Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of...
Transcript of Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of...
![Page 1: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/1.jpg)
Self Organisation in Random Geometric Graphmodels of Wireless Sensor Networks
Swaprava Nath
under the guidance ofProf. Anurag Kumar
ME Final Presentation
Department of Electrical Communication EngineeringIndian Institute of Science, Bangalore 560012, India
June 26, 2008
Swaprava (ECE, IISc) Self Organisation June 26, 2008 1 / 45
![Page 2: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/2.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 2 / 45
![Page 3: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/3.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 3 / 45
![Page 4: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/4.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 4 / 45
![Page 5: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/5.jpg)
Geometric Graph
b1 ≡ (0, 0)
Area to monitor, A
b2 ≡ (1, 0)
b3 ≡ (1, 1)b4 ≡ (0, 1)
sample deployment v
r
Neighbours in theGeometric GraphG(v, r)
Node locations can be arbitrary or random
Swaprava (ECE, IISc) Self Organisation June 26, 2008 5 / 45
![Page 6: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/6.jpg)
Definition of Hop Distance (HD)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Hop distance h
Area to monitor, A
Euclidean distance d
Swaprava (ECE, IISc) Self Organisation June 26, 2008 6 / 45
![Page 7: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/7.jpg)
Definition of Hop Distance (HD)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Hop distance h
Area to monitor, A
Euclidean distance d
Question: Relation between hop-distance (HD) and Euclidean distance (ED)?
Swaprava (ECE, IISc) Self Organisation June 26, 2008 6 / 45
![Page 8: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/8.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 7 / 45
![Page 9: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/9.jpg)
Assumption in HCRL
Hop Count Ratio-based Localisation (HCRL), proposed by Yang et al.[IEEE SECON 2007]
Assumption: d ∝ h, hence d1d2
= h1h2
Suppose, node location (x , y), Anchors (x01, y01) and (x02, y02)√
(x − x01)2 + (y − y01)2√
(x − x02)2 + (y − y02)2≈ h1
h2⇐ Equation of circle
0 200 400 600 800 10000
200
400
600
800
1000
B1
B2
B4
B3
s
Swaprava (ECE, IISc) Self Organisation June 26, 2008 8 / 45
![Page 10: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/10.jpg)
Assumption in PDM
Proximity Distance Map (PDM), proposed by Lim and Hou [IEEEInfocom 2005]
L anchors, node i has HD vector hi ∈ NL
Assumption: ED vector di = Thi
ED matrix between anchors, D = [d1, · · · ,dL], is known
HD matrix between anchors, H = [h1, · · · ,hL], is computed
D = TH ⇒ T = DHT(HHT)−1
This T is used for all non-anchor nodes
Node location estimated from the ED vector di
Swaprava (ECE, IISc) Self Organisation June 26, 2008 9 / 45
![Page 11: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/11.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 10 / 45
![Page 12: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/12.jpg)
HD-ED Relationship in Arbitrary GG
Setting:I n nodes placed on unit area A arbitrarilyI v = [v1, v2, · · · , vn] ∈ An
I Geometric graph G(v, r) is formed
Notation:I N = [n] = {1, 2, · · · , n}, the index set of the nodesI Hl,i(v) = HD of node i from l th anchor on G(v, r), for the deployment
vI Dl,i(v) = Euclidean distance of node i from anchor bl for the
deployment v.D l(v, hl) = max
{i∈N :Hl,i (v)=hl}Dl,i(v)
D l(v, hl) = min{i∈N :Hl,i (v)=hl}
Dl,i(v)
Swaprava (ECE, IISc) Self Organisation June 26, 2008 11 / 45
![Page 13: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/13.jpg)
Graphical Illustration
sample deployment v
Dl(v, hl)
Dl(v, hl)
bl
Area to monitor, A
These paths are on G(v, r)
Anchor can be anywhere in A, this is an examplelth anchor location
Swaprava (ECE, IISc) Self Organisation June 26, 2008 12 / 45
![Page 14: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/14.jpg)
HD-ED Relationship in Arbitrary GG (Contd.)
Lemma
For arbitrary v and hl ≥ 2, r < D l(v, hl ) ≤ D l(v, hl ) ≤ hl r and both
bounds are sharp.
