linear form and perform linear regression to get values of and • Ex: N=30

22. Characterising ConnectivityGoodness of Fit for N=30

23. Characterising Connectivity • We get a table of estimated and values

24. Characterising Connectivity • We perform a second level of regression on the estimated and • In Model I we choose simple third degree equations

25. Characterising Connectivity • In Model II we use two separate equations to model distinct parts of the curve

26. Characterising Connectivity

27. Characterising Connectivity

28. Characterising Connectivity • Comparison with model of Tang and others (Model III) • Model II is closer to simulated values than Model III in every case

29. Characterising Connectivity - Validation • 236 N,r pairs • N's chosen don't contribute to model • r chosen to ensure connectivity value between 0.05 and 0.95 • 10000 simulations with the chosen N,r pairs compared with Models I and II • For Model I • N < 30: Mean absolute error 0.069; maximum 0.1756 • N > 30: mean absolute error of 0.0116 with maximum seen being 0.044 • For Model 2 • Mean absolute error of 0.0089 with maximum of 0.0418

30. Reachability

31. Connectivity in Sparse MWNs • May not be an indicator of actual extent to which network can support communication • Can be unresponsive to fine changes in network parameters • As an alternative, we propose that reachability has better properties for dealing with sparse networks

32. Reachability • Reachability: fraction of connected node pairs in the network

33. Connectivity and Reachability 60 static nodes in 2000m x 2000m distributed uniformly at random

34. Connectivity and Reachability • When reachability is 0.4 • 40% of node pairs are connected • But connectivity still at0 • Connectivity remains at 0 from R = 50 to R = 320 m • Does not indicate actual extent of communication supported by the network • This gap increases with mobility and asynchronous communication

35. Calculating reachability Nodes Links • For a network with mobility, reachability is measured as the mean of frequent snapshots

36. Properties of Reachability • Reachability: • lies in the interval [0,1] • in a sparse network is not less than its connectivity • represents the probability that a randomly chosen pair of nodes in a network is connected • represents the long term maximal packet delivery ratio achievable between random-source destination pairs in the network - Application: Normalised Packet Delivery Ratio

37. Case Study - Sparse multi-hop wireless for voice communication

38. Simulation study • Village spread across 2km x 2km • Low population density • Devices capable of multi-hop voice communication to be deployed • Simulations performed using Simran - a simulator for topological properties of wireless multi-hop networks

39. Choosing N If a certain device has R fixed at 300m, how many nodes are needed to ensure that 60% of call attempts are successful? • Assumptions for simulations • Negligible mobility • Homogenous range assignment of R • Not a realistic propagation model • Results will be optimistic, but indicative • Average of 500 simulation results for each of several values of R

40. Choosing N • Around 70 nodes are required • When reachability is 0.6, connectivity is still at 0

41. Coverage • Are nodes connecting only to nearby nodes? • ForN=70, R=300m, average shortest path lengths between nodes in a run (from 500 runs) • Max = 9.24 • Average = 5.24 • Min = 2.01 • Shortest path length of 5 implies a piece-wise linear distance greater than 600m and upto 1500m

42. Adding mobility • For the previous case, (N=70, R=300m) we introduce mobility • Simulation time: 12 hours • Random way-point • Vmin=0.5 ms-1 • Vmax=2 ms-1 • Pause = 30 mins • Reachability increases from 0.6 to 0.71

43. Asynchronous Communication • N=60, varying R • Uniform velocity of 5ms-1 • Two nodes are connected at simulation time t if a path, possibly asynchronous, existed between them within time t+30 • That is, store-and-forward message passing can happen between the two nodes in 30 seconds • 20 simulations of 500 seconds each

44. Asynchronous communication • 80% of node pairs are connected before connectivity increases from 0 • Asynchronous communication helps sparse network achieve significant degree of communication

45. Characterising Reachability

46. Modeling Reachability • Static multihop network • N - number of nodes • R - uniform transmission range • l – side of square area • Reachability is a function of: • N • r – normalised transmission range • r = R/l • Mobility M, and number of dimensions, d • Denoted as

47. l N1 N1 R 2R R Reachability of 1-D static network (N=2) N=3

48. Modeling Reachability • If N nodes form k components with mi nodes in the ith component: • Asymptotic bounds for RchN,r may be possible to derive • We are interested in finite domain results and we model RchN,r using regression on simulated data

49. Modeling Reachability • Observations from simulations indicate that reachability grows logistically • The logistic curve • Frequently used to model populations • Models rapid growth beyond a threshold up to a stable maximum

50. The logistic curve k - limiting value of y - maximum rate of growth - constant of integration Point of inflection at /