l.decreusefond alsostarring(bychronologicalorderofappearance)...
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![Page 1: L.Decreusefond Alsostarring(bychronologicalorderofappearance) …helios.mi.parisdescartes.fr/~bierme/130626GeoSto/Laurent... · 2013. 6. 25. · Historique Algebraic topology Poisson](https://reader034.vdocuments.site/reader034/viewer/2022051911/6001d27671ce8133e6577bba/html5/thumbnails/1.jpg)
HistoriqueAlgebraic topology
Poisson homologiesOther applications
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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
L. Decreusefond Geometry of wireless networks 2 / 40
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Sensors : ambient or pervasive computing
L. Decreusefond Geometry of wireless networks 3 / 40
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Applications : intelligent vehicle, agriculture, house, ...
L. Decreusefond Geometry of wireless networks 4 / 40
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Coverage
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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
L. Decreusefond Geometry of wireless networks 5 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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).
L. Decreusefond Geometry of wireless networks 6 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Mathematical framework
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).
L. Decreusefond Geometry of wireless networks 6 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Mathematical framework
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).
L. Decreusefond Geometry of wireless networks 6 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Mathematical framework
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).
L. Decreusefond Geometry of wireless networks 6 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Construction of the Cech complex
L. Decreusefond Geometry of wireless networks 7 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Construction of the Cech complex
L. Decreusefond Geometry of wireless networks 7 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Construction of the Cech complex
L. Decreusefond Geometry of wireless networks 7 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Construction of the Cech complex
L. Decreusefond Geometry of wireless networks 7 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Construction of the Cech complex
L. Decreusefond Geometry of wireless networks 7 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Construction of the Cech complex
L. Decreusefond Geometry of wireless networks 7 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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 : ∅
L. Decreusefond Geometry of wireless networks 8 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Combinatorial structure of the Cech complex
a
b
c
d
e
Vertices : a, b, c, d, eEdges : ab, bc, ca, be, ec, ed
Triangles : becTetrahedron : ∅
L. Decreusefond Geometry of wireless networks 8 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Combinatorial structure of the Cech complex
a
b
c
d
e
Vertices : a, b, c, d, eEdges : ab, bc, ca, be, ec, ed
Triangles : bec
Tetrahedron : ∅
L. Decreusefond Geometry of wireless networks 8 / 40
![Page 19: L.Decreusefond Alsostarring(bychronologicalorderofappearance) …helios.mi.parisdescartes.fr/~bierme/130626GeoSto/Laurent... · 2013. 6. 25. · Historique Algebraic topology Poisson](https://reader034.vdocuments.site/reader034/viewer/2022051911/6001d27671ce8133e6577bba/html5/thumbnails/19.jpg)
HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Combinatorial structure of the Cech complex
a
b
c
d
e
Vertices : a, b, c, d, eEdges : ab, bc, ca, be, ec, ed
Triangles : becTetrahedron : ∅
L. Decreusefond Geometry of wireless networks 8 / 40
![Page 20: L.Decreusefond Alsostarring(bychronologicalorderofappearance) …helios.mi.parisdescartes.fr/~bierme/130626GeoSto/Laurent... · 2013. 6. 25. · Historique Algebraic topology Poisson](https://reader034.vdocuments.site/reader034/viewer/2022051911/6001d27671ce8133e6577bba/html5/thumbnails/20.jpg)
HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Combinatorial structure of the Cech complex
a
b
c
d
e
Vertices : a, b, c, d, eEdges : ab, bc, ca, be, ec, ed
Triangles : becTetrahedron : ∅
L. Decreusefond Geometry of wireless networks 8 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Combinatorial structure of the Cech complex
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
L. Decreusefond Geometry of wireless networks 8 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 9 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 9 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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 components
Same nb of holesSame Euler characteristic
L. Decreusefond Geometry of wireless networks 9 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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 holes
Same Euler characteristic
L. Decreusefond Geometry of wireless networks 9 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 9 / 40
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Poisson homologiesOther applications
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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 , · · · ]
L. Decreusefond Geometry of wireless networks 10 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Linear algebra : The boundary operators
Definition
∂k : Ck −→ Ck−1
[v0, · · · , vk−1] 7−→k∑
j=0(−1)j [v0, · · · , vj , · · · ]
Example∂2(bec) = ec − bc + be
L. Decreusefond Geometry of wireless networks 10 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Linear algebra : The boundary operators
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
L. Decreusefond Geometry of wireless networks 10 / 40
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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
L. Decreusefond Geometry of wireless networks 11 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 11 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 11 / 40
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Interpretation : The magic
β0 : number of connected components
β1 : number of holesβ2 : number of voidsto be continued
L. Decreusefond Geometry of wireless networks 12 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Interpretation : The magic
