L.Decreusefond Alsostarring(bychronologicalorderofappearance)...

HistoriqueAlgebraic topology

Poisson homologiesOther applications

References

Some geometrical aspects of wireless networks

L. Decreusefond

Also starring (by chronological order of appearance)

P. Martins, E. Ferraz, F. Yan, A. Vergne, I. Flint

June 2013

L. Decreusefond Geometry of wireless networks 1 / 40



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History

1870 1970 2000




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Sensors : ambient or pervasive computing




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Applications : intelligent vehicle, agriculture, house, ...




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Coverage

0 0.5 1 1.5 2 2.5 3 3.5 40

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0 0.5 1 1.5 2 2.5 3 3.5 40

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Figure 3: A sensors’ network and its associated Cech complex.

Definition 9 (Vietoris-Rips complex) Given (X, d) a metric space, ! a fi-nite set of points in X, and ✏ a real positive number. The Vietoris-Rips complexof parameter ✏ of !, denoted R✏(!), is the abstract simplicial complex whosek-simplices correspond to unordered (k + 1)-tuples of vertices in ! which arepairwise within distance less than ✏ of each other.

In general, unlike the Cech one, Vietoris-Rips complexes are not topologicallyequivalent to the coverage of an area. However, the following gives us the relationbetween coverage and Vietoris-Rips complexes:

Lemma 1 Given (X, d) a metric space, ! a finite set of points in X, and ✏ areal positive number,

Rp3✏(!) ⇢ C✏(!) ⇢ R2✏(!).

In the Erdös-Rényi model, which is a random graph model, there is nogeometric considerations, we extend the model to the homology:

Definition 10 (Erdös-Rényi complex) Given n an integer and p a real num-ber in [0, 1], the Erdös-Rényi complex of parameters n and p, denoted G(n, p),is an abstract simplicial complex with n vertices which are connected randomly.Each edge is included in the complex with probability p independent from everyother edge. Then a k-simplex, for k � 2, is included in the complex if and onlyif all its faces already are.

Only graph descritption is required to build a Vietoris-Rips or a Erdös-Rényicomplex. That is why here we will give examples only on these two complexes.

3 Moments of random variables of an abstractsimplicial complexe

By means of Malliavin calculus, we have computed explicitly the n-th ordermoment of the number of k-simplices. The computation of these moments arenot detailed here, only are given the main theorems.

4




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Mathematical framework

Geometry leads to a combinatorial object

Combinatorial object is equipped with a linearalgebra structureCoverage and connectivity reduce to compute therank of a matrixLocalisation of hole: reduces to the computation of abasis of a vector matrix, obtained by matrixreduction (as in Gauss algorithm).




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Geometry leads to a combinatorial objectCombinatorial object is equipped with a linearalgebra structure

Coverage and connectivity reduce to compute therank of a matrixLocalisation of hole: reduces to the computation of abasis of a vector matrix, obtained by matrixreduction (as in Gauss algorithm).




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Geometry leads to a combinatorial objectCombinatorial object is equipped with a linearalgebra structureCoverage and connectivity reduce to compute therank of a matrix

Localisation of hole: reduces to the computation of abasis of a vector matrix, obtained by matrixreduction (as in Gauss algorithm).




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Geometry leads to a combinatorial objectCombinatorial object is equipped with a linearalgebra structureCoverage and connectivity reduce to compute therank of a matrixLocalisation of hole: reduces to the computation of abasis of a vector matrix, obtained by matrixreduction (as in Gauss algorithm).




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Construction of the Cech complex




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Combinatorial structure of the Cech complex

a

b

c

d

e

Vertices : a, b, c, d, e

Edges : ab, bc, ca, be, ec, edTriangles : bec

Tetrahedron : ∅




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a

b

c

d

e

Vertices : a, b, c, d, eEdges : ab, bc, ca, be, ec, ed

Triangles : becTetrahedron : ∅




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a

b

c

d

e


Triangles : bec

Tetrahedron : ∅




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a

b

c

d

e






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a

b

c

d

e






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a

b

c

d

e

Vertices : { a, b, c, d, e } = C0

Edges : {ab, bc, ca, be, ec, ed } = C1

Triangles : {bec} = C2

Tetrahedron : ∅ = C3

Comput. Rips




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Cech complex

k-simplices

Ck =⋃{[x0, · · · , xk−1], xi ∈ ω,∩k

i=0B(xi , ε) 6= ∅}

Nerve theoremWe can read some topological properties of

⋃x∈ω B(x , ε) on

(Ck , k ≥ 0)

