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

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Outline A word about sensor nets Basic coverage criterion Better coverage criterion using persistence Introduce Persistent Homology Correspondence Theorem Computing the groups! Other Applications

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A Word About Sensor Nets

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August 29, 2005 Hurricane Katrina hits New Orleans

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Power and Communications Knocked Out

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Broken Levees

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City Flooded

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Inaccessible from the ground

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Law Enforcement

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Rescue Workers

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Replace live turkey with a parachute

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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!?

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Other uses of sensor networks Environmental monitoring Security systems Battlefield monitoring and communications Large mechanical systems Find Sarah Connor

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Hole in sensor coverage area Sarah Connor escapes!

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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

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Basic Coverage Criterion Part 1.2

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rcrc rbrb The problem to be solved: Each node has sensors that can cover a circular region of radius r c Each node can detect other nodes Within its broadcast radius r b r c ≥ r b /√(3)

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The problem to be solved: Each node has sensors that can cover a circular region of radius r c Each node can detect other nodes Within its broadcast radius r b r c ≥ r b /√(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 r b apart

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What we don’t have: Nodes don’t know their absolute or relative positions All we get is the connectivity graph

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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

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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 R rb ( 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

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Picture of a Rips Complex

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Recap: X = { set of nodes } r c = sensor radius r b = broadcast radius D = domain to be covered ∂ D = boundary of D X f = { fence nodes that lie on D } R = Rips complex of the communication graph U = Region covered by the sensors F = Fence subcomplex R

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Theorem (De Silva & Ghrist): For a set of nodes X in a planar domain D satisfying the assumptions (r c, r b, fence nodes etc), the sensor cover U c contains D if there exists [ ] H 2 ( R, F ) such that ∂ ≠ 0

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What about a generator of H 2 ( R, F )? A generator will look like some linear combination of 2-simplices i.e. Some triangulation of the domain D

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Theorem (De Silva & Ghrist): For a set of nodes X in a planar domain D satisfying the assumptions (r c, r b, fence nodes etc), the sensor cover U c contains D if there exists [ ] H 2 ( R, F ) such that ∂ ≠ 0 But why require ∂ ≠ 0 ?? Why not “if and only if” ??

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Pitfalls of the Rips complex Bound was r c ≥ r b /√(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 rbrb So the conditions of the theorem are sufficient, but not necessary, to guarantee coverage.

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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 H 2, but its boundary is zero!

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Better coverage criterion using persistence Part 1.3

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Eliminating the fence subcomplex The assumption of the nice fence sub-complex is unrealistic Can we replace it with some other assumptions?

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The new situation: Each node has sensors that can cover a circular region of radius r c Each node can detect its neighbors via a strong signal (r s ) or a weak signal (r w ). r c ≥ r s /√(2) r w ≥ r s √(10) rcrc rsrs rwrw Remember: strong “short” weak “ w long”

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The new situation (cont…): r c ≥ r s /√(2) r w ≥ r s √(10) Nodes lie in a compact connected domain D in R d Nodes can detect the presence of ∂D within distance r f The restricted domain D-C is connected, where C = {x D ||x- ∂D || ≤ r f + r s /√(2) The fence-detection hypersurface = {x D ||x- ∂D || = r f } Has internal injectivity radius ≥ r s /√(2) external injectivity radius ≥ r s

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The new situation (cont…): rfrf The fence “collar”, C restricted domain D - C The boundary ∂D Domain D

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New complexes We get two communication graphs now, corresponding to r s and r w One gives us the “strong” Rips Complex, R s The other gives the “weak” Rips complex R w Note that R s R w

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(more) New complexes We also get a subcomplex based on the nodes that lie within r f of ∂D Build this as a subcomplex of R s Call it the (strong) fence subcomplex F s rfrf

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What we’d like to see Conjecture: For a set of nodes X in a domain D R d satisfying the new assumptions (r c, r s, r w, r f, fence subcomplex etc), the sensor cover U contains D - C if there exists [ ] H d ( R s, F s ) such that ∂ ≠ 0

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Why it fails rfrf 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

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Theorem (De Silva & Ghrist): For a set of nodes X in a domain D in R d satisfying the new assumptions (r c, r s, r w, r f, fence subcomplex etc), the sensor cover U contains D - C if the homomorphism i * : H d ( R s, F s ) ----> H d ( R w, F w ) induced by the inclusion i: ( R s, F s ) ----> ( R w, F w ) is nonzero.

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The “Squeezing” Theorem For a set of points X in a domain D R d 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

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Persistence Part 2

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The Usual Homology Have a single topological space, X, and a PID, R Get a chain complex For k=0, 1, 2, … compute H k (X) H k =Z k /B k C k (X)C 1 (X)C 0 (X)C k-1 (X)0 ∂ ∂∂∂∂∂

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How about a filtration of spaces? X 1 X 2 X 3 … X n a b a b cd a b cd a b cd a b cd a b cd a, bc, d, ab, bccd, adacabcacd t=0t=1t=2t=3t=4t=5 We restrict to simplical complexes (so we can compute)

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Leads to a Persistence Complex X 1 X 2 X 3 … X n C0kC0k C01C01 C00C00 C 0 k-1 0 ∂ ∂∂∂∂∂ C1kC1k C11C11 C10C10 C 1 k-1 0 ∂ ∂∂∂∂∂ CnkCnk Cn1Cn1 Cn0Cn0 C n k-1 0 ∂ ∂∂∂∂∂ Columns are inclusion maps Inclusion is a chain map, and so induces a map on homology