β0 : number of connected componentsβ1 : number of holes
β2 : number of voidsto be continued
L. Decreusefond Geometry of wireless networks 12 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
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Interpretation : The magic
β0 : number of connected componentsβ1 : number of holesβ2 : number of voids
to be continued
L. Decreusefond Geometry of wireless networks 12 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Interpretation : The magic
β0 : number of connected componentsβ1 : number of holesβ2 : number of voidsto be continued
L. Decreusefond Geometry of wireless networks 12 / 40
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Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 13 / 40
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Number of holes
∂2 =
0−10110
Nb of holesdim ker∂1 = 2, range ∂2 = 1 hence β1 = 1
L. Decreusefond Geometry of wireless networks 14 / 40
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Poisson homologiesOther applications
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Euler characteristic
Definition
χ =d∑
j=0(−1)jβj
Discrete Morse inequality
− |Ck−1|+ |Ck | − |Ck+1| ≤ βk ≤ |Ck |
L. Decreusefond Geometry of wireless networks 15 / 40
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Poisson homologiesOther applications
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Euler characteristic
Definition
χ =d∑
j=0(−1)jβj =
∞∑j=0
(−1)j |Ck |
Discrete Morse inequality
− |Ck−1|+ |Ck | − |Ck+1| ≤ βk ≤ |Ck |
L. Decreusefond Geometry of wireless networks 15 / 40
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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
L. Decreusefond Geometry of wireless networks 16 / 40
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Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 16 / 40
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Poisson homologiesOther applications
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Difference RV vs Cech
For the l∞ distanceRV=Cech
Euclidean norm : false negativeRips complex may miss some holes
L. Decreusefond Geometry of wireless networks 17 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Difference RV vs Cech
For the l∞ distanceRV=Cech
Euclidean norm : false negativeRips complex may miss some holes
L. Decreusefond Geometry of wireless networks 17 / 40
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Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 18 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 18 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 18 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 18 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 18 / 40
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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
L. Decreusefond Geometry of wireless networks 19 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
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
L. Decreusefond Geometry of wireless networks 20 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
Goals and related works
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
Our resultsExact expressions of all moments of |Ck | and χ in any dimensionfor RG complex on a torus for the l∞ norm
L. Decreusefond Geometry of wireless networks 20 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
Goals and related works
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
Our resultsExact expressions of all moments of |Ck | and χ in any dimensionfor RG complex on a torus for the l∞ norm
L. Decreusefond Geometry of wireless networks 20 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
Goals and related works
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
Our resultsExact expressions of all moments of |Ck | and χ in any dimensionfor RG complex on a torus for the l∞ norm
L. Decreusefond Geometry of wireless networks 20 / 40
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Poisson homologiesOther applications
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Euler characteristicAsymptotic resultsRobust estimate
Random setting
x1a
a
a
a x1
ε
[0, a]× [0, a] T2a×a
L. Decreusefond Geometry of wireless networks 21 / 40
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Euler characteristicAsymptotic resultsRobust estimate
Euler characteristic (D.-Ferraz-Randriam-Vergne [1])
Euler characteristic
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
L. Decreusefond Geometry of wireless networks 22 / 40
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Euler characteristicAsymptotic resultsRobust estimate
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
L. Decreusefond Geometry of wireless networks 23 / 40
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Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
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
L. Decreusefond Geometry of wireless networks 23 / 40
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Euler characteristicAsymptotic resultsRobust estimate
Dimension 5
L. Decreusefond Geometry of wireless networks 24 / 40
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Poisson homologiesOther applications
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Euler characteristicAsymptotic resultsRobust estimate
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.
L. Decreusefond Geometry of wireless networks 25 / 40
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Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
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.
Tools
Chaos decomposition of |Ck |Chaos multiplication formula
L. Decreusefond Geometry of wireless networks 25 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
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.
ToolsChaos decomposition of |Ck |
Chaos multiplication formula
L. Decreusefond Geometry of wireless networks 25 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
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.
ToolsChaos decomposition of |Ck |Chaos multiplication formula
L. Decreusefond Geometry of wireless networks 25 / 40
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Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
Chaos multiplication formula
Theorem (Chaos representation)
|Ck | =1k!
k∑i=0
(ki
)λk−i Ii
(∫Xk−i
ϕ(d)k (v1, · · · , vk) dv1 . . . dvk−i
).