Same nb of connected componentsSame nb of holesSame Euler characteristic




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Cech complex

k-simplices

Ck =⋃{[x0, · · · , xk−1], xi ∈ ω,∩k

i=0B(xi , ε) 6= ∅}



(Ck , k ≥ 0)





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Cech complex

k-simplices

Ck =⋃{[x0, · · · , xk−1], xi ∈ ω,∩k

i=0B(xi , ε) 6= ∅}



(Ck , k ≥ 0)

Same nb of connected components

Same nb of holesSame Euler characteristic




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Cech complex

k-simplices

Ck =⋃{[x0, · · · , xk−1], xi ∈ ω,∩k

i=0B(xi , ε) 6= ∅}



(Ck , k ≥ 0)

Same nb of connected componentsSame nb of holes

Same Euler characteristic




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Cech complex

k-simplices

Ck =⋃{[x0, · · · , xk−1], xi ∈ ω,∩k

i=0B(xi , ε) 6= ∅}



(Ck , k ≥ 0)





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Linear algebra : The boundary operators

Definition

∂k : Ck −→ Ck−1

[v0, · · · , vk−1] 7−→k∑

j=0(−1)j [v0, · · · , vj , · · · ]




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Definition

∂k : Ck −→ Ck−1

[v0, · · · , vk−1] 7−→k∑

j=0(−1)j [v0, · · · , vj , · · · ]

Example∂2(bec) = ec − bc + be




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Definition

∂k : Ck −→ Ck−1

[v0, · · · , vk−1] 7−→k∑

j=0(−1)j [v0, · · · , vj , · · · ]

Example∂2(bec) = ec − bc + be∂1∂2(bec) = c − e − (c − b) + e − b = 0




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Theorem

∂k ◦ ∂k+1 = 0

Consequence

Im ∂k+1 ⊂ ker∂k

Definition

Hk = ker ∂k/Im∂k+1 and βk = dim ker ∂k − range ∂k+1




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Theorem

∂k ◦ ∂k+1 = 0

Consequence


Definition





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Theorem

∂k ◦ ∂k+1 = 0

Consequence


Definition





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Interpretation : The magic

β0 : number of connected components

β1 : number of holesβ2 : number of voidsto be continued




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β0 : number of connected componentsβ1 : number of holes

β2 : number of voidsto be continued




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β0 : number of connected componentsβ1 : number of holesβ2 : number of voids

to be continued




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β0 : number of connected componentsβ1 : number of holesβ2 : number of voidsto be continued




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Example

∂0 ≡ 0, ∂1 =

−1 0 1 −1 0 01 −1 0 0 0 −10 1 −1 0 1 00 0 0 0 0 10 0 0 1 −1 0

Nb of connected componentdim ker ∂0 = 5, range ∂1 = 4 hence β0 = 1




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Number of holes

∂2 =

0−10110

Nb of holesdim ker∂1 = 2, range ∂2 = 1 hence β1 = 1




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Euler characteristic

Definition

χ =d∑

j=0(−1)jβj

Discrete Morse inequality

− |Ck−1|+ |Ck | − |Ck+1| ≤ βk ≤ |Ck |




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Definition

χ =d∑

j=0(−1)jβj =

∞∑j=0

(−1)j |Ck |

Discrete Morse inequality

− |Ck−1|+ |Ck | − |Ck+1| ≤ βk ≤ |Ck |




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Alternative complex

Cech complex

[v0, · · · , vk ] ∈ Ck ⇐⇒ ∩kj=0B(xj , ε) 6= ∅

Rips-Vietoris complex

[v0, · · · , vk ] ∈ Rk ⇐⇒ B(xj , ε) ∩ B(xk , ε) 6= ∅

k simplex = clique of k + 1 points




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Alternative complex

Cech complex

[v0, · · · , vk ] ∈ Ck ⇐⇒ ∩kj=0B(xj , ε) 6= ∅

Rips-Vietoris complex

[v0, · · · , vk ] ∈ Rk ⇐⇒ B(xj , ε) ∩ B(xk , ε) 6= ∅

k simplex = clique of k + 1 points




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Difference RV vs Cech

For the l∞ distanceRV=Cech

Euclidean norm : false negativeRips complex may miss some holes




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Difference RV vs Cech

For the l∞ distanceRV=Cech

Euclidean norm : false negativeRips complex may miss some holes




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Cech vs Rips

Rε′(V) ⊂ Cε(V) ⊂ R2ε(V) whenever γ :=ε

ε′≥√

d2(d + 1)

