Beyond random graphs : random simplicial complexes. Applications to sensor networks

Transcription

1 Beyond random graphs : random simplicial complexes. Applications to sensor networks L. Decreusefond E. Ferraz P. Martins H. Randriambololona A. Vergne Telecom ParisTech Workshop on random graphs, April 5th, 2011

2 Plan Sensor networks Poisson homologies Euler characteristic Asymptotic results Robust estimate

3 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 Sensor networks A sensor A sensor is defined by 1. position 2. coverage radius 1208 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / December 2006 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 Sensor networks A sensor A sensor is defined by 1. position 2. coverage radius 1208 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / December 2006 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 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 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 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 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 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 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 Cech complex C k = {[x 0,, x k 1 ], x i ω, k i=0b(x i, ɛ) } Nerve theorem The set x ω B(x, ɛ) has the same homology groups as the simplicial complex (C k, k 0).

13 Rips complex

14 Rips complex

15 Rips complex

16 Rips complex

17 Rips complex

18 Rips complex

19 Rips complex

20 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.

21 Rips complex d b e a c Vertices : a, b, c, d, e. Edges : ab, bc, ca, ae, be, ec, bd, ed. Triangles : abc, abe, bec, aec, bed. Tetrahedron : abec.

22 Rips complex d b e a c Vertices : a, b, c, d, e. Edges : ab, bc, ca, ae, be, ec, bd, ed. Triangles : abc, abe, bec, aec, bed. Tetrahedron : abec.

23 Rips complex d b e a c Vertices : a, b, c, d, e. Edges : ab, bc, ca, ae, be, ec, bd, ed. Triangles : abc, abe, bec, aec, bed. Tetrahedron : abec.

24 Rips complex d b e a c Vertices : a, b, c, d, e. Edges : ab, bc, ca, ae, be, ec, bd, ed. Triangles : abc, abe, bec, aec, bed. Tetrahedron : abec.

25 Rips and Cech No hole in Cech implies no hole in Rips complex. The converse does not hold. For l distance, Rips=Cech.

26 Rips and Cech No hole in Cech implies no hole in Rips complex. The converse does not hold. For l distance, Rips=Cech.

27 Rips and Cech No hole in Cech implies no hole in Rips complex. The converse does not hold. For l distance, Rips=Cech.

28 Simplices algebra [Hat02, ZC05] ab = ba 3ab means three times the edge ab. Boundary operator n 1 a 1 a 2... a n = n ( 1) i a 1 a 2... â i... a n. i=1

29 Simplices algebra [Hat02, ZC05] ab = ba 3ab means three times the edge ab. Boundary operator n 1 a 1 a 2... a n = n ( 1) i a 1 a 2... â i... a n. i=1

30 Simplices algebra [Hat02, ZC05] ab = ba 3ab means three times the edge ab. Boundary operator n 1 a 1 a 2... a n = n ( 1) i a 1 a 2... â i... a n. i=1

31 Simplices algebra [Hat02, ZC05] ab = ba 3ab means three times the edge ab. Boundary operator n 1 a 1 a 2... a n = n ( 1) i a 1 a 2... â i... a n. i=1 Example : 2 abe = be ae + ab.

32 Main result Main observation n n+1 = abe = 1 (be ae + ab) = e b (e a) + b a = 0.

33 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. b e d a c

34 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. b e d a c

35 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. b e d a c

36 Betti numbers β 0 =number of connected components β 0 = Nb of vertices Nb of independent edges.

37 Betti numbers β 0 =number of connected components β 0 = Nb of vertices Nb of independent edges. β 0 = 5 3 = 2.

38 Betti numbers β 0 =number of connected components β 0 = Nb of vertices Nb of independent edges. β 0 = 5 4 = 1.

39 Betti numbers β 0 =number of connected components β 0 = Nb of vertices Nb of independent edges. β 0 = 5 4 = 1.

40 β 1 =number of «holes» β 1 = Nb of independent polygons Nb of independent triangles.

41 β 1 =number of «holes» β 1 = Nb of independent polygons Nb of independent triangles. β 1 = 1 1 = 0.

42 β 1 =number of «holes» β 1 = Nb of independent polygons Nb of independent triangles. β 1 = 2 1 = 1.

43 β 1 =number of «holes» β 1 = Nb of independent polygons Nb of independent triangles. β 1 = 3 3 = 0.

44 For larger n β n = 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.

45 For larger n β n = 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.

46 For larger n β n = 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.

47 For larger n β n = 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.

48 Poisson 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] a T 2 a a a x 1 a x 1 a

49 Poisson 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] a T 2 a a a x 1 a x 1 a

50 Poisson 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] a T 2 a a a x 1 a x 1 a

51 Poisson 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] a T 2 a a a x 1 a x 1 a

52 Poisson homologies Euler characteristic Euler characteristic d χ = ( 1) k β k. k=0 d=1 : {χ = 0 β 0 0} { circle is covered } d=2 : {χ = 0 β 0 β 1 } { domain is covered } d=3 : {χ = 0 β 0 + β 2 β 1 } { space is covered }

53 Poisson homologies Euler characteristic Euler characteristic d χ = ( 1) k β k. k=0 d=1 : {χ = 0 β 0 0} { circle is covered } d=2 : {χ = 0 β 0 β 1 } { domain is covered } d=3 : {χ = 0 β 0 + β 2 β 1 } { space is covered }