Chaos multiplication formula
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
L. Decreusefond Geometry of wireless networks 26 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
Chaos multiplication formula
Theorem (Chaos representation)
|Ck | =1k!
k∑i=0
(ki
)λk−i Ii
(∫Xk−i
ϕ(d)k (v1, · · · , vk) dv1 . . . dvk−i
).
Chaos multiplication formula
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
L. Decreusefond Geometry of wireless networks 26 / 40
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Poisson homologiesOther applications
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Euler characteristicAsymptotic resultsRobust estimate
Euler characteristic
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].
L. Decreusefond Geometry of wireless networks 27 / 40
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Poisson homologiesOther applications
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Euler characteristicAsymptotic resultsRobust estimate
Euler characteristic
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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Euler characteristicAsymptotic resultsRobust estimate
Asymptotic results
If λ→∞, βi (ω)p.s.−→ βi (Td ) =
(di).
L. Decreusefond Geometry of wireless networks 28 / 40
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Euler characteristicAsymptotic resultsRobust estimate
Limit theorems
CLT for Euler characteristic
distanceTV
(χ− E[χ]√
Vχ, N(0, 1)
)≤ c√
λ·
L. Decreusefond Geometry of wireless networks 29 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
Limit theorems
CLT for Euler characteristic
distanceTV
(χ− E[χ]√
Vχ, N(0, 1)
)≤ c√
λ·
Method
Stein method (Peccati/Schulte-Thäle)No combinatorics, only computations of some deterministicintegrals
L. Decreusefond Geometry of wireless networks 29 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
Limit theorems
CLT for Euler characteristic
distanceTV
(χ− E[χ]√
Vχ, N(0, 1)
)≤ c√
λ·
MethodStein method (Peccati/Schulte-Thäle)
No combinatorics, only computations of some deterministicintegrals
L. Decreusefond Geometry of wireless networks 29 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
Limit theorems
CLT for Euler characteristic
distanceTV
(χ− E[χ]√
Vχ, N(0, 1)
)≤ c√
λ·
MethodStein method (Peccati/Schulte-Thäle)No combinatorics, only computations of some deterministicintegrals
L. Decreusefond Geometry of wireless networks 29 / 40
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Poisson homologiesOther applications
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Euler characteristicAsymptotic resultsRobust estimate
Concentration inequality
Discrete gradient DxF (ω) = F (ω ∪ {x})− F (ω)
Dxβ0 ∈ {1, 0, −1, −2, −3}
L. Decreusefond Geometry of wireless networks 30 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
Concentration inequality
Discrete gradient DxF (ω) = F (ω ∪ {x})− F (ω)
Dxβ0 ∈ {1, 0, −1, −2, −3}
L. Decreusefond Geometry of wireless networks 30 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
References
Euler characteristicAsymptotic resultsRobust estimate
Concentration inequality
Discrete gradient DxF (ω) = F (ω ∪ {x})− F (ω)
Dxβ0 ∈ {1, 0, −1, −2, −3}
c > E[β0]
P(β0 ≥ c) ≤ exp[− c − E[β0]
6 log(1 +
c − E[β0]
3λ
)]
L. Decreusefond Geometry of wireless networks 30 / 40
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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
L. Decreusefond Geometry of wireless networks 31 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
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
L. Decreusefond Geometry of wireless networks 31 / 40
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HistoriqueAlgebraic topology
Poisson homologiesOther applications
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
L. Decreusefond Geometry of wireless networks 31 / 40
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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
L. Decreusefond Geometry of wireless networks 32 / 40
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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
.
L. Decreusefond Geometry of wireless networks 33 / 40
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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.
L. Decreusefond Geometry of wireless networks 34 / 40
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Three regimes (cont’d)
Critical regime : nθn → 1
(ln n)1−η < Hn < ln n, ∀η > 0.
L. Decreusefond Geometry of wireless networks 35 / 40
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Three regimes (cont’d)
Supercritical regime : nθ →∞
Hn ∼ nθn
L. Decreusefond Geometry of wireless networks 36 / 40
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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
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One solution
One solution
Add many points randomlyApply the reduction algorithm
L. Decreusefond Geometry of wireless networks 38 / 40
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One solution
One solutionAdd many points randomly
Apply the reduction algorithm
L. Decreusefond Geometry of wireless networks 38 / 40
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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
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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
L. Decreusefond Geometry of wireless networks 39 / 40
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Best choice : determinantal point processes
L. Decreusefond Geometry of wireless networks 40 / 40
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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/.
L. Decreusefond Geometry of wireless networks 40 / 40