= r
s
d
A regular hl + 1 sided polygon
γ > π3
∀hl > 2
= rall internal angles = γ
3
2
1
hl + 1
= r
s
d
r1 = r + δ′, δ′ > 0
hop distance between s and d = hl
this edge is deleted
all other angles
γ + δ
3
2
1
increase by δ
hl + 1
Figure: Node placement on the right achieves the lower bound of ED
HD does not give useful information about ED in Arbitrary GG
Swaprava (ECE, IISc) Self Organisation June 26, 2008 13 / 45
![Page 15: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/15.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 14 / 45
![Page 16: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/16.jpg)
Paradigms of HD-ED Relationship in RGG
HD-ED Relationship between Fixed Points
HD-ED Relationship between Random Nodes
HD-ED Relationship between Fixed Point andRandom Node
Swaprava (ECE, IISc) Self Organisation June 26, 2008 15 / 45
![Page 17: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/17.jpg)
HD-ED Relationship between Fixed Points
Setting:I n nodes placed on unit area A Uniform i.i.d.I Node location vector v = [v1, v2, · · · , vn] ∈ An
I Pn(.) is the probability measure
I Geometric graph G(v, r(n)) is formed
Notation:I Hb1b2(v) is the hop distance between any two points b1 and b2 on A,
for the sample deployment v
I We will take r(n) = c
√
log nn
, c >1√π, a constant, to guarantee
asymptotic connectivity (Gupta and Kumar, 1998)
Graphical illustration of hop−distancebetween two given points on the unit area
this is counted as one hop
d
r(n)
r(n)node location
this path is on the geometric graph
the minimum number of hops in all such possible pathsconnecting the two points is called the hop−distancenode location
2b
b1
Swaprava (ECE, IISc) Self Organisation June 26, 2008 16 / 45
![Page 18: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/18.jpg)
HD-ED Relationship between Fixed Points (Contd.)
Theorem
For all ε, 1 > ε > 0, if c2(ε) ≥ 2
q√
1−p2, where p and q are any two
constants satisfying 1 − ε < p < 1 and 0 < q < p − (1 − ε) and p ≥ 2q,
on a unit square A,
limn→∞
Pn
{
v : ∀z1, z2 ∈ A,z1z2
r(n, ε)≤ Hz1z2(v) <
z1z2
(1 − ε)r(n, ε)
}
= 1
where r(n, ε) = c(ε)√
log nn
.
i.e., with high probability, ED between any two points is roughly equalto HD × radius of the RGG, where the radius is larger than thecritical radius by a constant factor
Swaprava (ECE, IISc) Self Organisation June 26, 2008 17 / 45
![Page 19: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/19.jpg)
HD-ED Relationship between Random Nodes
Setting:I n nodes placed on unit area A Uniform i.i.d.I Node location vector v = [v1, v2, · · · , vn] ∈ An
I Pn(.) is the probability measure
I Geometric graph G(v, r(n)) is formed
Notation:I N = [n] = {1, 2, · · · , n}, the index set of the nodes, i.e., node i ∈ N
has a location vi on A.I Da,b(v): The Euclidean distance on A between two nodes a and b,
a, b ∈ N , for the sample deployment v.I Ha,b(v): The hop distance on G(v, r(n)) between two nodes a and b,
a, b ∈ N , for the sample deployment v.
D(v, h) = max{(a,b)∈N 2:Ha,b(v)=h}
Da,b(v)
D(v, h) = min{(a,b)∈N 2:Ha,b(v)=h}
Da,b(v)
Swaprava (ECE, IISc) Self Organisation June 26, 2008 18 / 45
![Page 20: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/20.jpg)
HD-ED Relationship between Random Nodes (Contd.)
We want bounds on the ED between any pair of nodes which are at ahop-distance h from each other
Area to monitor, A
Two different pairs of nodesboth h hops away from each other
We want the EDs of all such pairs to be bounded between
RGG is formed with radius r(n)
[kr(n), hr(n)] simultaneously, k < h is a constant
Swaprava (ECE, IISc) Self Organisation June 26, 2008 19 / 45
![Page 21: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/21.jpg)
HD-ED Relationship between Random Nodes (Contd.)