Theorem (In the plane)

γ ≤√3 : No hole in Rγε entails no holes in Cε

γ ≥ 2 : A hole in Rγε entails a hole in Cε√3 < γ < 2 : No guarantee whatsoever




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Cech vs Rips


ε′≥√

d2(d + 1)







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Cech vs Rips


ε′≥√

d2(d + 1)







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Cech vs Rips


ε′≥√

d2(d + 1)



γ ≥ 2 : A hole in Rγε entails a hole in Cε

√3 < γ < 2 : No guarantee whatsoever




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Cech vs Rips


ε′≥√

d2(d + 1)







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Explicit upper bound in the plane (D.-Feng-Martins [4])

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

1

2

3

4

5

6

7

8

9

intensity λ

p 2d(λ

)(%

)

simulation γ = 2.0lower bound γ = 2.0simulation γ = 2.2lower bound γ = 2.2simulation γ = 2.4lower bound γ = 2.4simulation γ = 2.6lower bound γ = 2.6simulation γ = 2.8lower bound γ = 2.8simulation γ = 3.0lower bound γ = 3.0

Figure : Probability to miss a hole using Rε and Rγε. Poissondistribution of points




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Euler characteristicAsymptotic resultsRobust estimate

Goals and related works

Evaluate Betti nb and Euler charac. in some random settings

Penrose : Asymptotics E[|Ck |m] for Euclidian-RG Ripscomplex on the whole space (m = 1, 2)Kähle : Asymptotics of E[βk ] for Euclidian-RG Cech complex(deterministic number of points) and ER

Our resultsExact expressions of all moments of |Ck | and χ in any dimensionfor RG complex on a torus for the l∞ norm




References



Evaluate Betti nb and Euler charac. in some random settingsPenrose : Asymptotics E[|Ck |m] for Euclidian-RG Ripscomplex on the whole space (m = 1, 2)

Kähle : Asymptotics of E[βk ] for Euclidian-RG Cech complex(deterministic number of points) and ER





References



Evaluate Betti nb and Euler charac. in some random settingsPenrose : Asymptotics E[|Ck |m] for Euclidian-RG Ripscomplex on the whole space (m = 1, 2)Kähle : Asymptotics of E[βk ] for Euclidian-RG Cech complex(deterministic number of points) and ER





References



Evaluate Betti nb and Euler charac. in some random settingsPenrose : Asymptotics E[|Ck |m] for Euclidian-RG Ripscomplex on the whole space (m = 1, 2)Kähle : Asymptotics of E[βk ] for Euclidian-RG Cech complex(deterministic number of points) and ER





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Random setting

x1a

a

a

a x1

ε

[0, a]× [0, a] T2a×a




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Euler characteristic (D.-Ferraz-Randriam-Vergne [1])


E [χ] = −λe−θ ad

θBd (−θ ad ) where θ = λ

(2εa

)d.

where Bd is the d-th Bell polynomial

Bd (x) =

{d1

}x +

{d2

}x2 + ...+

{dd

}xd




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k simplices

The key remark

|Ck | =

∫ϕ(d)k (x1, · · · , xk)dω(k)(x1, · · · , xk)

where

ϕ(d)k (v1, · · · , vk) =

∏1≤i<j≤k

1[0, 2ε)(‖(vi − vj)‖∞)

Theorem (First moments)

E[|Ck |] = λad (k + 1)d

(k + 1)!(adθ)k

where θ = λ(2ε/a)d




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k simplices

The key remark

|Ck | =

∫ϕ(d)k (x1, · · · , xk)dω(k)(x1, · · · , xk)

where

ϕ(d)k (v1, · · · , vk) =

∏1≤i<j≤k

1[0, 2ε)(‖(vi − vj)‖∞)

Theorem (First moments)

E[|Ck |] = λad (k + 1)d

(k + 1)!(adθ)k

where θ = λ(2ε/a)d




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Dimension 5




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Second order moments

Theorem

Cov(|Ck |, |Cl |) =

( 12ε

)d l−1∑i=0

1i!(k − l + i)!(l − i)!

