Connectivity Properties for Topology design
in Sparse Wireless Multi-hop Networks
Ph.D. Defense
Srinath Perur
Advisor: Sridhar Iyer
IIT Bombay
Introduction
2
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
3
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
4
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
5
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
6
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]
7
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 tradeoffs while designing networks?
8
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
9
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
10
Instances of a network
11
Connectivity for Sparse Networks
12
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 (ln n) n
the network is almost surely connected as
n 
13
Connectivity
• Assumption of threshold function does not
hold for small N
• We require a finite domain connectivity
model valid for entire operating range
14
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 twodimensions for N between 3 and 125
and connectivity between 0.5 and 0.99
• We present a more general and accurate
empirical model
15
Characterising Connectivity
• We characterise C(N,r) in terms of
• N - number of nodes
• r - normalised transmission range
for 3  N  500 and 0.05  C ( N , r )  0.95
• 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
16
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
17
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
18
How many simulations?
• The mean of n runs is known to have be
within an error of 1.96s / n 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
19
Connectivity Characterisation
• We obtained a table for each of the 44
values of N chosen
20
Connectivity Characterisation
• We convert the Gompertz equation for
C(N,r) to a linear form and perform linear
regression to get values of  N and  N
• Ex: N=30
21
Characterising Connectivity
Goodness of Fit for N=30
22
Characterising Connectivity
• We get a table of estimated  N and  N
values
23
Characterising Connectivity
• We perform a second level of regression
on the estimated  N and  N
• In Model I we choose simple third
degree equations
24
Characterising Connectivity
• In Model II we use two separate
equations to model distinct parts of the
curve
25
Characterising Connectivity
26
Characterising Connectivity
27
Characterising Connectivity
• Comparison with model of Tang and
others (Model III)
• Model II is closer to simulated values
than Model III in every case
28
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
29
Reachability
30
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
31
Reachability
• Reachability: fraction of connected node
pairs in the network
No. of connected node pairs
Reachabili ty 
No. of possible node pairs
32
Connectivity and Reachability
60 static nodes in 2000m x 2000m distributed uniformly at
random
33
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
34
Calculating reachability
Nodes
Links
NumConnectedPairs
Rch. 
N C2
17
Rch. 
 0.378
10 C 2
• For a network with mobility, reachability is
measured as the mean of frequent
snapshots
35
Properties of Reachability
•
Reachability:
1. lies in the interval [0,1]
2. in a sparse network is not less than its
connectivity
3. represents the probability that a randomly
chosen pair of nodes in a network is
connected
4. represents the long term maximal packet
delivery ratio achievable between randomsource destination pairs in the network
- Application: Normalised Packet Delivery Ratio
36
Case Study - Sparse multi-hop
wireless for voice communication
37
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
38
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
39
Choosing N
• Around 70 nodes are required
• When reachability is 0.6, connectivity is
still at 0
40
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
41
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
42
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
43
Asynchronous communication
• 80% of node pairs are connected before connectivity
increases from 0
• Asynchronous communication helps sparse network
achieve significant degree of communication
44
Characterising Reachability
45
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 RchNM,r,d
46
Reachability of 1-D static network (N=2)
l
N1
N1
R
2R
R
2
2 R 3R (l  2 R)
2
Rl

R
Coverage( N1 ) 
.

.2 R 
l 2
l
l
Rch
1
2,r
2R R 2

 2
l
l
Rch21,r  2r  r 2
N=3
47
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
48
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
49
The logistic curve
k
y
  x
1 e
k - limiting value of y
Point of inflection at  / 
 - maximum rate of growth
 - constant of integration
50
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
51
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
52
Characterising Reachability
•  and  fitted in terms of N:
 N  3.815(1  e
4.091102 N
)  15.4(1  e
2.055103 N
)  3.004
2  N  500
 N  5.141  0.9421N  2.597 10 3 N 2  8.42 10 6 N 3
 1.37 10 8 N 4  1.058 10 11 N 5  3.209 10 15 N 6
2  N  500
• 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
53
Characterising Reachability
54
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
55
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]
56
Edge effects on Connectivity
Properties
57
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
58
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
59
Effect of rectangularity on connectivity
properties
• Constant node density (N=30, Area = 2 sq.units, R = 0.4 units)
• Area of operation stretched
60
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
61
62
Coverage in Region 1
• Coverage in Region 1:
63
Coverage in Region 2
Area of a circular segment:
64
Coverage in Region 2
Average area outside the rectangle:
Therefore:
Simplifying:
65
Coverage of Region 3
R 2
  3  R 2
• Clearly
4
• Several cases, unwieldy to analyse
• We convert this to an equivalent
problem
66
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
67
Coverage of Region 3
• Run with: N c  N p  10000 ; l , b  2 R ; R  1

• We get:
68
Combined coverage
• Substituting for  1, 2 ,  3 and simplifying:
• For validation, we use the property that
connectivity of two nodes of range R in a l x b
area is given by:
69
Validation
• We set l=b=1 and compare with existing
simulation data for C2,r
70
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
71
Simran
72
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
73
The Simran Simulation Environment
74
Simran: Asynchronous Communication
T=1
p
q
r
75
Simran: Asynchronous Communication
T=2
p
q
r
76
Simran: Asynchronous Communication
T=3
p
q
r
• 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
77
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
78
More details about Simran
79
Conclusion
80
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
81
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
82
Questions from Reviewer 1
1.
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.
83
Questions from Reviewer 1
3. What are the time and space complexities of the Temporal
Transitive Closure algorithm?
Time – (
T
PN 3 )
dt
2
Space – ( PN )
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.
84
Publications
Publications 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.)
85
Thank you
86
Supplementary slides
87
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 measures
i. The network's ability to possess routes between
nodes; and
ii. The routing protocol's ability to exploit those
routes
88
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
89
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
gN 
N
90
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:
 N  16.16(1  e
1.947103 N
)  6.658
 N  27.8844  0.5522 N
500  N  1000
500  N  1000
91
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
92
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
93
Reachability of a 1-D static network (N=3)
94
Reachability of a 1-D static network (N=3)
• We weight the Rch for each case with its
probability of occurrence to get:
1
Rch31,r  P(c)  P(d )  P(b)
3
• We calculate P(a), P(c) and P(d) to get:
2
7
r
14
r
P(a)  (1  4r  4r 2 )(1  4r  2r 2 )  (2r  3r 2 )(1 

)
2
3
r
P(c)  (2r  r 2 )( r  r 2 )  (2r  3r 2 )( )
2
2
3
r
3
r
P(d )  (2r  r 2 )( 
)
2
8
95
Reachabilty of a 1-D static network (N=3)
Since
P(a) + P(b) + P(c) + P(d) = 1,
We can write
1
3, r
Rch
1
 [1  P(a )  2 P(c)  2 P(d )]
3
Back to modelling reachability
96
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
97
98
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 ( N 3 )
• Inconvenient when N is a few hundred nodes
• If we do not require shortest path
• Calculate reachabilty from connected
components data: (N )
• Can go up to N in thousands
99
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
Back
100