Topology of wireless networks L. Decreusefond Institut Also - - PowerPoint PPT Presentation

Institut Mines-Telecom

Topology of wireless networks

• L. Decreusefond

Also starring (by chronological order of appearance)

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

Flint, N.K. Le, A. Vasseur, (T. Bonis, B. Robert) GANDI: Graphs ANalysis for Data and Information

Algebraic topology Poisson homologies Persistence

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

2/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Outline

Algebraic topology Poisson homologies Persistence

3/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Coverage

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

4/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Mathematical framework

Geometry leads to a combinatorial object Combinatorial object is equipped with a Linear algebra structure Coverage and connectivity reduce to compute the rank of a matrix Localisation of hole: reduces to the computation of a basis of a vector matrix, obtained by matrix reduction (as in Gauss algorithm).

5/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

a b c d e Vertices : a, b, c, d, e Edges : ab, bc, ca, be, ec, ed Triangles : bec Tetrahedron : ;

7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

a b c d e Vertices : a, b, c, d, e Edges : ab, bc, ca, be, ec, ed Triangles : bec Tetrahedron : ;

7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

a b c d e Vertices : a, b, c, d, e Edges : ab, bc, ca, be, ec, ed Triangles : bec Tetrahedron : ;

7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

a b c d e Vertices : a, b, c, d, e Edges : ab, bc, ca, be, ec, ed Triangles : bec Tetrahedron : ;

7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex

a b c d e Vertices : a, b, c, d, e Edges : ab, bc, ca, be, ec, ed Triangles : bec Tetrahedron : ;

7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

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 7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Hypergraphs

A simplicial complex = hypergraph = boolean monotone function

8/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Hypergraphs

A simplicial complex = hypergraph = boolean monotone function The Embedded Homology of Hypergraphs and Applications Stephane Bressan, Shiquan Ren, Jie Wu arXiv:1610.00890

8/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech complex k-simplices

Ck = [ {[x0, · · · , xk1], xi 2 !, \k

i=0B(xi, ✏) 6= ;}

Nerve theorem

We can read some topological properties of S

x2! B(x, ✏) on

(Ck, k 0)

I Same nb of connected components I Same nb of holes I Same Euler characteristic

9/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Boundary operator Definition

@k : Ck ! Ck1 [v0, · · · , vk1] 7 !

k

X

j=0

(1)j[v0, · · · , ˆ vj, · · · ]

10/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Boundary operator Definition

@k : Ck ! Ck1 [v0, · · · , vk1] 7 !

k

X

j=0

(1)j[v0, · · · , ˆ vj, · · · ]

Example

@2(bec) = ec bc + be

10/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Boundary operator Definition

@k : Ck ! Ck1 [v0, · · · , vk1] 7 !

k

X

j=0

(1)j[v0, · · · , ˆ vj, · · · ]

Example

@2(bec) = ec bc + be @1@2(bec) = c e (c b) + e b = 0

10/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Theorem

@k @k+1 = 0

Consequence

Im @k+1 ⇢ ker@k

Definition

Hk = ker @k/Im@k+1 and k = dim ker @k range @k+1

11/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Interpretation : The magic

I 0 : number of connected components I 1 : number of holes I 2 : number of voids I to be continued

12/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Example

@0 ⌘ 0, @1 = B B B B @ 1 1 1 1 1 1 1 1 1 1 1 1 1 C C C C A

Nb of connected components

dim ker @0 = 5, range @1 = 4 hence 0 = 1

13/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Number of holes

@2 = B B B B B B @ 1 1 1 1 C C C C C C A

Nb of holes

dim ker@1 = 2, range @2 = 1 hence 1 = 1

14/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Polygons=cycles

1 = Nb of independent polygons Nb of independent triangles.

15/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Polygons=cycles

1 = Nb of independent polygons Nb of independent triangles.

15/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Polygons=cycles

1 = Nb of independent polygons Nb of independent triangles. 1 = 2 1 = 1.