Define,
Eh(n, ε)
=
{
v :
(
(1 − ε)(h − 1) − 1
2
)
r(n, ε) ≤ D(v, h) ≤ D(v, h) ≤ hr(n, ε)
}
Theorem
For 1 > ε > 0, if c2(ε) ≥ 1g(ε) , where
g(ε) = q(ε)√
1 − p2(ε), and
p(ε) =1−ε+
√(1−ε)2+84 ,
q(ε) =−3(1−ε)+
√(1−ε)2+8
4 ,
Pn(Eh(n, ε)) = 1 −O
(
n1−c2(ε)g(ε)
lnn
)
Thus, limn→∞
Pn(Eh(n, ε)) = 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
g(e
psi
lon
)epsilon
g(epsilon) vs epsilon plot
Swaprava (ECE, IISc) Self Organisation June 26, 2008 20 / 45
![Page 22: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/22.jpg)
HD-ED Relationship between Fixed Point and Random
Node
Setting:I n nodes placed on unit area A Uniform i.i.d.I Node location vector v = [v1, v2, · · · , vn] ∈ An
I Pn(.) is the probability measure
I Geometric graph G(v, r(n)) is formed
Notation:I N = [n] = {1, 2, · · · , n}, the index set of the nodesI Hl,i(v) = HD of node i from l th anchor on G(v, r(n)), for the
deployment vI Dl,i(v) = Euclidean distance of node i from anchor bl for the
deployment v.D l(v, hl) = max
{i∈N :Hl,i (v)=hl}Dl,i(v)
D l(v, hl) = min{i∈N :Hl,i (v)=hl}
Dl,i(v)
Swaprava (ECE, IISc) Self Organisation June 26, 2008 21 / 45
![Page 23: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/23.jpg)
HD-ED Relationship between Fixed Point and Random
Node (Contd.)
We want bounds on the ED between a fixed point and all nodes at ahop-distance h from the point
We want an upper and a lower bound on the ED
Dl(v, hl)
Dl(v, hl)
bl
Area to monitor, A
Swaprava (ECE, IISc) Self Organisation June 26, 2008 22 / 45
![Page 24: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/24.jpg)
HD-ED Relationship between Fixed Point and Random
Node (Contd.)
Define, Ehl(n) = {v : (1− ε)(hl − 1)r(n) ≤ D l(v, hl ) ≤ D l (v, hl ) ≤ hl r(n)}
Theorem
For a given 1 > ε > 0, and r(n) = c
√
ln nn
, c >1√π,
Pn(Ehl
(n)) = 1 −O(
1
ng(ε)c2
)
where g(ε) = q(ε)√
1 − p2(ε), p(ε) =1−ε+
√(1−ε)2+8
4 ,
q(ε) =−3(1−ε)+
√(1−ε)2+8
4 .
Hence, limn→∞
Pn(Ehl
(n)) = 1Since g(ε) ↓ as ε ↓, the rate of convergence slows down
Swaprava (ECE, IISc) Self Organisation June 26, 2008 23 / 45
![Page 25: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/25.jpg)
Proof Techniques (1/4)
hlr(n)
. . .
J(n) blades. . .
hl strips in each blade
bl
blade Blj
A
Figure: Construction using the blades cutting the circumference of the circle ofradius hl r(n).
Swaprava (ECE, IISc) Self Organisation June 26, 2008 24 / 45
![Page 26: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/26.jpg)
Proof Techniques (2/4)
Blade
· · ·· · ·
r(n)r(n)
hl1 2
u(n) =√
1 − p2r(n)
t(n) = qr(n)
pr(n)
(p − q)(hl − 1)r(n)
hl − 1bl
Blj
all nodes that fall here will have hop distance ≤ hl − 1 from bl
Figure: The construction with hl hops.