θk+i

×(

k + i + 2 i(k − l + i)l − i + 1

)d.




References



Theorem

Cov(|Ck |, |Cl |) =

( 12ε

)d l−1∑i=0

1i!(k − l + i)!(l − i)!

θk+i

×(

k + i + 2 i(k − l + i)l − i + 1

)d.

Tools

Chaos decomposition of |Ck |Chaos multiplication formula




References



Theorem

Cov(|Ck |, |Cl |) =

( 12ε

)d l−1∑i=0

1i!(k − l + i)!(l − i)!

θk+i

×(

k + i + 2 i(k − l + i)l − i + 1

)d.

ToolsChaos decomposition of |Ck |

Chaos multiplication formula




References



Theorem

Cov(|Ck |, |Cl |) =

( 12ε

)d l−1∑i=0

1i!(k − l + i)!(l − i)!

θk+i

×(

k + i + 2 i(k − l + i)l − i + 1

)d.

ToolsChaos decomposition of |Ck |Chaos multiplication formula




References



Theorem (Chaos representation)

|Ck | =1k!

k∑i=0

(ki

)λk−i Ii

(∫Xk−i

ϕ(d)k (v1, · · · , vk) dv1 . . . dvk−i

).


Ii (fi )Ij(fj)

=

2(i∧j)∑s=0

Ii+j−s

∑s≤2t≤2(s∧i∧j)

t!

(it

)(jt

)(t

s − t

)fi ◦s−t

t fj




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Theorem (Chaos representation)

|Ck | =1k!

k∑i=0

(ki

)λk−i Ii

(∫Xk−i

ϕ(d)k (v1, · · · , vk) dv1 . . . dvk−i

).


Ii (fi )Ij(fj)

=

2(i∧j)∑s=0

Ii+j−s

∑s≤2t≤2(s∧i∧j)

t!

(it

)(jt

)(t

s − t

)fi ◦s−t

t fj




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Corollary (Variance of Euler characteristic)

var[χ] =

( a2ε

)d ∞∑n=1

cdn θ

n,

where

cdn =∑n

j=d(n+1)/2e

[2∑j

i=n−j+1(−1)i+j

(n−j)!(n−i)!(i+j−n)!

(n+ 2(n−i)(n−j)

1+i+j−n)d

− 1(n−j)!2(2j−n)!

(n+ 2(n−j)2

1+2j−n

)d].




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Corollary (Variance of Euler characteristic)

var[χ] =

( a2ε

)d ∞∑n=1

cdn θ

n,

where

cdn =∑n

j=d(n+1)/2e

[2∑j

i=n−j+1(−1)i+j

(n−j)!(n−i)!(i+j−n)!

(n+ 2(n−i)(n−j)

1+i+j−n)d

− 1(n−j)!2(2j−n)!

(n+ 2(n−j)2

1+2j−n

)d].

In dimension 1,Var(χ) =

(θe−θ − 2θ2e−2θ

)L. Decreusefond Geometry of wireless networks 27 / 40



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Asymptotic results

If λ→∞, βi (ω)p.s.−→ βi (Td ) =

(di).




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Limit theorems

CLT for Euler characteristic

distanceTV

(χ− E[χ]√

Vχ, N(0, 1)

)≤ c√

λ·




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Limit theorems


distanceTV

(χ− E[χ]√

Vχ, N(0, 1)

)≤ c√

λ·

Method

Stein method (Peccati/Schulte-Thäle)No combinatorics, only computations of some deterministicintegrals




References


Limit theorems


distanceTV

(χ− E[χ]√

Vχ, N(0, 1)

)≤ c√

λ·

MethodStein method (Peccati/Schulte-Thäle)

No combinatorics, only computations of some deterministicintegrals




References


Limit theorems


distanceTV

(χ− E[χ]√

Vχ, N(0, 1)

)≤ c√

λ·

MethodStein method (Peccati/Schulte-Thäle)No combinatorics, only computations of some deterministicintegrals




References


Concentration inequality

Discrete gradient DxF (ω) = F (ω ∪ {x})− F (ω)

Dxβ0 ∈ {1, 0, −1, −2, −3}




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Dxβ0 ∈ {1, 0, −1, −2, −3}




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Dxβ0 ∈ {1, 0, −1, −2, −3}

c > E[β0]

