We show how to quantify coverage in a random sensor network.

1. Random homologies and sensor networks
Random homologies and sensor networks
L. Decreusefond E. Ferraz H. Randriambololona
Telecom ParisTech
Avignon, June 4th, 2010

2. Random homologies and sensor networks
Plan
Sensor networks
Mathematical framework
Applications
A primer on homology theory
Poissonian homologies
Euler characteristic
Asymptotic results
Robust estimate

3. Random homologies and sensor networks
Sensor networks
Sensor networks
A wireless sensor network (WSN) is a wireless network consisting of
spatially distributed autonomous devices using sensors to
cooperatively monitor physical or environmental conditions.
Wikipedia

4. Random homologies and sensor networks
Sensor networks
A sensor
A sensor is defined by
1. position
12082. THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / December 2006
coverage radius
at each time.
Fig. 1. A collection of sensor nodes generates a cover in the workspace [bottom]. The Rips complex of the network is an
abstract simplicial complex which has no localization or coordinate data [top]. In the example illustrated, the Rips complex
encodes the communication network as one closed 3-simplex, eleven closed 2-simplices, and seven closed 1-simplices
connected as shown. The ‘holes’ in this Rips complex reflect the holes in the sensor cover, below.

5. Random homologies and sensor networks
Sensor networks
A sensor
A sensor is defined by
1. position
12082. THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / December 2006
coverage radius
at each time.
Fig. 1. A collection of sensor nodes generates a cover in the workspace [bottom]. The Rips complex of the network is an
abstract simplicial complex which has no localization or coordinate data [top]. In the example illustrated, the Rips complex
encodes the communication network as one closed 3-simplex, eleven closed 2-simplices, and seven closed 1-simplices
connected as shown. The ‘holes’ in this Rips complex reflect the holes in the sensor cover, below.

6. Random homologies and sensor networks
Sensor networks
Some questions
All positions known : Domain covered ?
Some positions known : Optimal locations of other sensors
Positions varying with time : Creation of holes ?
Fault-tolerance/ Power-saving : Can we support failure (or
switch-off) without creating holes ?

7. Random homologies and sensor networks
Sensor networks
Some questions
All positions known : Domain covered ?
Some positions known : Optimal locations of other sensors
Positions varying with time : Creation of holes ?
Fault-tolerance/ Power-saving : Can we support failure (or
switch-off) without creating holes ?

8. Random homologies and sensor networks
Sensor networks
Some questions
All positions known : Domain covered ?
Some positions known : Optimal locations of other sensors
Positions varying with time : Creation of holes ?
Fault-tolerance/ Power-saving : Can we support failure (or
switch-off) without creating holes ?

9. Random homologies and sensor networks
Sensor networks
Some questions
All positions known : Domain covered ?
Some positions known : Optimal locations of other sensors
Positions varying with time : Creation of holes ?
Fault-tolerance/ Power-saving : Can we support failure (or
switch-off) without creating holes ?

10. Random homologies and sensor networks
Mathematical framework
Mathematical framework
Homology : Algebraization of the topology
Coverage : reduces to compute the rank of a matrix
Detection of hole, redundancy : reduces to the computation of a
basis of a vector matrix, obtained by matrix reduction
(as in Gauss algorithm).

11. Random homologies and sensor networks
Mathematical framework
Mathematical framework
Homology : Algebraization of the topology
Coverage : reduces to compute the rank of a matrix
Detection of hole, redundancy : reduces to the computation of a
basis of a vector matrix, obtained by matrix reduction
(as in Gauss algorithm).

