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

Srinath Perur

Advisor: Sridhar Iyer

IIT Bombay

Ph.D. Defense

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 n

n)(ln

n

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

5003 N 95.0),(05.0 rNC

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

ns /96.1

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 NN

• Ex: N=30

22

Characterising ConnectivityGoodness of Fit for N=30

23

Characterising Connectivity

• We get a table of estimated and

valuesN N

24

Characterising Connectivity

• We perform a second level of regression on the estimated and

• In Model I we choose simple third degree equations

N N

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

pairs node possible of No.

pairs node connected of No.tyReachabili

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 at 0

• 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

• For a network with mobility, reachability is measured as the mean of frequent snapshots

LinksNodes

CNedPairsNumConnect

Rch2

.

378.010

17.

2

C

Rch

36

Properties of Reachability

• Reachability:1. lies in the interval [0,1]2. in a sparse network is not less than its

connectivity3. represents the probability that a randomly

chosen pair of nodes in a network is connected4. 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?

• For N=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 dMrNRch ,

,

47

Reachability of 1-D static network (N=2)

N=3

R R2R

N1N1

l

Rl

RlR

l

RNCoverage 2.

)2(

2

3.

2)( 1

l

RRl 22

2

21

,2

2

l

R

l

RRch r

21,2 2 rrRch r

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

xe

ky

1

Point of inflection at /

51

Characterising Reachability

• For fixed N, reachability varies logistically with r:

• r – transmission range normalized with side of square

• and are estimated by fitting to simulation results of runs for various values of N

52

Characterising Reachability

• Simulations• 55 values of N between 2 and 500• For each N, several values of r to span

reachability from 0 to 1• Each simulation run on 1000 randomly

generated network graphs• Mean error within 0.018 with 95% confidence

• Yields a table of r vs. Rch(N,r) for one value of N

53

Characterising Reachability

• and fitted in terms of N:

• Average relative error around 3.5% for cases that didn’t contribute to the model

• Equations for and together with logistic equation characterize reachability

• Model extended for N > 500

004.3)1(4.15)1(815.332 10055.210091.4

NNN ee 5002 N

3623 1042.810597.29421.0141.5 NNNN

61551148 10209.310058.11037.1 NNN 5002 N

54

Characterising Reachability

55

Spanner

• Design tool for sparse MWNs• Given three values from N, R, l and Rch,

computes the fourth

• Uses reachability model

• Particularly useful for finding N• Cannot solve directly because and

are functions of N• Binary search

56

Mobility

• Our models for reachability and connectivity are most useful when nodes are mobile

• Can be used with mobility models that retain uniform random distribution of nodes assumed in the model

• Ex: Random direction [RMSM01]

57

Edge effects on Connectivity Properties

58

Common assumptions in MWN topology design

• Square or d-cube area of operation• Allows generalising results to 1-, 2-, and 3-d

• Toroidal area of operation• No edge effects to handle

• Using node density as a parameter• Subsumes both N and area of operation

59

However…

• Many practical deployment areas are rectangular

• Connectivity properties are not geometry invariant • Two networks with similar nodes and equal

node densities can have different values of connectivity

• Significant in sparse networks

60

Effect of rectangularity on connectivity properties

• Constant node density (N=30, Area = 2 sq.units, R = 0.4 units)• Area of operation stretched

61

Edge effect

• Part of transmission range not being used for connectivity

• We define coverage as the effective transmission area of a node

• Aim:• To determine exptected coverage, , for a

single node with transmission range, R, in an l x b rectangle

• To obtain an effective transmission range, Rlb to use with existing results for square areas

62

63

Coverage in Region 1

• Coverage in Region 1:

64

Coverage in Region 2

Area of a circular segment:

65

Coverage in Region 2

Average area outside the rectangle:

Therefore:

Simplifying:

66

Coverage of Region 3

• Clearly • Several cases, unwieldy to analyse• We convert this to an equivalent

problem

23

2

4R

R

67

Coverage of Region 3

• Equivalent problem:

In a Cartesian co-ordinate system, consider a circle of radius R centred between (0,0) and (R,R). Find the average fraction of this circle's area lying in the rectangle formed by (0,0) and (l,b).

