persistent homology and sensor networks

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Persistent Homology and Sensor Networks Persistent homology motivated by an application to sensor nets

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Persistent Homology and Sensor Networks. Persistent homology motivated by an application to sensor nets. Outline. A word about sensor nets Basic coverage criterion Better coverage criterion using persistence Introduce Persistent Homology Correspondence Theorem Computing the groups! - PowerPoint PPT Presentation

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Page 1: Persistent Homology and Sensor Networks

Persistent Homology and Sensor Networks

Persistent homology motivated by an application to sensor nets

Page 2: Persistent Homology and Sensor Networks

Outline• A word about sensor nets

• Basic coverage criterion

• Better coverage criterion using persistence

• Introduce Persistent Homology

• Correspondence Theorem

• Computing the groups!

• Other Applications

Page 3: Persistent Homology and Sensor Networks

A Word About Sensor Nets

Page 4: Persistent Homology and Sensor Networks
Page 5: Persistent Homology and Sensor Networks

August 29, 2005

Hurricane Katrina hits New Orleans

Page 6: Persistent Homology and Sensor Networks

Power and Communications Knocked Out

Page 7: Persistent Homology and Sensor Networks

Broken Levees

Page 8: Persistent Homology and Sensor Networks

City Flooded

Page 9: Persistent Homology and Sensor Networks

Inaccessible from the ground

Page 10: Persistent Homology and Sensor Networks

Law Enforcement

Page 11: Persistent Homology and Sensor Networks

Rescue Workers

Page 12: Persistent Homology and Sensor Networks
Page 13: Persistent Homology and Sensor Networks
Page 14: Persistent Homology and Sensor Networks
Page 15: Persistent Homology and Sensor Networks
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Page 17: Persistent Homology and Sensor Networks
Page 18: Persistent Homology and Sensor Networks
Page 19: Persistent Homology and Sensor Networks

Replace live turkey with a parachute

Page 20: Persistent Homology and Sensor Networks

Result: Useful sensor network

• Measure conditions on the ground at many locations

• Relay messages to and from rescue workers

• Instant infrastructure

• Low power/auto-power

• Cheap!?

Page 21: Persistent Homology and Sensor Networks

Other uses of sensor networks

• Environmental monitoring• Security systems• Battlefield monitoring and communications• Large mechanical systems• Find Sarah Connor

Page 22: Persistent Homology and Sensor Networks
Page 23: Persistent Homology and Sensor Networks

Hole in sensor coverage area

Sarah Connor escapes!

Page 24: Persistent Homology and Sensor Networks

Identifying holes in the network

• De Silva and Ghrist have developed a method for identifying gaps in sensor coverage

• Method is based on Algebraic Topology• Computing and examining Simplical

Homology groups• Theoretical underpinings allow you to do so

much more

Page 25: Persistent Homology and Sensor Networks

Basic Coverage Criterion

Part 1.2

Page 26: Persistent Homology and Sensor Networks

rc

rb

The problem to be solved:

Each node has sensors that cancover a circular region of radius rc

Each node can detect other nodesWithin its broadcast radius rb

rc ≥ rb/√(3)

Page 27: Persistent Homology and Sensor Networks

The problem to be solved:

Each node has sensors that cancover a circular region of radius rc

Each node can detect other nodesWithin its broadcast radius rb

rc ≥ rb/√(3)

Nodes lie in compact connected planar domain with piecewise linear boundary. Fence nodes at the vertices

All fence nodes know their neighbors’ identities and are no more than rb apart

Page 28: Persistent Homology and Sensor Networks

What we don’t have:• Nodes don’t know their absolute or relative

positions

• All we get is the connectivity graph

Page 29: Persistent Homology and Sensor Networks

It would be nice to have the Cech Complex

Def: For a collection of sets U={U}, the Cech Complex C(U) is the simplical complex where each non-empty intersection of (k+1) of the U correspond to a k-simplex.