P(β0 ≥ c) ≤ exp[− c − E[β0]

6 log(1 +

c − E[β0]

3λ

)]




References

Green networking (D.-Martins-Vergne [2])

ObjectiveSwitch off some sensors keeping the coverage

Height of an edgeRank of the highest simplex it belongs to

Index of a vertexInfimum of the height of its adjacent edges




References








References








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ExampleWe can see in Figure 5 the realisation of the coverage algorithm on a Vietoris-

Rips complex of parameter ✏ = 1 based on a Poisson point process of intensity� = 4.2 on a square of side length 2, with a fixed boundary of vertices on thesquare perimeter. The boundary vertices are circled in red.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 5: A Vietoris-Rips complex before and after the coverage reduction al-gorithm.

For this configuration, on average on 200 runs, the algorithm removed 69.22%of the non-boundary vertices, and computed in 206.01 seconds.

We can see in Figure 6 the realisation of the connectivity algorithm on aErdös-Rényi complex of parameter n = 15 and p = 0.3, with random activevertices. We chose a small number of vertices for the figure to be readable.A vertex is active with probability pa = 0.5 independantly from every othervertices. The graph key is the same as before.

!1 !0.5 0 0.5 1 1.5 2 2.5 3 3.5 4!1

!0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

!1 !0.5 0 0.5 1 1.5 2 2.5 3 3.5 4!1

!0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 6: A Erdös-Rényi complex before and after the connectivity reductionalgorithm.

15

Complexity bounded by 2H




References

Depoissonization

Theorem

E[|Ck−1| | |C0| = n] =

(nk

)kdθk−1

and,

var|Ck−1| =

k∑i=1

(n

2k − i

)(2k − ik + 1

)(ki

)θ2k−i−1

(2k − i + 1(k − i)2

i + 1

)d

.




References

Three regimes

Theorem (Subcritical : nθn → 0)If for some k ≥ 1,

θ′k =k

1+η−dk−1

nk

k−1≤ θn ≤ θk =

k−1+η+d

k−1

nk

k−1,

ThenHn

n→∞−−−→ k, a.s.




References

Three regimes (cont’d)

Critical regime : nθn → 1

(ln n)1−η < Hn < ln n, ∀η > 0.




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Three regimes (cont’d)

Supercritical regime : nθ →∞

Hn ∼ nθn




References

Disaster repair (D-Flint-Martins-Vergne)

After an earthquake, a tsunami, ....How to reconstruct a network ?

Figure : A network to be repaired ...L. Decreusefond Geometry of wireless networks 37 / 40



References

One solution

One solution

Add many points randomlyApply the reduction algorithm




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One solution

One solutionAdd many points randomly

Apply the reduction algorithm




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One solution

One solutionAdd many points randomlyApply the reduction algorithm

Figure : Random points vs repulsive pointsL. Decreusefond Geometry of wireless networks 38 / 40



References

Performance [3]

% of area initially covered 20% 40% 60% 80%

Uniform 32 29 24 16Determinantal 16 14 12 8

Table : Mean number of added vertices




References

Best choice : determinantal point processes




References

L. Decreusefond et al. “Simplicial Homology of RandomConfigurations”. In: Journal of Advances in AppliedProbability. (Mar. 2013). url: http://hal-institut-mines-telecom.archives-ouvertes.fr/hal-00578955.A. Vergne, L. Decreusefond, and P. Martins. “Reductionalgorithm for simplicial complexes”. In: IEEE INFOCOM.2013. url:http://hal.archives-ouvertes.fr/hal-00688919.A. Vergne et al. “Homology based algorithm for disasterrecovery in wireless networks”. Anglais. Mar. 2013. url:http://hal.archives-ouvertes.fr/hal-00800520.F. Yan, P. Martins, and L. Decreusefond. “Accuracy ofHomology based Approaches for Coverage Hole Detection inWireless Sensor Networks”. In: ICC 2012. June 2012. url:http://hal.archives-ouvertes.fr/hal-00646894/.


http://hal-institut-mines-telecom.archives-ouvertes.fr/hal-00578955

http://hal-institut-mines-telecom.archives-ouvertes.fr/hal-00578955

http://hal.archives-ouvertes.fr/hal-00688919

http://hal.archives-ouvertes.fr/hal-00800520

http://hal.archives-ouvertes.fr/hal-00646894/

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