15/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Polygons=cycles

1 = Nb of independent polygons Nb of independent triangles. 1 = 2 2 = 0.

15/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Open question

What is the interpretation of the Betti numbers for hypergraphs or boolean monotone functions ?

16/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Open question

What is the interpretation of the Betti numbers for hypergraphs or boolean monotone functions ? Find the single minimal triangulation = construct the minimum weight basis of H2

16/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Euler characteristic (S A + F) Definition

=

d

X

j=0

(1)jj

Discrete Morse inequality

|Ck1| + |Ck| |Ck+1|  k  |Ck|

17/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Euler characteristic (S A + F) Definition

=

d

X

j=0

(1)jj =

1

X

j=0

(1)j |Cj|

Discrete Morse inequality

|Ck1| + |Ck| |Ck+1|  k  |Ck|

17/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Alternative complex Cech complex

[v0, · · · , vk1] 2 Ck ( ) \k

j=0B(xj, ✏) 6= ;

Rips-Vietoris complex

[v0, · · · , vk1] 2 Rk ( ) B(xj, ✏) \ B(xl, ✏) 6= ; k simplex = clique of k + 1 points

18/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Difference RV vs Cech For the l1 distance

RV=Cech

Euclidean norm : false negative

Rips complex may miss some holes

19/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Difference RV vs Cech For the l1 distance

RV=Cech

Euclidean norm : false negative

Rips complex may miss some holes

19/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Cech vs Rips

R✏0(V) ⇢ ˇ C✏(V) ⇢ R2✏(V) whenever ✏ ✏0 s d 2(d + 1)

Euclidean distance (D.-Feng-Martins)

I Coverage radius RS I Communication radius RC = RS

20/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Lower-bound of the error Theorem ( p 3  γ  2)

p2dl() =2⇡2 Z Rc/

p 3 Rs

r0dr0 Z 'u(r0)

'l(r0)

d'1 Z R1(r0,'1)

r0

e⇡r2 ⇥ e|S+(r0,'1)|(1 e|S(r0,r1,'1)|)r1dr1 (1) where

'l(r0)=2 arccos(Rc/(2r0)), 'u(r0)=2 arcsin(Rc/(2r0))2 arccos(Rc/(2r0)) R1(r0,'1)=min(p R2

c r2 0 sin2 '1r0 cos '1

p

R2

c r2 0 sin2('1+'l(r0))+r0 cos('1+'l(r0))) 21/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 1 2 3 4 5 6 7 8 9

intensity λ p2d(λ)(%)

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

Probability to miss a hole using RRS and RRC

22/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Goals and related works

I Evaluate Betti nb and Euler charac. in some random settings I Penrose : Asymptotics of E[|Ck|m] for Euclidian-RG Rips

complex on the whole space (m = 1, 2)

I Kähle : Asymptotics of E[k] for Euclidian-RG Cech complex

(deterministic number of points) and ER

23/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Goals and related works

I Evaluate Betti nb and Euler charac. in some random settings I Penrose : Asymptotics of E[|Ck|m] for Euclidian-RG Rips

complex on the whole space (m = 1, 2)

I Kähle : Asymptotics of E[k] for Euclidian-RG Cech complex

(deterministic number of points) and ER

Our results

Exact expressions of all moments of |Ck| and in any dimension for RG complex on a torus for the l1 norm

23/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Outline

Algebraic topology Poisson homologies Euler characteristic Asymptotic results Robust estimate Persistence

24/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Random setting

x1 a a a a x1 " [0, a] ⇥ [0, a] T2

a⇥a

25/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Euler characteristic

I d=1 : { = 0 \ 0 6= 0} , { circle is covered } I d=2 : { = 0 \ 0 6= 1} , { domain is covered } I d=3 : { = 0 \ 0 + 2 6= 1} , { space is covered }

26/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Euler characteristic (D.-Ferraz-Randriam-Vergne) Euler characteristic