3-simplex

Page 30: Persistent Homology and Sensor Networks

We have just enough to build the Rips Complex

•Let X be a collection of points in a metric space

•Rips complex R(X) contains a simplex for every collection of points that are pairwise within distance

•Even though our domain is planar, a dense graph can lead to simplices with arbitrary dimension

• In our case, we are building Rrb(X)

• Every complete k-subgraph of the communication graph becomes a simplex in the Rips Complex

• Also, it’s the maximal simplicial complex that has the connectivity graph as its 1-skeleton

Page 31: Persistent Homology and Sensor Networks

Picture of a Rips Complex

Page 32: Persistent Homology and Sensor Networks

Recap:X= { set of nodes }

rc = sensor radius

rb = broadcast radius

D = domain to be covered

∂D = boundary of D

Xf= { fence nodes that lie on D }

R= Rips complex of the communication graph

U= Region covered by the sensors

F= Fence subcomplex R

Page 33: Persistent Homology and Sensor Networks

Theorem (De Silva & Ghrist):

For a set of nodes X in a planar domain D satisfying the assumptions (rc, rb, fence nodes etc), the sensor cover Uc

contains D if there exists [] H2(R,F) such that ∂ ≠ 0

Page 34: Persistent Homology and Sensor Networks

What about a generator of H2(R,F)?

A generator will look like some linear combination of 2-simplices

i.e. Some triangulation of the domain D

Page 35: Persistent Homology and Sensor Networks

Theorem (De Silva & Ghrist):

For a set of nodes X in a planar domain D satisfying the assumptions (rc, rb, fence nodes etc), the sensor cover Uc contains D if there exists [] H2(R,F) such that ∂ ≠ 0

But why require ∂ ≠ 0 ??

Why not “if and only if” ??

Page 36: Persistent Homology and Sensor Networks

Pitfalls of the Rips complex

Bound was rc ≥ rb/√(3)

1/√ (3) ≈ 0.57

Therefore it’s possible to have a rectangle that is completely covered, but not triangulated in the communication graph

rbrb

So the conditions of the theorem are sufficient, but not necessary, to guarantee coverage.

Page 37: Persistent Homology and Sensor Networks

Pitfalls of the Rips complex

It’s possible to have an arrangement of nodes whose Rips complex is the surface of an octahedron.

This has non-zero H2, but its boundary is zero!

Page 38: Persistent Homology and Sensor Networks

Better coverage criterion using persistence

Part 1.3

Page 39: Persistent Homology and Sensor Networks

Eliminating the fence subcomplex• The assumption of the nice fence sub-complex is

unrealistic• Can we replace it with some other assumptions?

Page 40: Persistent Homology and Sensor Networks

The new situation:

Each node has sensors that cancover a circular region of radius rc

Each node can detect its neighborsvia a strong signal (rs) or a weaksignal (rw).

rc ≥ rs/√(2)rw ≥ rs √(10)

rc

rs

rw

Remember:strong <---> “short”weak <---> “wlong”

Page 41: Persistent Homology and Sensor Networks

The new situation (cont…):rc ≥ rs/√(2) rw ≥ rs √(10)

Nodes lie in a compact connected domain D in Rd

Nodes can detect the presence of ∂D within distance rf

The restricted domain D-C is connected, whereC = {x D ||x-∂D|| ≤ rf + rs/√(2)

The fence-detection hypersurface = {x D ||x-∂D|| = rf}Has internal injectivity radius ≥ rs/√(2) external injectivity radius ≥ rs

Page 42: Persistent Homology and Sensor Networks

The new situation (cont…):

rf

The fence “collar”, C

restricted domainD-C

The boundary ∂D

Domain D

Page 43: Persistent Homology and Sensor Networks

New complexes

• We get two communication graphs now, corresponding to rs and rw

• One gives us the “strong” Rips Complex, Rs

• The other gives the “weak” Rips complex Rw

• Note that Rs Rw

Page 44: Persistent Homology and Sensor Networks

(more) New complexes

• We also get a subcomplex based on the nodes that lie within rf of ∂D

• Build this as a subcomplex of Rs

• Call it the (strong) fence subcomplex Fs

rf

Page 45: Persistent Homology and Sensor Networks

What we’d like to see

Conjecture:For a set of nodes X in a domain D Rd satisfying the new assumptions (rc, rs, rw, rf, fence subcomplex etc), the sensor cover

