L.Decreusefond Alsostarring(bychronologicalorderofappearance)...
Transcript of L.Decreusefond Alsostarring(bychronologicalorderofappearance)...
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
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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
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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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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).
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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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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).
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Construction of the Cech complex
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Construction of the Cech complex
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Construction of the Cech complex
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Construction of the Cech complex
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Construction of the Cech complex
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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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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 : ∅
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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 : ∅
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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 : ∅
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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 : ∅
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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
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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= ∅}
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= ∅}
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
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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 holes
Same Euler characteristic
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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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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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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
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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 , · · · ]
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
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
Im ∂k+1 ⊂ ker∂k
Definition
Hk = ker ∂k/Im∂k+1 and βk = dim ker ∂k − range ∂k+1
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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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Interpretation : The magic
β0 : number of connected componentsβ1 : number of holes
β2 : number of voidsto be continued
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Interpretation : The magic
β0 : number of connected componentsβ1 : number of holesβ2 : number of voids
to be continued
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Interpretation : The magic
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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Euler characteristic
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
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
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
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
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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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
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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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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
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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
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Euler characteristicAsymptotic resultsRobust estimate
Random setting
x1a
a
a
a x1
ε
[0, a]× [0, a] T2a×a
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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
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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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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
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Euler characteristicAsymptotic resultsRobust estimate
Dimension 5
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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.
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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.
Tools
Chaos decomposition of |Ck |Chaos multiplication formula
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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.
ToolsChaos decomposition of |Ck |
Chaos multiplication formula
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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.
ToolsChaos decomposition of |Ck |Chaos multiplication formula
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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
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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
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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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References
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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References
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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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
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
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
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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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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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
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
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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Poisson homologiesOther applications
References
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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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
.
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.
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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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Poisson homologiesOther applications
References
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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References
Best choice : determinantal point processes
L. Decreusefond Geometry of wireless networks 40 / 40
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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/.
L. Decreusefond Geometry of wireless networks 40 / 40