E [] = e✓ ad ✓ Bd(✓ ad) where ✓ = ✓2✏ a ◆d . where Bd is the d-th Bell polynomial Bd(x) = ⇢d 1

• x +

⇢d 2

• x2 + ... +

⇢d d

• xd

27/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

k simplices The key remark

|Ck| = Z h(x1, · · · , xk)d!(k)(x1, · · · , xk) where h(x1, · · · , xk) , 1 k! Y

i6=j

1{kxixjk<✏}

First moments

E[|Ck|] = ad (k + 1)d (k + 1)! (ad✓)k

28/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Dimension 5

29/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Asymptotic results

If ! 1, i(!)

p.s.

• ! i(Td) =

d

i

• .

30/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Limit theorems CLT for Euler characteristic

distanceTV E[] p V , N(0, 1) !  c p

• ·

31/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Limit theorems CLT for Euler characteristic

distanceTV E[] p V , N(0, 1) !  c p

• ·

Method

I Stein method I Malliavin calculus for Poisson process

31/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Concentration inequality

I Discrete gradient DxF(!) = F(! [ {x}) F(!) I Dx0 2 {1, 0, 1, 2, 3}

32/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Concentration inequality

I Discrete gradient DxF(!) = F(! [ {x}) F(!) I Dx0 2 {1, 0, 1, 2, 3}

32/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Concentration inequality

I Discrete gradient DxF(!) = F(! [ {x}) F(!) I Dx0 2 {1, 0, 1, 2, 3}

c > E[β0]

P(0 c)  exp  c E[0] 6 log ✓ 1 + c E[0] 3 ◆

32/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Complexity An important remark

I Construction of the complex is exponential (worst case)

33/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Complexity An important remark

I Construction of the complex is exponential (worst case) I Computations of Betti numbers is polynomial

33/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Further application (D.-Martins-Vergne) Green networking

Switch off some sensors keeping the coverage

Height of an edge

Rank of the highest simplex it belongs to

Index of a vertex

Infimum of the height of its adjacent edges

34/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

v0 v1 v2 v3 v4 D[v0, v1, v2]=D[v0, v1, v3]=D[v0, v2, v3]=D[v1, v2, v3]=3 D[v1, v3, v4] = 2 I[v0]=I[v2]=3 and I[v1]=I[v3]=I[v4]=2

35/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Example

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

I Complexity C bounded by 2H

36/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Complexity θn = (rn/a)d

✓0

k = k

1+ηd k1

n

k k1

, ✓k = k 1+η+d

k1

n

k k1

✓n 2 [✓0

k, ✓k] =

) C

n!1

• ! k

37/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Other regimes Theorem (Critical: nθn ! 1)

C = O(n3 ln n).

38/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence Euler characteristic Asymptotic results Robust estimate

Other regimes Theorem (Critical: nθn ! 1)

C = O(n3 ln n).

Theorem (Super-critiqual: nθn ! 1)

Cn = O(2nn3)

38/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Outline

Algebraic topology Poisson homologies Persistence

39/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

And then ? Topological algebra

I Algebraic procedure to determine

40/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

And then ? Topological algebra

I Algebraic procedure to determine I 0= nb of connected components

40/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

And then ? Topological algebra

I Algebraic procedure to determine I 0= nb of connected components I 1= nb of holes

40/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

And then ? Topological algebra

I Algebraic procedure to determine I 0= nb of connected components I 1= nb of holes

Problem

I Continuous structure (the cloud)

40/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

And then ? Topological algebra

I Algebraic procedure to determine I 0= nb of connected components I 1= nb of holes

Problem

I Continuous structure (the cloud) I Discrete result (Betti numbers)

40/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

And then ? Topological algebra

I Algebraic procedure to determine I 0= nb of connected components I 1= nb of holes

Problem

I Continuous structure (the cloud) I Discrete result (Betti numbers) I No continuity