U contains D-C if there exists [] Hd(Rs,Fs) such that ∂ ≠ 0

Page 46: Persistent Homology and Sensor Networks

Why it fails

rf

By comparing to the “weak” Rips complex, we can see which of these cycles are phantom and which are legitimate

• It’s possible to get “phantom” d-cycles in the relative homology that have non-zero boundary

Page 47: Persistent Homology and Sensor Networks

Theorem (De Silva & Ghrist):

For a set of nodes X in a domain D in Rd satisfying the new assumptions (rc, rs, rw, rf, fence subcomplex etc), the sensor cover U contains D-C if the homomorphism

i*: Hd(Rs,Fs) ----> Hd(Rw,Fw) induced by the inclusion i: (Rs,Fs) ----> (Rw,Fw) is nonzero.

Page 48: Persistent Homology and Sensor Networks

The “Squeezing” Theorem

For a set of points X in a domain D Rd

R(X) C(X) R(X)

whenever / ≥ √(2d/(d+1))

•Note that for d=2 this means ≥ 1.15

• This means that if you can enlarge (or shrink) the radius of your Rips complex a little, and the complex doesn’t change, then you actually have a Cech complex

Page 49: Persistent Homology and Sensor Networks

Persistence

Part 2

Page 50: Persistent Homology and Sensor Networks

The Usual Homology

• Have a single topological space, X, and a PID, R

• Get a chain complex

• For k=0, 1, 2, …

compute Hk(X)

• Hk=Zk/Bk

Ck(X) C1(X) C0(X)Ck-1(X) 0∂ ∂∂ ∂ ∂ ∂

Page 51: Persistent Homology and Sensor Networks

How about a filtration of spaces?

X1 X2 X3 … Xn

a b a b

cd

a b

cd

a b

cd

a b

cd

a b

cd

a, b c, d, ab, bc cd, ad ac abc acd

t=0 t=1 t=2 t=3 t=4 t=5

• We restrict to simplical complexes (so we can compute)

Page 52: Persistent Homology and Sensor Networks

Leads to a Persistence Complex

X1 X2 X3 … Xn

C0k C0

1 C00C0

k-10

∂∂∂ ∂ ∂ ∂

C1k C1

1 C10C1

k-10

∂∂∂ ∂ ∂ ∂

Cnk Cn

1 Cn0Cn

k-10

∂∂∂ ∂ ∂ ∂

• Columns are inclusion maps

• Inclusion is a chain map, and so induces a map on homology

Page 53: Persistent Homology and Sensor Networks

Induces a map on homology

a b a b

cd

a b

cd

a b

cd

a b

cd

a b

cda, b c, d, ab, bc cd, ad ac abc acd

t=0 t=1 t=2 t=3 t=4 t=5

• For each dimension k=0,1,2,…

• Consider a generator []Hik

• We may want to consider where in the filtration that generator first appears (created), and when it first becomes bounding (destroyed)

H0k H1

k

i*H2

k

i*Hn

k

i*

Page 54: Persistent Homology and Sensor Networks

Concept: P-interval

a b a b

cd

a b

cd

a b

cd

a b

cd

a b

cda, b c, d, ab, bc cd, ad ac abc acd

t=0 t=1 t=2 t=3 t=4 t=5

• A P-interval is an ordered pair (i, j) with 0≤i<j ≤∞

• Consider a generator []Hik

• We can encode information about the creation and destruction time of [] as a P-interval

•For example [ab+bc-ac] H1 has P-interval (3, 4)