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Induces a map on homology a b a b cd a b cd a b cd a b cd a b cd a, bc, d, ab, bccd, adacabcacd t=0t=1t=2t=3t=4t=5 For each dimension k=0,1,2,… Consider a generator [ ] H i k We may want to consider where in the filtration that generator first appears (created), and when it first becomes bounding (destroyed) H0kH0k H1kH1k i*i* H2kH2k i*i* HnkHnk i*i*

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Concept: P -interval a b a b cd a b cd a b cd a b cd a b cd a, bc, d, ab, bccd, adacabcacd t=0t=1t=2t=3t=4t=5 A P -interval is an ordered pair (i, j) with 0≤i

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Definition: Persistent Homology a b a b cd a b cd a b cd a b cd a b cd a, bc, d, ab, bccd, adacabcacd t=0t=1t=2t=3t=4t=5 H k i,p = Start with the k-cycles at t=i ZkiZki “fast-forward” the boundaries to some future time, i+p B k i+p Intersect the denominator with Z k i so it’s well-defined Zki Zki

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Too much work? a b a b cd a b cd a b cd a b cd a b cd a, bc, d, ab, bccd, adacabcacd t=0t=1t=2t=3t=4t=5 This is interesting, but for an N-step filtration of dimension D, this means we have to compute O(N 2 D) 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! H k i,p = ZkiZki B k i+p Zki Zki

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Definition: Persistence Module Let R be a commutative PID A persistence module is a collection of R-modules, M i, together with R-module homomorphisms i such that i :M i ---> M i+1 M = {M i, i } A persistence module M is said to be of finite type if the individual M i are finitely generated, and N such that n ≥ N i :M i M i+1

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Correspondence Theorem Let R be a commutative PID, and M = {M i, i } a persistence module of finite type over R Define a functor Where the R-module structure on the M i is the sum of the individual components, and the action of t is given by t·(m 0, m 1, m 2, …) = (0, 0 (m 0 ), 1 (m 1 ), 2 (m 2 ), …) R-persistence modules of finite type Finitely generated non- negatively graded R[t] modules i=0 ∞ ( M ) = M i Proof: “the Artin-Rees theory in commutative algebra”?

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Correspondence Theorem Let R be a commutative PID, and M = {M i, 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 ) = M i i=1 n = _i F[t] _j F[t]/(t n_j ) m j=1 free parttorsion part

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Example: Homology of a filtration The homology groups H k l (for a fixed k) of a finite filtration {X l }, along with the maps induced by inclusions are a persistence module of finite type. H k = {H k l, i * l } In the corresponding graded R[t] module M= ( H k ), 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 cd a, bc, d, ab, bccd, adacabcacd t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 The element [ab+bc-ac] H 1 3 has degree 3 But t·[ab+bc-ac] 0 in H 1 4

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Visualization: “Barcodes” a b a b cd a b cd a b cd a b cd a b cd a, bc, d, ab, bccd, adacabcacd t0t0 t1t1 t2t2 t3t3 t4t4 t5t5

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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. C k (X)C 1 (X)C 0 (X)C k-1 (X)0 ∂ ∂∂∂∂∂ By the standard basis we mean the basis where individual simplices are represented as the unit vectors in R k a b cd C 0 = = C 1 = =

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Computing simplicial homology 2 The boundary map k :C k ---> C k-1 is represented by the R-matrix M k a b cd C 0 = = C 1 = = M 1 =

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Computing simplicial homology 3 Then M k can be reduced by elementary operations to a matrix, M k in Smith Normal Form a b cd M 1 = ~ abcdabcd ab bc cd ad ac r 0 0 M k = ~ b 1... b r b r+1... b m a 1... a r z z n-r The i ’s that are >1 are the torsion coefficients of H k-1 z 1,..., z r are a basis for kerM k = Z k 1 b 1,..., r b r are a basis for imM k = B k-1 So between M k and M k+1 we have enough information to compute H k, betti numbers M 1 = ~ a a+b b+c c+d -ab -bc -cd z1 z2 z1 = ab+bc-ac z2 = ac+cd-ad

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Computing persistent homology To 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 cd a, bc, d, ab, bccd, adacabcacd t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 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 C k deg( ) = deg(∂ k ) For example, ∂ k ac) = t 2 ·c - t 3 ·a

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Computing persistent homology 2 a b a b cd a b cd a b cd a b cd a b cd a, bc, d, ab, bccd, adacabcacd t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 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 -t 2 -t 3 t -t t 0 t t t 0 M 1 = abcdabcd ab bc cd ad ac t t t t t 0 0 M 1 = ~ dcbadcba cd bc ab z1 z2 z1 = ad - cd - t·bc - t·ab z2 = ac - t 2 ·bc - t 2 ·ab

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Computing persistent homology 3 a b a b cd a b cd a b cd a b cd a b cd a, bc, d, ab, bccd, adacabcacd t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t t t t t 0 0 M 1 = ~ dcbadcba cd bc ab z1 z2 z1 = ad - cd - t·bc - t·ab z2 = ac - t 2 ·bc - t 2 ·ab Torsion terms in persistent homology! A torsion coefficient t i corresponding to a basis element of degree j gives a term in the persistent homology group: j F[t]/(t i ) 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,∞)

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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

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