12. Random homologies and sensor networks
Mathematical framework
Rips complex

13. Random homologies and sensor networks
Mathematical framework
Rips complex

14. Random homologies and sensor networks
Mathematical framework
Rips complex

15. Random homologies and sensor networks
Mathematical framework
Rips complex

16. Random homologies and sensor networks
Mathematical framework
Rips complex

17. Random homologies and sensor networks
Mathematical framework
Rips complex

18. Random homologies and sensor networks
Mathematical framework
Rips complex

19. Random homologies and sensor networks
Mathematical framework
Rips complex of sensor network (cf. [dSG07, dSG06, GM05])
1208 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / December 2006
Fig. 1. A collection of sensor nodes generates a cover in the workspace [bottom]. The Rips complex of the network is an
abstract simplicial complex which has no localization or coordinate data [top]. In the example illustrated, the Rips complex
encodes the communication network as one closed 3-simplex, eleven closed 2-simplices, and seven closed 1-simplices
connected as shown. The ‘holes’ in this Rips complex reflect the holes in the sensor cover, below.

20. Random homologies and sensor networks
A primer on homology theory
Rips complex
d
b e
c
a
Vertices : a, b, c, d, e.
Edges : ab, bc, ca, ae, be, ec, bd, ed.
Triangles : abc, abe, bec, aec, bed.
Tetrahedron : abec.

21. Random homologies and sensor networks
A primer on homology theory
Rips complex
d
b e
c
a
Vertices : a, b, c, d, e.
Edges : ab, bc, ca, ae, be, ec, bd, ed.
Triangles : abc, abe, bec, aec, bed.
Tetrahedron : abec.

22. Random homologies and sensor networks
A primer on homology theory
Rips complex
d
b e
c
a
Vertices : a, b, c, d, e.
Edges : ab, bc, ca, ae, be, ec, bd, ed.
Triangles : abc, abe, bec, aec, bed.
Tetrahedron : abec.

23. Random homologies and sensor networks
A primer on homology theory
Rips complex
d
b e
c
a
Vertices : a, b, c, d, e.
Edges : ab, bc, ca, ae, be, ec, bd, ed.
Triangles : abc, abe, bec, aec, bed.
Tetrahedron : abec.

24. Random homologies and sensor networks
A primer on homology theory
Simplices algebra [Hat02, ZC05]
ab = −ba
3ab means three times the edge ab.
Boundary operator
n
∂n−1 a1 a2 . . . an = (−1)i a1 a2 . . . ai . . . an .
i=1

25. Random homologies and sensor networks
A primer on homology theory
Simplices algebra [Hat02, ZC05]
ab = −ba
3ab means three times the edge ab.
Boundary operator
n
∂n−1 a1 a2 . . . an = (−1)i a1 a2 . . . ai . . . an .
i=1

26. Random homologies and sensor networks
A primer on homology theory
Simplices algebra [Hat02, ZC05]
ab = −ba
3ab means three times the edge ab.
Boundary operator
n
∂n−1 a1 a2 . . . an = (−1)i a1 a2 . . . ai . . . an .
i=1

27. Random homologies and sensor networks
A primer on homology theory
Simplices algebra [Hat02, ZC05]
ab = −ba
3ab means three times the edge ab.
Boundary operator
n
∂n−1 a1 a2 . . . an = (−1)i a1 a2 . . . ai . . . an .
i=1
Example :
∂2 abe = be − ae + ab.

28. Random homologies and sensor networks
A primer on homology theory
Main result
Main observation
∂n ∂n+1 = 0
∂1 ∂2 abe = ∂1 (be − ae + ab)
= e − b − (e − a) + b − a = 0.

29. Random homologies and sensor networks
A primer on homology theory
Cycles and boundaries
A triangle is a cycle of edges.
A tetrahedron is a cycle of triangles.
A triangle is a boundary of a tetrahedron.
d
b e
c
a

30. Random homologies and sensor networks
A primer on homology theory
Cycles and boundaries
A triangle is a cycle of edges.
A tetrahedron is a cycle of triangles.
A triangle is a boundary of a tetrahedron.
d
b e
c
a

31. Random homologies and sensor networks
A primer on homology theory
Cycles and boundaries
A triangle is a cycle of edges.
A tetrahedron is a cycle of triangles.
A triangle is a boundary of a tetrahedron.
d
b e
c
a

32. Random homologies and sensor networks
A primer on homology theory
Betti numbers
β0 =number of connected components
β0 = Nb of independent vertices − Nb of independent edges.

