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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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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The problem to be solved
Each node has sensors that can cover a circular
region of radius rc
Each node can detect other nodes Within its
broadcast radius rb
rc rb/v(3)
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The problem to be solved
Each node has sensors that can cover a circular
region of radius rc
Each node can detect other nodes Within its
broadcast radius rb
rc rb/v(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
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What we dont have

• Nodes dont 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 UUa, the Cech
Complex C(U) is the simplical complex where each
non-empty intersection of (k1) of the Ua
correspond to a k-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 Re(X) contains a simplex for every
  collection of points that are pairwise within
  distance e
• 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, its 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
rc sensor radius
rb broadcast radius
D domain to be covered
?D boundary of D
Xf fence nodes that lie on dD
U Region covered by the sensors
R Rips complex of the communication graph
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 (rc, rb, fence nodes
etc), the sensor cover Uc contains D if there
exists a ? H2(R,F) such that ?a ? 0
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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
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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 a ? H2(R,F) such that ?a ? 0
But why require ?a ? 0 ??
Why not if and only if ??
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Pitfalls of the Rips complex
Bound was rc rb/v(3) 1/v (3) 0.57
rb
rb
Therefore its possible to have a rectangle that
is completely covered, but not triangulated in
the communication graph
So the conditions of the theorem are sufficient,
but not necessary, to guarantee coverage.
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Pitfalls of the Rips complex
Its 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!
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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 rc
Each node can detect its neighbors via a strong
signal (rs) or a weak signal (rw).
rc rs/v(2) rw rs v(10)
Remember strong lt---gt short weak lt---gt
wlong
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The new situation (cont)
rc rs/v(2) rw rs v(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, where C
x ? D ? x-?D rf rs/v(2)

• The fence-detection hypersurface
• x ? D ? x-?D rf
• Has internal injectivity radius rs/v(2)
• external injectivity radius rs

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The new situation (cont)
Domain D
The fence collar, C
The boundary ?D
rf
restricted domain D-C
S
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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

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(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
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What wed 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 a ? Hd(Rs,Fs) such
that ?a ? 0
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Why it fails

• Its possible to get phantom d-cycles in the
  relative homology that have non-zero boundary

By comparing to the weak Rips complex, we can
see which of these cycles are phantom and which
are legitimate
rf
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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)
----gt Hd(Rw,Fw) induced by the inclusion i
(Rs,Fs) ----gt (Rw,Fw) is nonzero.
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The Squeezing Theorem
For a set of points X in a domain D ? Rd Re?(X) ?
Ce(X) ? Re(X) whenever e/e? v(2d/(d1))

• Note that for d2 this means e 1.15 e?
• This means that if you can enlarge (or shrink)
  the radius of your Rips complex a little, and the
  complex doesnt 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 k0, 1, 2, compute Hk(X)
• HkZk/Bk

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How about a filtration of spaces?
X1 ? X2 ? X3 ? ? Xn

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

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Leads to a Persistence Complex
X1 ? X2 ? X3 ? ? Xn

• 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

• For each dimension k0,1,2,
• 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)

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Concept P-interval

• A P-interval is an ordered pair (i, j) with
  0iltj 8
• Consider a generator ??Hik
• We can encode information about the creation and
  destruction time of ? as a P-interval
• For example abbc-ac ?H1 has P-interval (3, 4)

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Definition Persistent Homology
Hki,p
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Too much work?

• This is interesting, but for an N-step
  filtration of dimension D, this means we have to
  compute O(N2?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!

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Definition Persistence Module
Let R be a commutative PID A persistence module
is a collection of R-modules, Mi, together with
R-module homomorphisms fi such that fiMi ---gt
Mi1 M Mi, fi
A persistence module M is said to be of finite
type if the individual Mi are finitely generated,
and ? N such that n N ? fiMi ? Mi1
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Correspondence Theorem
Let R be a commutative PID, and M Mi, fi a
persistence module of finite type over R Define a
functor a 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, f0(m0), f1(m1), f2(m2), )
Proof the Artin-Rees theory in commutative
algebra?
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Correspondence Theorem
Let R be a commutative PID, and M Mi, fi a
persistence module over R Define a functor
a If RF is a field, then Ft is a graded
PID and we have a structure theorem for its
finitely-generated graded modules
n
m
M ? Sa_i Ft ? ? Sg_j Ft/(tn_j)
i1
j1
free part torsion part
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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, il In the corresponding
graded Rt module Ma(Hk), each stage in the
filtration corresponds to a particular degree.
The element abbc-ac ? H13 has degree 3 But
tabbc-ac ? 0 in H14
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Visualization Barcodes
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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.
By the standard basis we mean the basis where
individual simplices are represented as the unit
vectors in Rk
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Computing simplicial homology 2
The boundary map ?kCk ---gt Ck-1 is represented
by the R-matrix Mk
-1 0 0 -1 -1 1 -1 0 0 0 0
1 -1 0 1 0 0 1 1 0
M1
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Computing simplicial homology 3
ab bc cd ad ac
-1 0 0 -1 -1 1 -1 0 0 0 0
1 -1 0 1 0 0 1 1 0
a b c d
M1

Then Mk can be reduced by elementary operations
to a matrix, Mk in Smith Normal Form
The sis that are gt1 are the torsion coefficients
of Hk-1 z1, ..., zr are a basis for kerMk
Zk s1b1, ..., srbr are a basis for imMk Bk-1
So between Mk and Mk1 we have enough information
to compute Hk , betti numbers
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Computing persistent homology
To compute persistent homology over a field, F,
do the same thing except work over the ring Ft
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 cant map across the
grading So for a simplex a?Ck deg(a)
deg(?ka) For example, ?k(ac) t2c - t3a
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Computing persistent homology 2
For a given dimension, k, there is a single
boundary operator, ?k , encoding information for
the entire filtration. Note that basis elements
are homogenous.
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Computing persistent homology 3
An extra basis element of degree j at the bottom
gives a free term Sj Ft IOW a P-interval (j,8)
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 Sj Ft/(ti) Or in other
words, a P-interval (j, ij)
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Applications

• When your only tool is persistent homology, every
  problems starts to look like a filtered
  simplicial complex
• That sensor nets thing
• Point cloud data
• Dimension estimation

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