Page 55: Persistent Homology and Sensor Networks

Definition: Persistent Homology

a b a b

cd

a b

cd

a b

cd

a b

cd

a b

cda, b c, d, ab, bc cd, ad ac abc acd

t=0 t=1 t=2 t=3 t=4 t=5

Hki,p =

• Start with the k-cycles at t=i

Zki

• “fast-forward” the boundaries to some future time, i+p

Bki+p

• Intersect the denominator with Zki so it’s well-defined

Zki

Page 56: Persistent Homology and Sensor Networks

Too much work?

a b a b

cd

a b

cd

a b

cd

a b

cd

a b

cda, b c, d, ab, bc cd, ad ac abc acd

t=0 t=1 t=2 t=3 t=4 t=5

• This is interesting, but for an N-step filtration of dimension D, this means we have to compute O(N2D) homology groups!

• And how can we tell what a generator at one time step becomes at the next timestep?

• We need compatible bases for the whole filtration!

Hki,p = Zk

i

Bki+p Zk

i

Page 57: Persistent Homology and Sensor Networks

Definition: Persistence ModuleLet R be a commutative PID

A persistence module is a collection of R-modules, Mi, together with R-module homomorphisms i such that i:Mi ---> Mi+1

M = {Mi, i}

A persistence module M is said to be of finite type if the individual Mi are finitely generated, and N such that n ≥ N i:Mi Mi+1

Page 58: Persistent Homology and Sensor Networks

Correspondence TheoremLet R be a commutative PID, and M = {Mi, i} a persistence module of finite type over R

Define a functor

Where the R-module structure on the Mi is the sum of the individual components, and the action of t is given by

t·(m0, m1, m2, …) = (0, 0(m0), 1(m1), 2(m2), …)

R-persistence modules of finite type

Finitely generated non-negatively graded R[t] modules

i=0

∞(M) = Mi

Proof: “the Artin-Rees theory in commutative algebra”?

Page 59: Persistent Homology and Sensor Networks

Correspondence TheoremLet R be a commutative PID, and M = {Mi, i} a persistence module over R

Define a functor

If R=F is a field, then F[t] is a graded PID and we have a structure theorem for its finitely-generated graded modules

R-persistence modules of finite type

Finitely generated non-negatively graded R[t] modules

i=0

(M) = Mi

i=1

n

= _i F[t] _j F[t]/(tn_j) m

j=1

free part torsion part

Page 60: Persistent Homology and Sensor Networks

Example: Homology of a filtration

The homology groups Hkl (for a fixed k) of a finite filtration {Xl}, along with the maps

induced by inclusions are a persistence module of finite type.

Hk = {Hkl, i*

l}

In the corresponding graded R[t] module M=(Hk), each stage in the filtration corresponds to a particular degree.

a b a b

cd

a b

cd

a b

cd

a b

cd

a b

cda, b c, d, ab, bc cd, ad ac abc acd

t0 t1 t2 t3 t4 t5

The element [ab+bc-ac] H13 has degree 3

But t·[ab+bc-ac] 0 in H14

Page 61: Persistent Homology and Sensor Networks

Visualization: “Barcodes”a b a b

cd

a b

cd

a b

cd

a b

cd

a b

cda, b c, d, ab, bc cd, ad ac abc acd

t0 t1 t2 t3 t4 t5

Page 62: Persistent Homology and Sensor Networks

Computing simplicial homology

The boundary operators of the chain complex are linear operators operating on chain groups which are free R-modules

Therefore they can be represented as matrices relative to some bases.