33. Random homologies and sensor networks
A primer on homology theory
Betti numbers
β0 =number of connected components
β0 = Nb of independent vertices − Nb of independent edges.
β0 = 5 − 3 = 2.

34. Random homologies and sensor networks
A primer on homology theory
Betti numbers
β0 =number of connected components
β0 = Nb of independent vertices − Nb of independent edges.
β0 = 5 − 4 = 1.

35. Random homologies and sensor networks
A primer on homology theory
Betti numbers
β0 =number of connected components
β0 = Nb of independent vertices − Nb of independent edges.
β0 = 5 − 4 = 1.

36. Random homologies and sensor networks
A primer on homology theory
β1 =number of « holes »
β1 = Nb of independent polygons − Nb of independent triangles.

37. Random homologies and sensor networks
A primer on homology theory
β1 =number of « holes »
β1 = Nb of independent polygons − Nb of independent triangles.
β1 = 1 − 1 = 0.

38. Random homologies and sensor networks
A primer on homology theory
β1 =number of « holes »
β1 = Nb of independent polygons − Nb of independent triangles.
β1 = 2 − 1 = 1.

39. Random homologies and sensor networks
A primer on homology theory
β1 =number of « holes »
β1 = Nb of independent polygons − Nb of independent triangles.
β1 = 3 − 3 = 0.

40. Random homologies and sensor networks
A primer on homology theory
For larger n
βn = dim ker ∂n − dim range ∂n+1 .
Hard to visualize the intuitive meaning
But relatively easy to compute.
Existence of tetrahedron in Rips complex means over-coverage.

41. Random homologies and sensor networks
A primer on homology theory
For larger n
βn = dim ker ∂n − dim range ∂n+1 .
Hard to visualize the intuitive meaning
But relatively easy to compute.
Existence of tetrahedron in Rips complex means over-coverage.

42. Random homologies and sensor networks
A primer on homology theory
For larger n
βn = dim ker ∂n − dim range ∂n+1 .
Hard to visualize the intuitive meaning
But relatively easy to compute.
Existence of tetrahedron in Rips complex means over-coverage.

43. Random homologies and sensor networks
A primer on homology theory
For larger n
βn = dim ker ∂n − dim range ∂n+1 .
Hard to visualize the intuitive meaning
But relatively easy to compute.
Existence of tetrahedron in Rips complex means over-coverage.

44. Random homologies and sensor networks
Poissonian homologies
Random setting
Sensors = Poisson process (λ)
Domain = d dimensional torus of width a
Coverage = square of width ε
r -dilation of P.P.(λ)=P.P.(λr −d ), one can choose a = 1.
[0, a] × [0, a]
T2
a×a
a
a
x1 a
a
x1

45. Random homologies and sensor networks
Poissonian homologies
Random setting
Sensors = Poisson process (λ)
Domain = d dimensional torus of width a
Coverage = square of width ε
r -dilation of P.P.(λ)=P.P.(λr −d ), one can choose a = 1.
[0, a] × [0, a]
T2
a×a
a
a
x1 a
a
x1

46. Random homologies and sensor networks
Poissonian homologies
Random setting
Sensors = Poisson process (λ)
Domain = d dimensional torus of width a
Coverage = square of width ε
r -dilation of P.P.(λ)=P.P.(λr −d ), one can choose a = 1.
[0, a] × [0, a]
T2
a×a
a
a
x1 a
a
x1

47. Random homologies and sensor networks
Poissonian homologies
Random setting
Sensors = Poisson process (λ)
Domain = d dimensional torus of width a
Coverage = square of width ε
r -dilation of P.P.(λ)=P.P.(λr −d ), one can choose a = 1.
[0, a] × [0, a]
T2
a×a
a
a
x1 a
a
x1

48. Random homologies and sensor networks
Poissonian homologies
Euler characteristic
Euler characteristic
d
χ= (−1)k βk .
k=0
d=1 : {χ = 0 ∩ β0 = 0} ⇔ { circle is covered }
d=2 : {χ = 0 ∩ β0 = β1 } ⇔ { domain is covered }
d=3 : {χ = 0 ∩ β0 + β2 = β1 } ⇔ { space is covered }

