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2005 Workshop on Ad Hoc Networks 1
Fundamental Properties ofWireless Ad Hoc Networks
Li-Hsing YenChung Hua University
Dec. 9, 2005
2005 Workshop on Ad Hoc Networks 2
Outline
•Link Probability•Node Degree•Network Coverage•Connectedness•Clustering Coefficient•Quantity of Hidden Terminals
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Network Model
•n, r, l, m-network–A set of n nodes placed in an l m
rectangle area–The position of each node is a
random variable uniformly distributedover the given area.
–Each node has a transmission radiusof r unit length.
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Connectivity
•Any two nodes that are within thetransmission range of each other will have alink connecting them
. .
link
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Border Effects
•A node placed near the boundary of therectangle area will cover less system areathen expected
Desired Area‧
‧
border effectsborder effects
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Torus Convention
•turns the system area into a torus•the region covered by any node is considered
completely within the system•border effects are gone
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Link Probability
•with torus convention; location independent
•without torus convention–location dependent; must consider border
effects
d(x, y): the area covered by a node located at (x, y)
mlr
p
2
)/2,min(when mlr
Ayx
p yx),(d
, Px,y: link probability if node is at (x, y)
A = l m
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Location-Dependent LinkProbability
r = 250
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Expected Link ProbabilityConsidering Border Effects
•Way 1–Compute the expected coverage of a node
•R: the deployment region.–Link probability = expected coverage /
overall system area
R
xyyxA
N d)d,(d1
][E
E[N] / A
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Results of Way 1
•Expected probability of link occurrence
22
2334
34
34
21
lm
mlrmr-lr-rp
In a 1000 1000 network
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Computing Expected LinkProbability: Way 2
•Let the location of node i be (Xi, Yi)
•Derive Pr[Ui+Vi r2], the probability of the linkthat connects nodes i and j
where Ui = (XiXj)2, Vi = (YiYj)2
•Find first the pdf of Ui and Vi, and then theirjoint pdf
•The probability can be derived by taking anappropriate integration
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Results of Way 2
•The pdf of Ui
•The pdf of Vi
22
5.0
0,1
)( lul
luuf
22
5.0
0,1
)( mvm
mvvg
22
2334
0 0
2
34
34
21
dd),(]Pr[2 2
lm
mlrmrlrr
uvvuhrVUr ur
ii
joint pdf of Ui and Vi= f(u) g(v)
We got thesame result!
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Node Degree: Expected Value
•Let random variable Li,j be the number of linksconnecting nodes i and j (Li,j = 1 or 0).
•Let Di = i Li,j be the node degree of i
•E[Di] = E[i Li,j] = i E[Li,j] no matter Li,j’s areindependent or not
•E[Li,j] = p
•E[Di] = (n1)p
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Expected Node Degree
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Node Degree Distribution
•The probability that a node has k links•With torus convention; location independent
•When n(or equivalently, p0) but np = remainsto be a constant, the binomial distribution becomes aPoisson distribution with parameter
knki pp
kn
kD
1)1(
1]Pr[ for all i
where p = r2 / A
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Node Degrees Are NotIndependent
•Consider Di and Dj
•Pr[Di = 0, Dj = n1] = 0
•Pr[Di = 0] Pr[Dj = n1] 0
•Di and Dj are not independent for all i, j
•The joint pdf of Di and Dj are hard toderive
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Network Links: Expected Value
•Total number of links in a network•L = Di / 2•E[L] = E[Di] = E[Di] no matter Di’s
are independent or not
•E[Di] = (n1)p
•E[L] = n(n1) p / 2
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Network Coverage
•A piece of area is said to be covered if every point inthis area is within the communication range of somenode.
•Network coverage: the area collectively covered by aset of nodes
Desired AreaCoverage < 100%
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Network Coverage: Two Factors
•region covered by each node mayoverlap one another in a stochastic way
•a node placed near the border of thedeployment region will cover less areathan nodes placed midway (bordereffects)
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Network Coverage Estimate (1)
•The deployment of n nodes can be modeled as astochastic process that places nodes one by oneaccording to a uniform distribution over R
•When a node is placed, only a portion of its nodecoverage gives extra network coverage
...