Ck(X) C1(X) C0(X)Ck-1(X) 0∂ ∂∂ ∂ ∂ ∂

By the standard basis we mean the basis where individual simplices are represented as the unit vectors in Rk

a b

cd

C0 = < a, b, c, d >

= < , , , >1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

C1 = < ab, bc, cd, ad, ac >

= < , , , , >

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

Page 63: Persistent Homology and Sensor Networks

Computing simplicial homology 2

The boundary map

k:Ck ---> Ck-1

is represented by the R-matrix Mk

a b

cd

C0 = < a, b, c, d >

= < , , , >1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

C1 = < ab, bc, cd, ad, ac >

= < , , , , >

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

-1 0 0 -1 -1

1 -1 0 0 0

0 1 -1 0 1

0 0 1 1 0

M1 =

Page 64: Persistent Homology and Sensor Networks

Computing simplicial homology 3

Then Mk can be reduced by elementary operations to a matrix, Mk in Smith Normal Form

a b

cd

-1 0 0 -1 -1

1 -1 0 0 0

0 1 -1 0 1

0 0 1 1 0

M1 =

~

a

b

c

d

ab bc cd ad ac

1 0 0

0 ... 0 0

0 0 r

0 0

Mk = ~b1

...

br

br+1

...

bm

a1 ... ar z1 .... zn-r The i’s that are >1 are the torsion coefficients of Hk-1

z1, ..., zr are a basis for kerMk = Zk

1b1, ..., rbr are a basis for imMk = Bk-1

So between Mk and Mk+1 we have enough information to compute Hk , betti numbers

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 0 0

M1 = ~a

a+b

b+c

c+d

-ab -bc -cd z1 z2

z1 = ab+bc-ac

z2 = ac+cd-ad

Page 65: Persistent Homology and Sensor Networks

Computing persistent homologyTo compute persistent homology over a field, F, do the same thing except work over the ring F[t]

a b a b

cd

a b

cd

a b

cd

a b

cd

a b

cda, b c, d, ab, bc cd, ad ac abc acd

t0 t1 t2 t3 t4 t5

Each simplex is assigned a degree according to when it got added to the complex

For example, deg(a)=0 deg(abc)=4

The boundary operator can’t map across the grading

So for a simplex Ck deg() = deg(∂k)

For example, ∂kac) = t2·c - t3·a

Page 66: Persistent Homology and Sensor Networks

Computing persistent homology 2

a b a b

cd

a b

cd

a b

cd

a b

cd

a b

cda, b c, d, ab, bc cd, ad ac abc acd

t0 t1 t2 t3 t4 t5

For a given dimension, k, there is a single boundary operator, ∂k encoding information for the entire filtration.

Note that basis elements are homogenous.

-t 0 0 -t2 -t3

t -t 0 0 0

0 1 -t 0 t2

0 0 t t 0

M1 =

a

b

c

d

ab bc cd ad ac t 0 0 0 0

-t 1 0 0 0

0 -t t 0 0

0 0 -t 0 0

M1 = ~d

c

b

a

cd bc ab z1 z2

z1 = ad - cd - t·bc - t·ab

z2 = ac - t2·bc - t2·ab

Page 67: Persistent Homology and Sensor Networks

Computing persistent homology 3a b a b

cd

a b

cd

a b

cd

a b

cd

a b

cda, b c, d, ab, bc cd, ad ac abc acd

t0 t1 t2 t3 t4 t5

t 0 0 0 0

-t 1 0 0 0

0 -t t 0 0

0 0 -t 0 0

M1 = ~d

c

b

a

cd bc ab z1 z2

z1 = ad - cd - t·bc - t·ab

z2 = ac - t2·bc - t2·ab

Torsion terms in persistent homology!

A torsion coefficient ti corresponding to a basis element of degree j gives a term in the

persistent homology group: j F[t]/(ti)

Or in other words, a P-interval (j, i+j)

An extra basis element of degree j at the bottom gives a free term:

j F[t]

IOW a P-interval (j,∞)

Page 68: Persistent Homology and Sensor Networks

Applications• When your only tool is persistent homology, every problems

starts to look like a filtered simplicial complex

1. That sensor nets thing

2. Point cloud data

3. Dimension estimation

Page 69: Persistent Homology and Sensor Networks
Page 70: Persistent Homology and Sensor Networks