49. Random homologies and sensor networks
Poissonian homologies
Euler characteristic
Euler characteristic
d
χ= (−1)k βk .
k=0
d=1 : {χ = 0 ∩ β0 = 0} ⇔ { circle is covered }
d=2 : {χ = 0 ∩ β0 = β1 } ⇔ { domain is covered }
d=3 : {χ = 0 ∩ β0 + β2 = β1 } ⇔ { space is covered }

50. Random homologies and sensor networks
Poissonian homologies
Euler characteristic
Euler characteristic
d
χ= (−1)k βk .
k=0
d=1 : {χ = 0 ∩ β0 = 0} ⇔ { circle is covered }
d=2 : {χ = 0 ∩ β0 = β1 } ⇔ { domain is covered }
d=3 : {χ = 0 ∩ β0 + β2 = β1 } ⇔ { space is covered }

51. Random homologies and sensor networks
Poissonian homologies
Euler characteristic
Euler characteristic
d
χ= (−1)k βk .
k=0
d=1 : {χ = 0 ∩ β0 = 0} ⇔ { circle is covered }
d=2 : {χ = 0 ∩ β0 = β1 } ⇔ { domain is covered }
d=3 : {χ = 0 ∩ β0 + β2 = β1 } ⇔ { space is covered }

52. Random homologies and sensor networks
Poissonian homologies
Euler characteristic
Euler characteristic
d
χ= (−1)k βk .
k=0
d=1 : {χ = 0 ∩ β0 = 0} ⇔ { circle is covered }
d=2 : {χ = 0 ∩ β0 = β1 } ⇔ { domain is covered }
d=3 : {χ = 0 ∩ β0 + β2 = β1 } ⇔ { space is covered }
∞
χ= (−1)k sk .
k=1

53. Random homologies and sensor networks
Poissonian homologies
Euler characteristic
Euler characteristic
Bd (x) : Bell polynomial
d d d
Bd (x) = x+ x 2 + ... + xd
1 2 d
Euler characteristic
λe −θ
E [χ] = − Bd (−θ) where θ = λ d .
θ

54. Random homologies and sensor networks
Poissonian homologies
Euler characteristic
Euler characteristic
Bd (x) : Bell polynomial
d d d
Bd (x) = x+ x 2 + ... + xd
1 2 d
Euler characteristic
λe −θ
E [χ] = − Bd (−θ) where θ = λ d .
θ

55. Random homologies and sensor networks
Poissonian homologies
Euler characteristic
k simplices
1
Define h(x1 , ..., xk ) k! I{ xi −xj < ,i=j} (x1 , ..., xk )
Then (Campbell) :
sk = λk+1 ··· h(x1 , ..., xk+1 )dxk+1 ...dx1
T T
(k + 1)d
= λθk
(k + 1)!

56. Random homologies and sensor networks
Poissonian homologies
Euler characteristic
k simplices
1
Define h(x1 , ..., xk ) k! I{ xi −xj < ,i=j} (x1 , ..., xk )
Then (Campbell) :
sk = λk+1 ··· h(x1 , ..., xk+1 )dxk+1 ...dx1
T T
(k + 1)d
= λθk
(k + 1)!

57. Random homologies and sensor networks
Poissonian homologies
Euler characteristic
Dimension 5
Domination regions of βk when d = 5 in function of λ
10
8 Dominating βk
6 β0
4 β0 and β1
2 β1
β1 and β2
0
β2
χ
-2
β2 and β3
-4 β3
-6 β3 and β4
-8 β4
-10 No dominance
-12
0 2 4 6 8 10 12 14 16
λ

58. Random homologies and sensor networks
Poissonian homologies
Asymptotic results
Asymptotic results
p.s. d
If λ → ∞, βi (ω) −→ βi (Td ) = i .

