fundamental properties of wireless ad hoc networkslhyen/files/presents/manet05.pdf2005 workshop on...

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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


Outline

•Link Probability•Node Degree•Network Coverage•Connectedness•Clustering Coefficient•Quantity of Hidden Terminals

2


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.


Connectivity

•Any two nodes that are within thetransmission range of each other will have alink connecting them

．．

link

3


Border Effects

•A node placed near the boundary of therectangle area will cover less system areathen expected

Desired Area‧

‧

border effectsborder effects


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


Location-Dependent LinkProbability

r = 250

5


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


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


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


Expected Node Degree

8


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


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


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)


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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•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



•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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•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[


Expected Network Coverage

13


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


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


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


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


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

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•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



•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


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

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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


HT-Triples: Theoretical Results

20


Conclusions

•We have analyzed–Link Probability–Node Degree–Network Coverage–Clustering Coefficient–Quantity of Hidden Terminals

•Exact expression for connectedness remainsunsolved


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.

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