Manet 05

8/13/2019 Manet 05

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1

2005 Workshop on Ad Hoc Networks 1

Fundamental Properties of Wireless Ad Hoc Networks

Li-Hsing Yen

Chung Hua University

Dec. 9, 2005


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 mrectangle area

– The position of each node is arandom variable uniformly distributedover the given area.

– Each node has a transmission radiusof r unit length.


Connectivity

• Any two nodes that are within the

transmission range of each other will have a

link connecting them

．．

link

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Border Effects

• A node placed near the boundary of the

rectangle area will cover less system area

then 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)

ml

r p

2

)/2,min(when ml r

A

y x p y x

),(d, P x, y: link probability if node is at ( x, y)

A = l m


Location-Dependent Link Probability

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

x y y x A

N d)d,(d1

][E

E[ N ] / A


Results of Way 1

• Expected probability of link occurrence

22

2334

3

4

3

4

2

1

l m

ml r mr -lr -r

p

In a 1000 1000 network

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Computing Expected Link Probability: Way 2

• Let the location of node i be ( X i, Y i)

• Derive Pr[U i+V i r 2], the probability of the linkthat connects nodes i and j

where U i = ( X i X j)2, V i = (Y i Y j)

2

• Find first the pdf of U i and V

i, and then their

joint pdf

• The probability can be derived by taking anappropriate integration


Results of Way 2

• The pdf of U i

• The pdf of V i

2

2

5.0

0,1

)( l ul

luu f

2

2

5.0

0,1)( mvm

mvv g

22

2334

0 0

2

3

4

3

4

2

1

dd),(]Pr[2 2

l m

ml r mr lr r

uvvuhr V U r ur

ii

joint pdf of U i and V i= f (u) g (v)

We got the

same result!

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Node Degree: Expected Value

• Let random variable Li, j be the number of links

connecting 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 are

independent or not• E[ Li, j] = p

• E[ Di] = (n1) p


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 a

Poisson distribution with parameter

k nk

i p pk

nk D

1)1(

1]Pr[ for all i

where p = r 2 / A


Node Degrees Are Not

Independent

• Consider Di and D j

• Pr[ Di = 0 , D j = n1] = 0

• Pr[ Di = 0] Pr[ D j = n1] 0

• Di and D j are not independent for all i, j

• The joint pdf of Di and D j are hard to

derive

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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 in

this area is within the communication range of some

node.

• Network coverage: the area collectively covered by aset of nodes

Desired Area

Coverage < 100%

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Network Coverage: Two Factors

• region covered by each node may

overlap one another in a stochastic way

• a node placed near the border of the

deployment region will cover less areathan nodes placed midway (border

effects)


Network Coverage Estimate (1)

• The deployment of n nodes can be modeled as a

stochastic process that places nodes one by one

according to a uniform distribution over R

• When a node is placed, only a portion of its node

coverage gives extra network coverage

．．

．

Extra

coverage

Extracoverage

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• Let C i be the random variable denoting the size of the

covered region collectively offered by i randomly

placed nodes

• Let X i denote the extra network coverage contributed

by the i-th placed node

]E[]E[]E[ 11 N X C

expected node

coverage (p.9)

C i = C i1 + X i for all i, 2 i n

E[C i] = E[C i1 + X i ] = E[C i1] + E[ X i]

for all i, 2 i n



• Let N i be the node coverage of the i-th

placed node

• Let F i = X i / N i

• If border effects are ignored

N i = r 2, a constant independent of F i,

so E[ F i N i] = E[ F i] E[ N i]

E[C i] = E[C i1]+ E[ F i N i] for all i, 2 i n

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• As nodes are uniformly distributed, F i is expected to

be the proportion of the uncovered area to the whole

• It turns out that

• Since E[C 1] = E[ N ]

A

C A F i

i

]E[]E[ 1

]E[

]E[

1]E[]E[ 1

1 N A

C

C C i

ii

A A

N C

n

n

]E[

11]E[


Expected Network Coverage

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k -Coverage Problem

• Find the area that can be covered by at

least 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 dynamic

programming

]E[)1(]E[0

t d j

d i

t t d d

t

j

i C p pt

d C

: the size of the j-covered area after i nodes

have been randomly placed

j

iC

holds for any integer d , 0 d i j

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Expected k -Coverage

r = 100

A = 10001000


Connectedness

• A network is said to be connected if it

contains no isolated node

• phase transitions – an ad hoc network possesses many graph

properties with a rather small increase in

the expected number of edges

B. Krishnamachari et al., “Critical Density Thresholds in Distributed

Wireless 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 E c

mi: the num of i’s neighbors

E i: the num of links that exist among i’s neighbors

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Clustering Coefficient: Examples

102

45

2

12

ii

i

mm ,mC

6i E 60

10

6

2.

,mC

E c

i

ii


Clustering Coefficient Estimate (1)

• Given any node A with m ≥ 2 neighbors, let

N ( A) = { X 1 , X 2 , · · ·, X m} be the set of A’s

neighbors

• For any X i∈ N ( A), let N ( A)i = { X j |X j∈ N ( A)∧ X j∈ N ( X i)}

• The expected number of links connecting any

two neighbors of A is

m

i

i A N 1

)(E2

1

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• The expected area jointly covered by two

neighboring nodes is

• It follows that

4

332

r

m

i

i

m

i

i A N A N 11

])(E[2

1)(E

2

1

4

331)1(|])([| m A N E i for all i

(assuming torus conv.) ．．

expected

area



• Therefore,

• Dividing this value by C(m,2) yields the

expected clustering coefficient

4

331

2

)1(])(E[

2

1

1

mm A N

m

i

i

4

331c a constant

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Clustering Coefficient:Experimental Results

With torus convention Without torus convention


Hidden Terminal Triples

• X, Y,Z forms an HT-triple if Y located within X ’ scoverage region and Z located within Y ’ s coverage

region but not within X ’ s

• the probability of HT-triple X, Y,Z is

Y

X Z

An HT-triple

Z must be in

this region

X’s coverage

Y’s coverage

2

22

2

)1(4

33

pclm

r r

lm

r

Y’s locatedwithin X’s

coverage

Z’s located within

region Y - X

joint

node

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Quantity of HT-Triples

• There are C (n, 3) ways to select three nodes

from n nodes without order

• Any selection may yield three possible HT-

triples, each corresponding to a distinct joint

node

• Total number of HT-triples

22)2)(1(

2

1)1(

33 pnnn

c pc

n

η∝ n3 p2


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 remains

unsolved


References

• L.-H. Yen and C. W. Yu, “Link probability, network coverage,and related properties of wireless ad hoc networks,” The 1st IEEE 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 wireless

ad hoc networks and the quantity of hidden terminals,” IEEE Communications Letters, 9(3): 234-236, Mar. 2005.

• C. W. Yu and L.-H. Yen, “Computing subgraph probability of random geometric graphs: Quantitative analyses of wireless adhoc networks,” 25th IFIP WG 6.1 Int'l Conf. on Formal Techniques 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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