Ali ,j = {v : ∃ at least one node in the i th strip of Bl
j}
{∩J(n)j=1 ∩hl−1
i=1 Ali ,j}
⊆ {v : (p − q)(hl − 1)r(n) ≤ D l(v, hl ) ≤ D l(v, hl ) ≤ hl r(n)}
Swaprava (ECE, IISc) Self Organisation June 26, 2008 25 / 45
![Page 27: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/27.jpg)
Proof Techniques (3/4)
Pn(
∩J(n)j=1 ∩hl−1
i=1 Ali ,j
)
= 1 − Pn(
∪J(n)j=1 ∪hl−1
i=1 Ali ,j
c)
≥ 1 −J(n)∑
j=1
hl−1∑
i=1
Pn(
Ali ,j
c)
≥ 1 − (hl − 1)
⌈
πhl
2√
1 − p2
⌉
(1 − u(n)t(n))n
≥ 1 − (hl − 1)
⌈
πhl
2√
1 − p2
⌉
e−nu(n)t(n)
= 1 − (hl − 1)
⌈
πhl
2√
1 − p2
⌉
e−nq√
1−p2r2(n)
= 1 − (hl − 1)
⌈
πhl
2√
1 − p2
⌉
n−q√
1−p2c2 n→∞−→ 1
Swaprava (ECE, IISc) Self Organisation June 26, 2008 26 / 45
![Page 28: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/28.jpg)
Proof Techniques (4/4)
We take p − q = 1 − ε, and maximise q√
1 − p2
Gives p(ε) =1−ε+
√(1−ε)2+8
4 , q(ε) =−3(1−ε)+
√(1−ε)2+8
4
Define g(ε) = q(ε)√
1 − p2(ε), Hence,
Pn{v : (1 − ε)(hl − 1)r(n) ≤ D l (v, hl ) ≤ D l (v, hl ) ≤ hl r(n)}
= 1 −O(
1
ng(ε)c2
)
Swaprava (ECE, IISc) Self Organisation June 26, 2008 27 / 45
![Page 29: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/29.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 28 / 45
![Page 30: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/30.jpg)
1000 nodes : 5 hops
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1000 nodes, prob lower bound = 0.37, epsilon = 0.4
Predicted region for h = 5, on G(V,r(n)), Uniform i.i.d.
Figure: The dashed line shows the ED bounds given by the point-node theorem,the solid line shows ED (h1 − 1)r(n) from b1 for 1000 nodes, 5 hops, ε = 0.4,
Pn(E1(n)) ≥ 0.37. r(n) = 4√
π
√
ln nn
.
Swaprava (ECE, IISc) Self Organisation June 26, 2008 29 / 45
![Page 31: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/31.jpg)
5000 nodes : 5 hops
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
5000 nodes, prob lower bound = 0.79, epsilon = 0.4
Predicted region for h = 5, on G(V,r(n)), Uniform i.i.d.
Figure: The dashed line shows the ED bounds given by the point-node theorem,the solid line shows ED (h1 − 1)r(n) from b1 for 5000 nodes, 5 hops, ε = 0.4,
Pn(E1(n)) ≥ 0.79. r(n) = 4√
π
√
ln nn
.
Swaprava (ECE, IISc) Self Organisation June 26, 2008 30 / 45
![Page 32: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/32.jpg)
5000 nodes : 10 hops
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
5000 nodes, prob lower bound = 0.80, epsilon = 0.48
Predicted region for h = 10, on G(V,r(n)), Uniform i.i.d.
Figure: The dashed line shows the ED bounds given by the point-node theorem,the solid line shows ED (h1 − 1)r(n) from b1 for 5000 nodes, 10 hops,
Pn(E1(n)) ≥ 0.80. r(n) = 4√
π
√
ln nn
.
Swaprava (ECE, IISc) Self Organisation June 26, 2008 31 / 45
![Page 33: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/33.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 32 / 45
![Page 34: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/34.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 33 / 45
![Page 35: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/35.jpg)
Theorem Used
For localisation we need multiple (say L) anchors
hop-distance vector, h = [h1, · · · , hl , · · · , hL] ∈ NL
For L anchors, all possible h vectors are not feasible
H(n): set of all feasible h vectors (it depends on n)
Recall, Ehl(n) = {v : (1 − ε)(hl − 1)r(n) ≤ D l(v, hl ) ≤ D l(v, hl ) ≤ hl r(n)}
Theorem
For a given 1 > ε > 0, and r(n) = c
√
ln nn
, c >1√π,
∀h = [h1, · · · , hl , · · · , hL] ∈ H(n),
Pn(
∩Ll=1Ehl
(n))
= 1 −O(
1
ng(ε)c2
)
where g(ε) = q(ε)√
1 − p2(ε),
p(ε) =1−ε+
√(1−ε)2+8
4 , q(ε) =−3(1−ε)+
√(1−ε)2+8
4
Swaprava (ECE, IISc) Self Organisation June 26, 2008 34 / 45
![Page 36: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/36.jpg)
Illustration
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
���������������������������������������������������������������
���������������������������������������������������������������
b4
b2
b3
b1
h4r(n)
h1r(n
) h2r(n)
h3r(n
)
(1−
ε)(h2− 1)r(n)
(1−
ε)(h4−
1)r(n)
(1−
ε)(
h3−
1)r(n
)
(1 −ε)(h1
−1)r(n
)
the node lies inthis region w. p.1 −O(n−g(ε)c2)
Swaprava (ECE, IISc) Self Organisation June 26, 2008 35 / 45
![Page 37: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/37.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 36 / 45
![Page 38: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/38.jpg)
Algorithm: Hop Count-derived Distance-based Localisation
(HCDL)1
STEP 1: (Initialisation) Each node finds the hop-distance vectorh = [h1, · · · , hL]
STEP 2: (Region of Intersection) For a certain node, set an ε,small enough, and find the region of intersection formed by the annuliof radii [(1− ε)(hl − 1)r(n), hl r(n)] centred at the l th anchor location,l = 1, · · · ,L
STEP 3: (Terminating Condition)
I IF there is an intersection, declare the centroid of the region ofintersection as the estimate of the node. GO TO STEP 4.