• We find this area by Monte Carlo simulation• Probabilistic method for evaluating expressions• Often used to evaluate inconvenient definite

integrals

68

Coverage of Region 3

• Run with: ; ;

• We get:

1

RRbl 2, 10000 pc NN

69

Combined coverage

• Substituting for , , and simplifying: 1 32

• For validation, we use the property that connectivity of two nodes of range R in a l x b area is given by:

70

Validation

• We set l=b=1 and compare with existing simulation data for C2,r

71

Equivalent square network

• Expected number of neighbours per node is a known invariant for reachability and connectivity [NC94]

• Since coverage determines number of neighbours we equate coverage equations for a square and rectangle to get:

• Solving for a gives the side of the equivalent square network

72

Simran

73

Simran - Design Goals

• Simran - simulator for topological simulations of MWNs

• Support for metrics significant to design of sparse MWNs• Connectivity, reachability, size and number of

connected components, average number of neighbours, shortest paths, etc.

• Mobility support• Easy introduction of new mobility models

• Support for asynchronous communication• Ease of running comparative simulations

74

The Simran Simulation Environment

75

Simran: Asynchronous Communication

T=1

p q r

76

Simran: Asynchronous Communication

T=2

p q r

77

Simran: Asynchronous Communication

T=3

p rq

• p and r are connected within a patience factor of 3 time units• Patience factor corresponds to packet lifetime

• Directional connectivity - r and p are not connected

78

Simran: Temporal Transitive Closure (TTC)

• Modification of Floyd-Warshall transitive closure algorithm

• For a patience factor of P• Maintain sliding window of network state

for last P-1 steps

• Qt - transitive closure of adjacency matrix at present time, t

• TTC - collapses Qt-p to Qt in into a single matrix in direction of time

79

More details about Simran

80

Conclusion

81

Contributions

• Connectivity for sparse networks• Empirical, finite domain characterisation

• Reachability• Definition and properties• Applications • Characterisation in the finite domain

• 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

82

Some future directions• Analytical models for reachability• Topology design in three dimensional

networks• Are existing metrics sufficient?• Characterisations for 3D networks

• Models to interpret analytical results for deployment purposes

• Simulation techniques• Realistic propagation models• Temporal network graph representations

83

Questions from Reviewer 11. Do you think this study could be applied to wireless sensor

networks? If so, could you enumerate a few applications? If not, why?

- It is applicable in any randomly deployed mobile sensor network.

2. Can you extend this work to coverage in WSNs where a sensor field does not have to be fully covered, i.e., consider coverage instead of reachability?

- This can be formulated as a problem in which discs of radii equal to the sensors' sensing range are dropped randomly to cover an area. It is very likely that the principle behind the usefulness of sparse networks is applicable here – it would be more economical if a small area left uncovered could be tolerated, and there could be a tradeoff between sensing range, number of sensors and covered area. An added factor of interest is a simultaneous requirement of some level of network communication. But it is not clear that our work can be directly extended to this problem.

84

Questions from Reviewer 1

3. What are the time and space complexities of the Temporal Transitive Closure algorithm?

Time –

Space –

where N is the number of nodes in the network, T is the simulation time, dt is the simulation granularity and P is the number of time steps for which the TTC is to be computed.

)( 3PNdt

T

)( 2PN

85

PublicationsPublications from the work presented (with Sridhar Iyer):• Characterization of a connectivity measure for sparse wireless multi-hop networks.

Workshop on Wireless Ad hoc and Sensor Networks (WWASN), in conjunction with ICDCS, Lisboa, July 2006. (Expanded version appears in Ad Hoc and Sensor Wireless Networks Journal)

• Designing sparse wireless multi-hop networks. Student workshop paper at IEEE INFOCOM, Barcelona, April 2006.

• Reachability: An alternative to connectivity for sparse wireless multi-hop networks. Poster at IEEE INFOCOM, Barcelona, April 2006.

• Sparse multi-hop wireless for voice communication in rural India. National Conference on Communications (NCC), New Delhi, January 2006.

Other publications:• Bridging the gap between reality and simulation: An Ethernet case study. (To appear) Conference on

Information Technology (CIT), Bhubaneswar, December 2006. (With Punit Rathod and Raghuraman Rangarajan.)