59. Random homologies and sensor networks
Poissonian homologies
Asymptotic results
Asymptotic results (cont’d)
(k + 1)d (j + 1)d −d
Cov(sk , sj ) ∼ θk+j+1
(k + 1)!(j + 1)!
Euler characteristic
d
1 1 −2θ
Var(χ) ∼ e Bd (−θ)2 = Vχ
θ

60. Random homologies and sensor networks
Poissonian homologies
Asymptotic results
Asymptotic results (cont’d)
(k + 1)d (j + 1)d −d
Cov(sk , sj ) ∼ θk+j+1
(k + 1)!(j + 1)!
Euler characteristic
d
1 1 −2θ
Var(χ) ∼ e Bd (−θ)2 = Vχ
θ

61. Random homologies and sensor networks
Poissonian homologies
Asymptotic results
Limit theorems
CLT for Euler characteristic
χ − E[χ] λ→∞
− − N(0, 1).
−→
Vχ

62. Random homologies and sensor networks
Poissonian homologies
Asymptotic results
Limit theorems
CLT for Euler characteristic
χ − E[χ] λ→∞
− − N(0, 1).
−→
Vχ
Method
Stein method
Malliavin calculus for Poisson process

63. Random homologies and sensor networks
Poissonian homologies
Asymptotic results
Limit theorems
CLT for Euler characteristic
χ − E[χ] λ→∞
− − N(0, 1).
−→
Vχ
Method
Stein method
Malliavin calculus for Poisson process

64. Random homologies and sensor networks
Poissonian homologies
Asymptotic results
Limit theorems
CLT for Euler characteristic
χ − E[χ] λ→∞
− − N(0, 1).
−→
Vχ
Method
Stein method
Malliavin calculus for Poisson process

65. Random homologies and sensor networks
Poissonian homologies
Robust estimate
Concentration inequality
Discrete gradient Dx F (ω) = F (ω ∪ {x}) − F (ω)
Dx β0 ∈ {1, 0, −1, −2, −3}

66. Random homologies and sensor networks
Poissonian homologies
Robust estimate
Concentration inequality
Discrete gradient Dx F (ω) = F (ω ∪ {x}) − F (ω)
Dx β0 ∈ {1, 0, −1, −2, −3}

67. Random homologies and sensor networks
Poissonian homologies
Robust estimate
Concentration inequality
Discrete gradient Dx F (ω) = F (ω ∪ {x}) − F (ω)
Dx β0 ∈ {1, 0, −1, −2, −3}
c > E[β0 ]
c − E[β0 ] c − E[β0 ]
P(β0 ≥ c) ≤ exp − log 1 +
6 3λ

68. Random homologies and sensor networks
Poissonian homologies
Robust estimate
Dimension 2
β0 ≤ Nb of points in a MHC process
1 − eθ
E[β0 ] ≤ λ =τ
θ

69. Random homologies and sensor networks
Poissonian homologies
Robust estimate
Dimension 2
β0 ≤ Nb of points in a MHC process
1 − eθ
E[β0 ] ≤ λ =τ
θ
c −τ c −τ
P(β0 ≥ c) ≤ exp − log 1 +
6 3λ

70. Random homologies and sensor networks
Poissonian homologies
Robust estimate
Références I
V. de Silva and R. Ghrist.
Coordinate-free coverage in sensor networks with controlled
boundaries via homology.
Intl. Journal of Robotics Research, 25(12) :1205–1222, 2006.
V. de Silva and R. Ghrist.
Coverage in sensor networks via persistent homology.
Algebr. Geom. Topol., 7 :339–358, 2007.
R. Ghrist and A. Muhammad.
Coverage and hole-detection in sensor networks via homology.
Information Processing in Sensor Networks, 2005. IPSN 2005.
Fourth International Symposium on, pages 254–260, 2005.

71. Random homologies and sensor networks
Poissonian homologies
Robust estimate
Références II
A. Hatcher.
Algebraic topology.
Cambridge University Press, Cambridge, 2002.
A. Zomorodian and G. Carlsson.
Computing persistent homology.
Discrete Comput. Geom., 33(2) :249–274, 2005.