54 Poisson homologies Euler characteristic Euler characteristic d χ = ( 1) k β k. k=0 d=1 : {χ = 0 β 0 0} { circle is covered } d=2 : {χ = 0 β 0 β 1 } { domain is covered } d=3 : {χ = 0 β 0 + β 2 β 1 } { space is covered }

55 Poisson homologies Euler characteristic Euler characteristic d χ = ( 1) k β k. k=0 d=1 : {χ = 0 β 0 0} { circle is covered } d=2 : {χ = 0 β 0 β 1 } { domain is covered } d=3 : {χ = 0 β 0 + β 2 β 1 } { space is covered }

56 Poisson homologies Euler characteristic Euler characteristic d χ = ( 1) k β k. k=0 d=1 : {χ = 0 β 0 0} { circle is covered } d=2 : {χ = 0 β 0 β 1 } { domain is covered } d=3 : {χ = 0 β 0 + β 2 β 1 } { space is covered } χ = ( 1) k s k. k=1

57 Poisson homologies Euler characteristic Euler characteristic B d (x) : Bell polynomial { } d B d (x) = x + 1 { } d x { } d x d d Euler characteristic E [χ] = λe θ B d ( θ) where θ = λɛ d. θ

58 Poisson homologies Euler characteristic Euler characteristic B d (x) : Bell polynomial { } d B d (x) = x + 1 { } d x { } d x d d Euler characteristic E [χ] = λe θ B d ( θ) where θ = λɛ d. θ

59 Poisson homologies Euler characteristic k simplices Define h(x 1,..., x k ) 1 k! I { x i x j <ɛ,i j}(x 1,..., x k ) Then (Campbell) : s k = λ k+1 = T (k + 1)d (k + 1)! λθk h(x 1,..., x k+1 )dx k+1...dx 1 T

60 Poisson homologies Euler characteristic k simplices Define h(x 1,..., x k ) 1 k! I { x i x j <ɛ,i j}(x 1,..., x k ) Then (Campbell) : s k = λ k+1 = T (k + 1)d (k + 1)! λθk h(x 1,..., x k+1 )dx k+1...dx 1 T

61 Poisson homologies Euler characteristic Dimension 5

62 Poisson homologies Euler characteristic Second order moments Cov(s k, s l ) = ( ) 1 d l 1 2ɛ i=0 1 i!(k l + i)!(l i)! θk+i ( ) i(k l + i) d k + i + 2. l i + 1

63 Poisson homologies Euler characteristic Euler characteristic Euler characteristic Var(χ) = ( ) 1 d c i θ i d i=1

64 Poisson homologies Euler characteristic Euler characteristic Euler characteristic In dimension 1, Var(χ) = ( ) 1 d c i θ i d i=1 Var(χ) = (θe θ 2θ 2 e 2θ)

65 Poisson homologies Asymptotic results Asymptotic results If λ, β i (ω) p.s. β i (T d ) = ( d i ).

66 Poisson homologies Asymptotic results Limit theorems CLT for Euler characteristic ( ) χ E[χ] distance TV, N(0, 1) Vχ c λ

67 Poisson homologies Asymptotic results Limit theorems CLT for Euler characteristic ( ) χ E[χ] distance TV, N(0, 1) Vχ c λ Method Stein method Malliavin calculus for Poisson process

68 Poisson homologies Asymptotic results Limit theorems CLT for Euler characteristic ( ) χ E[χ] distance TV, N(0, 1) Vχ c λ Method Stein method Malliavin calculus for Poisson process

69 Poisson homologies Asymptotic results Limit theorems CLT for Euler characteristic ( ) χ E[χ] distance TV, N(0, 1) Vχ c λ Method Stein method Malliavin calculus for Poisson process

70 Poisson homologies Robust estimate Concentration inequality Discrete gradient D x F (ω) = F (ω {x}) F (ω) D x β 0 {1, 0, 1, 2, 3}

71 Poisson homologies Robust estimate Concentration inequality Discrete gradient D x F (ω) = F (ω {x}) F (ω) D x β 0 {1, 0, 1, 2, 3}

72 Poisson homologies Robust estimate Concentration inequality Discrete gradient D x F (ω) = F (ω {x}) F (ω) D x β 0 {1, 0, 1, 2, 3} c > E[β 0 ] [ P(β 0 c) exp c E[β ( 0] log 1 + c E[β )] 0] 6 3λ

73 Poisson homologies Robust estimate Dimension 2 β 0 Nb of points in a MHC process E[β 0 ] λ 1 e θ θ = τ

74 Poisson homologies Robust estimate Dimension 2 β 0 Nb of points in a MHC process E[β 0 ] λ 1 e θ θ = τ [ P(β 0 c) exp c τ ( log 1 + c τ )] 6 3λ

75 Poisson 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) : , V. de Silva and R. Ghrist. Coverage in sensor networks via persistent homology. Algebr. Geom. Topol., 7 : , R. Ghrist and A. Muhammad. Coverage and hole-detection in sensor networks via homology. Information Processing in Sensor Networks, IPSN Fourth International Symposium on, pages , 2005.

76 Poisson homologies Robust estimate Références II A. Hatcher.. Cambridge University Press, Cambridge, A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete Comput. Geom., 33(2) : , 2005.