Extracoverage
Extracoverage
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Network Coverage Estimate (2)
•Let Ci be the random variable denoting the size of thecovered region collectively offered by i randomlyplaced nodes
•Let Xi denote the extra network coverage contributedby the i-th placed node
]E[]E[]E[ 11 NXC expected nodecoverage (p.9)
Ci = Ci1 + Xi for all i, 2 i n E[Ci] = E[Ci1 + Xi ] = E[Ci1] + E[Xi]for all i, 2 i n
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Network Coverage Estimate (3)
•Let Ni be the node coverage of the i-thplaced node
•Let Fi = Xi / Ni
•If border effects are ignoredNi = r2, a constant independent of Fi,so E[Fi Ni] = E[Fi] E[Ni]
E[Ci] = E[Ci1]+ E[Fi Ni] for all i, 2 i n
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Network Coverage Estimate (4)
•As nodes are uniformly distributed, Fi is expected tobe the proportion of the uncovered area to the whole
• It turns out that
•Since E[C1] = E[N]
ACA
F ii
]E[]E[ 1
]E[]E[
1]E[]E[ 11 N
AC
CC iii
AAN
Cn
n
]E[11]E[
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Expected Network Coverage
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k-Coverage Problem
•Find the area that can be covered by atleast k out of n randomly placed nodes
‧
‧
‧1-coverage
2-coverage 3-coverage
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k-Coverage Estimate
•Similar to the 1-coverage case
•It can be computed by way of dynamicprogramming
]E[)1(]E[0
tdjdi
ttdd
t
ji Cpp
td
C
: the size of the j-covered area after i nodeshave been randomly placed
jiC
holds for any integer d, 0 d i j
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Expected k-Coverage
r = 100A = 10001000
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Connectedness
•A network is said to be connected if itcontains no isolated node
•phase transitions–an ad hoc network possesses many graph
properties with a rather small increase inthe expected number of edges
B. Krishnamachari et al., “Critical Density Thresholds in DistributedWireless Networks,”Communications, Information and Network Security,Kluwer Publishers, 2002.
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Connectedness: ExperimentalResults
1250 1250
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Clustering Coefficient
•the extent to which a node’s neighbors arealso neighbors to each other
•Node i’s clustering coefficient
•The clustering coefficient of the whole networkis the average of all individual ci’s
)2,( i
ii mC
Ec
mi: the num of i’s neighborsEi: the num of links that exist among i’s neighbors
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Clustering Coefficient: Examples
10
245
21
2
iii
mm,mC
6iE 60106
2.
,mCE
ci
ii
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Clustering Coefficient Estimate (1)
•Given any node A with m ≥2 neighbors, letN(A) = {X1, X2, · · ·, Xm} be the set of A’sneighbors
•For any Xi ∈ N(A), let N(A)i = {Xj |Xj ∈ N(A) ∧Xj ∈ N(Xi)}
•The expected number of links connecting anytwo neighbors of A is
m
iiAN
1
)(E21
17
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Clustering Coefficient Estimate (2)
•The expected area jointly covered by twoneighboring nodes is
•It follows that
4332 r
m
ii
m
ii ANAN
11
])(E[21
)(E21
433
1)1(|])([| mANE i for all i
(assuming torus conv.) . .
expectedarea
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Clustering Coefficient Estimate (3)
•Therefore,
•Dividing this value by C(m,2) yields theexpected clustering coefficient
433
12
)1(])(E[
21
1
mmAN
m
ii
433
1c a constant
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Clustering Coefficient:Experimental Results
With torus convention Without torus convention
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Hidden Terminal Triples
•X, Y,Zforms an HT-triple if Y located within X’scoverage region and Z located within Y’s coverageregion but not within X’s
• the probability of HT-triple X, Y,Zis
Y
XZ
An HT-tripleZ must be inthis region
X’s coverage
Y’s coverage
2
22
2
)1(4
33
pclm
rr
lmr
Y’s locatedwithin X’scoverage
Z’s located withinregion Y - X
jointnode
19
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Quantity of HT-Triples
•There are C(n, 3) ways to select three nodesfrom n nodes without order
•Any selection may yield three possible HT-triples, each corresponding to a distinct jointnode
•Total number of HT-triples
22 )2)(1(2
1)1(
33 pnnn
cpc
n
η ∝ n3p2
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HT-Triples: Theoretical Results
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Conclusions
•We have analyzed–Link Probability–Node Degree–Network Coverage–Clustering Coefficient–Quantity of Hidden Terminals
•Exact expression for connectedness remainsunsolved
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References
• L.-H. Yen and C. W. Yu, “Link probability, network coverage,and related properties of wireless ad hoc networks,”The 1stIEEE Int'l Conf. on Mobile Ad-hoc and Sensor Systems, Oct.2004, pp. 525-527.
• L.-H. Yen and Y.-M. Cheng, “Clustering coefficient of wirelessad hoc networks and the quantity of hidden terminals,”IEEECommunications Letters, 9(3): 234-236, Mar. 2005.
• C. W. Yu and L.-H. Yen, “Computing subgraph probability ofrandom geometric graphs: Quantitative analyses of wireless adhoc networks,”25th IFIP WG 6.1 Int'l Conf. on FormalTechniques for Networked and Distributed Systems, Oct. 2005,LNCS, vol. 3731, pp. 458-472.
• L.-H. Yen, C. W. Yu, and Y.-M. Cheng, “Expected k-coverage inwireless sensor networks,”Ad Hoc Networks, to appear.