• Improving the performance of MANET routing protocols using cross-layer feedback. Conference on Information Technology (CIT), Bhubaneswar, December 2003. (With Leena Chandran-Wadia and Sridhar Iyer.)

• Router handoff: A preemptive route repair strategy for AODV. IEEE International Conference on Personal Wireless Computing (IEEE ICPWC), New Delhi, December 2002. (With Abhilash P. and Sridhar Iyer.)

• Router handoff: preemptive route repair in mobile ad hoc networks. International Conference on High Performance Computing (HiPC), Bangalore, December 2002. (With Abhilash P. and Sridhar Iyer.)

86

Thank you

87

Supplementary slides

88

Normalised PDR for sparse networks

• In a dense (unsaturated) network • Packet Delivery Ratio (PDR) measures the

ability of the routing protocol to deliver packets to the intended destination

• In a sparse network, PDR measuresi. The network's ability to possess routes between

nodes; and

ii. The routing protocol's ability to exploit those routes

89

Normalised PDF for Sparse Networks

• Using the property of Rch as maximal PDR, we can identify only the routing contribution by normalising PDR with reachability

• NPDR = PDR/Rch• PDR value can be obtained from packet-level

simulations or test-bed experiments• Rch for the network can be obtained from

simulations or from a model

Back to properties of reachability

90

Characterising Reachability

• For larger values of N• RchN,r resembles a step function

• RchN,r increase from 0.1 to 0.9 requires 0.3 increase in r when N = 10, but only 0.015 increase in r when N = 500

• Characterization is equivalent to finding the transition point, gN. This is given by the point of inflection for the logistic curve

N

NNg

91

Characterising Reachability

• Since the shape of and beta curves is relatively stable beyond N=200• We estimate alpha and beta for N between

500 and 1000 by extrapolating from data points between 200 and 500:

658.6)1(16.16310947.1

NN e

NN 5522.08844.27 1000500 N

1000500 N

92

Characterising Reachability

• Setting r = gN - 0.01 results in RchN,r

close to 0, and setting r = gN + 0.01 results in RchN,r close to 1

93

Reachability of a 1-D static network (N=3)

• The ways in which three nodes can be positioned are:

a - All three nodes are isolated

b - One node is isolated and two are connected

c - All three nodes are connected with one intermediate hop

d - All three nodes are directly connected to each other

94

Reachability of a 1-D static network (N=3)

95

Reachability of a 1-D static network (N=3)

• We weight the Rch for each case with its probability of occurrence to get:

• We calculate P(a), P(c) and P(d) to get:

)(3

1)()(1

,3 bPdPcPRch r

)8

3

2

3)(2()(

22 rrrrdP

)2

)(32())(2()( 222 rrrrrrrcP

)3

14

2

71)(32()241)(441()(

2222 rrrrrrrraP

96

Reachabilty of a 1-D static network (N=3)

SinceP(a) + P(b) + P(c) + P(d) = 1,

We can write

Back to modelling reachability

)](2)(2)(1[3

11,3 dPcPaPRch r

97

Simran - Important Data Structures

struct mobilityModel: <mmType, Xmax, Ymax, Zmax, Vmin, Vmax,pauseTime>

struct Node: <x, y, z, dxBydt, dyBydt, dzBydt, stopTime, lastUpdated>

Adj: adjacency matrix; Adj[i][j] is

set to 1 if nodes I and j are within R of each other.

Dist: matrix with shortest distances between all pairs of nodes.

Pre: matrix with precursor node on shortest path.

connC: list of connected components

cSize: list of connected component sizes

98

99

Complexity and Scalability• We get number of connected node pairs

from the Dist matrix for reachability calculation

• However, Floyd-Warshall all-pairs shortest path run in time• Inconvenient when N is a few hundred nodes

• If we do not require shortest path• Calculate reachabilty from connected

components data:• Can go up to N in thousands

)( 3N

)(N

100

Complexity and Scalability

• User chooses• Simulation time - T• Simulation granularity - dt

• dt can be set carefully to reduce execution time• Low mobility simulations can have large dt

• dt can be used to trade-off precision for execution time• Fewer snapshots of network state

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