40/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Persistence diagram of 10 pts on a circle

• 40
• 20

20 40

• 40
• 20

20 40 x1 x2 0e+00 1e-04 2e-04 3e-04 4e-04 5e-04 0e+00 1e-04 2e-04 3e-04 4e-04 5e-04 Birth Death

dim Birth Death 6.50655e-22 5.124479e-04 2.00255e-15 2.956910e-07 1.15748e-14 1.118440e-06 1.78540e-14 1.039640e-08 1.18076e-13 9.427080e-13 4.58441e-09 3.829860e-08 2.99134e-08 3.758330e-08 1 3.37283e-07 1.587770e-04 1 3.80028e-06 5.124480e-04 1 9.12595e-05 4.632000e-04 1 9.31748e-05 1.577760e-04 1 1.23796e-04 1.596950e-04 41/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Persistence diagram of 500 pts on a circle

• 1.0
• 0.5

0.0 0.5 1.0

• 1.0
• 0.5

0.0 0.5 1.0 x1 x2 1 2 3 4 5 1 2 3 4 5 Birth Death

42/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Comparison of persistence diagrams Definition

If |D1| > |D2|, ˜ D1=D1\{ the |D1| |D2| pts of D1 closest to the diagonal } ⇢(D1, D2) = T1( ˜ D1, D2)

43/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Comparison of persistence diagrams Definition

If |D1| > |D2|, ˜ D1=D1\{ the |D1| |D2| pts of D1 closest to the diagonal } ⇢(D1, D2) = T1( ˜ D1, D2) X

43/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Comparison of persistence diagrams Definition

If |D1| > |D2|, ˜ D1=D1\{ the |D1| |D2| pts of D1 closest to the diagonal } ⇢(D1, D2) = T1( ˜ D1, D2) X

43/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Costs on configuration space Definition (Total variation)

CTV(!, ⌘) = sup

A compact

|!(A) ⌘(A)| = ( nb of 6= pts) where ! =

n

X

j=1

"xi, ⌘ =

m

X

k=1

"yk

44/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Costs on configuration space Definition (Total variation)

CTV(!, ⌘) = sup

A compact

|!(A) ⌘(A)| = ( nb of 6= pts) where ! =

n

X

j=1

"xi, ⌘ =

m

X

k=1

"yk

Definition (Quadratic cost)

C2(!, ⌘) = 1 2 inf ⇢Z dE(x, y)2d(x, y), 2 Σ!,⌘

• ,

= 8 < : +1 if m 6= n inf

2Sn 1 2

Pn

j=1 dE(xj, y(j))2

if m = n < +1.

44/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Finite point processes on E = Rd Theorem (LD’08)

TC2(µ, ⌫) < 1 iff µ(⌘(E) = n) = ⌫(!(E) = n), 8n 0 X

n1

TCe(jµ

n , j⌫ n )2 µ(⌘(E) = n) < +1

Moreover, the optimal map T is described by T : Γ(n)

E

• ! Γ(n)

E

! =

n

X

j=1

"xi 7 ! tjµ

n ,jν n (x1, · · · , xn) 45/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Poisson process Theorem

I Te(1, 2) < +1 I t1,2 the transport map of MKP(1, 2, Ce) I Then

T : E ! E X

x2!

✏x 7 ! X

x2!

✏tσ1,σ2(x) is the transport map from ⇡1 to ⇡2 and TC2(⇡1, ⇡2) = 1(E) TCe(1, 2).

46/47 February, 2017 Institut Mines-Telecom Topology of wireless networks

Algebraic topology Poisson homologies Persistence

Persistence diagrams of point processes Theorem (LD, A. Vasseur’15)

I µ and ⌫ 2 point processes I D#µ = distribution of the µ-persistence diagram

T⇢(D#µ, D#⌫)  TC2(µ, ⌫)

47/47 February, 2017 Institut Mines-Telecom Topology of wireless networks