I ELSE increase ε by an amount k , 0 < k < 1. GO TO STEP 2.
STEP 4: (Repetition) Repeat STEP 2 to STEP 3 for all n nodes.
STEP 5: STOP
1This is a joint work with Venkatesan N.E. and Prof. P. Vijay KumarSwaprava (ECE, IISc) Self Organisation June 26, 2008 37 / 45
![Page 39: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/39.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 38 / 45
![Page 40: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/40.jpg)
HCRL (Yang et al. 2007): Localisation Error Pattern
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1000 nodes, avg error = 0.084
Localization error pattern using Gcrit
, HCRL
Swaprava (ECE, IISc) Self Organisation June 26, 2008 39 / 45
![Page 41: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/41.jpg)
PDM (Lim and Hou, 2005): Localisation Error Pattern
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1000 nodes, avg error = 0.055
Localization error pattern using Gcrit
, PDM
Swaprava (ECE, IISc) Self Organisation June 26, 2008 40 / 45
![Page 42: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/42.jpg)
HCDL: Localisation Error Pattern
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1000 nodes, 50 ´ 50 bins, avg error = 0.045
Localization error pattern using Gcrit
, HCDL
Swaprava (ECE, IISc) Self Organisation June 26, 2008 41 / 45
![Page 43: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/43.jpg)
Cumulative Distribution of Error
0 0.05 0.1 0.15 0.2 0.250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Location error CDF for the 3 approaches
location error (1000 nodes, 20 runs) ======>
Em
piric
al C
DF
HCDL
PDM
HCRL
Swaprava (ECE, IISc) Self Organisation June 26, 2008 42 / 45
![Page 44: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/44.jpg)
Outline of Talk
1 TheoryReview of Geometric Graph (GG)Assumptions in LiteratureMotivation for Random GGParadigms of HD-ED Relationship in RGG
2 Simulations illustrating Point-Node Theorem
3 Application in LocalisationTheorem UsedAlgorithm: Hop Count-derived Distance-based Localisation (HCDL)
4 Performance Comparison
5 Conclusion and Future Work
Swaprava (ECE, IISc) Self Organisation June 26, 2008 43 / 45
![Page 45: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/45.jpg)
Conclusion and Future Work
Assumed a Geometric Graph model of the Wireless Sensor Network
HD is not a good measure for ED for arbitrary node placements
Three paradigms of HD-ED proportionality for random nodeplacements
I Sufficiency theorems for ED-HD relationships in point-point, node-nodeand point-node paradigms
I For point-point and node-node cases, the radius of the GG is largerthan the critical radius
I For point-node case, theorem is valid for critical radius too
For point-node theory, given HD = h, (1 − ε)(h − 1)r < ED ≤ hr
with high probability
Proposed algorithm HCDL based on this theory
Performs better than HCRL and PDM
Swaprava (ECE, IISc) Self Organisation June 26, 2008 44 / 45
![Page 46: Self Organisation in Random Geometric Graph models of ... · ME Final Presentation Department of Electrical Communication Engineering Indian Institute of Science, Bangalore 560012,](https://reader034.fdocuments.net/reader034/viewer/2022042921/5f6ace8425c67b3e3778ffdb/html5/thumbnails/46.jpg)
Thank You
Swaprava (ECE, IISc) Self Organisation June 26, 2008 45 / 45