Using Persistent Homology To Recover Spatial Information From Encounter Traces

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Using Persistent Homology To Recover Spatial
Information From Encounter Traces

• Brenton Walker
• University of Maryland

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Encounter Traces
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Mobile Ad-Hoc Networks

• Wireless network
• No infrastructure!

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Delay-Tolerant Networks

• No end-to-end connectivity
• Messages are handed off and (maybe) delivered
  later

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Mobility of Nodes Is Important

• But its hard to understand
• Instead people like to study encounter patterns

???
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Encounter Pattern Experiments

• Haggle Project (Intel Cambridge) for example
• Record encounters between motes

Source Intel Mote Overview http//www.intel.com/r
esearch/exploratory/motes.htm
Source Haggle Project (?) http//www.haggleprojec
t.org/
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What Can You Do With Data Like These?
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Spatial Reconstruction

• In a connected network it would not be
  unreasonable to try and reconstruct the geometry
  of the network.

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There is not much geometric information here!
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Persistent Homology
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Topological invariant Betti numbers
?1 1
?1 2
?1 0 ?2 1
?1 2 ?2 1
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Filtration Example
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Filtration Example
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Filtration Example
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Filtration Example
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Filtration Example
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Filtration Example
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Filtration Example
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Filtration Example
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Noisier Data
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The idea of Persistence

• Topological features that appear and disappear
  quickly are not interesting
• Topological features that persist across the
  filtration are interesting

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The idea of Persistence

• Topological features that appear and disappear
  quickly are not interesting
• Topological features that persist across the
  filtration are interesting

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The idea of Persistence

• Visualize this in a persistence diagram or
  barcode

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The Math of Persistence

• Computing Persistent Homology Zamorodian and
  Carlsson

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How to process an encounter trace?

• We want to build a topological object from the
  data.
• We need data points with a metric.

Topological Object
Data
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Consider Guys Traveling on A Graph

• A line

A
C
B
A cant get to C without encountering B
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Consider Guys Traveling on A Graph

• A loop

B
A
C
Now A can get to C without encountering B
The Point The topology of the space has an
effect on what encounter patterns are possible.
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What Can You Do With Data Like These?
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Make Some Deductions
node A encounters node B at time t1, coordinates
(x1,y1)
node B encounters node C at time t2, coordinates
(x2,y2)
Conclusion
If t1- t2 is small, then the points where the
encounters took place must be close t1-t2 lt T
gt dist((x1,y1),(x2,y2)) lt D
A
C
B
B
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This Gives Us A Weighted Graph

• (1st degree) Encounter Complex
• Rule
• Each encounter ei is a vertex (0-simplex)
• If ei and ej have a node in common and ti-tj
• Then connect ei and ej with with weight ti-tj

A
B
C
D
e1
e1
t1-t2
t1-t3
e2
e2
e3
t
e3
t2-t4
e4
t3-t4
e4
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The space the data lie in

• Two encounters can be close in space, but distant
  in this graph-based metric
• Picture the data points lying in the physical
  space crossed with time X?R

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How to triangulate these data?

• We have data points with a metric.
• We want to build a topological object from the
  data.
• The object should reveal something about the
  encounter data

Topological Object
Data
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Witness Complex

• Vin deSilva A Weak Characterization of the
  Delaunay Triangulation
• Like a Delaunay triangulation for discrete spaces
• Except it preserves the topology of the data

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Experiments and Results
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Example Linear graph
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Example Linear graph
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Example Loop
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Example Loop
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Example Figure-8
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Success!

• In this simplified model we can distinguish
  between a line, a loop and a multi-loop.
• Problems
• What are the key mobility assumptions?
• How many guys need to be on the graph?
• What has its mobility restricted to unknown
  graphs?

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Naked Mole Rats
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Naked Mole Rats

• Rufus from Kim Possible
• 25 of the Errol Morris film Fast, Cheap Out of
  Control
• Naked mole rats are going to be the next mercats
• Clearly there is an urgent need to study their
  burrowing habits

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What if the graph changes?

• Could we detect emergence and disappearance of a
  loop?
• An expanding and contracting loop would manifest
  itself as non-trivial H2(C)

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Expanding contracting loop
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What About Spaces That Arent Graphs?

• Somewhat made up for if the motes have larger
  radio range
• Its a harder problem
• The guys can just go around each other
• Can we do anything?

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Room Boundary Cross-section

• Look at the room when everyone is at the
  perimeter
• It should look like a 1-cycle

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Bounded rectangle vs torus

• Simulations and mathematical models of wireless
  networks are usually done on a torus
• The encounter patterns are detectably different

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Haggle Day 1
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Haggle Day 2
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Haggle Day 3
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Questions?
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Localized Homology and the Problem of Missing Data
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Idea Is there some other way to tell the
difference between these loops and these loops?
Computation gets more difficult
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The Math of Persistence

• Computing Persistent Homology Zamorodian and
  Carlsson

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Filtration of spaces
X1 ? X2 ? X3 ? ? Xn
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Persistence Complex

• Columns are inclusion maps
• Inclusion is a chain map, and so induces a map on
  homology

X0 ? X1 ? X2 ? ? Xn
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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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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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The Result

• We can compute the homology of the entire
  filtration at once by computing it over a
  polynomial ring over a field
• There is a fast algorithm for computing it
• Faster than your typical Smith Normal Form
  algorithm

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Simplicial Complex

• Simplicial complexes are built from simplices
• (k1) points form a k-simplex
• This can be a purely combinatorial construction

k 1 0-simplex
a
k 2 1-simplex
a b
k 3 2-simplex
a b c
k 4 3-simplex
a b c d
k 5 4-simplex
etc
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Example 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
• Every complete k-subgraph of the communication
  graph becomes a simplex in the Rips Complex

?
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Better Witness Complex

• Invented and studied by Vin deSilva
• Like a Delaunay triangulation for discrete spaces
• Much more efficient than Rips complex

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Better Witness Complex

• Step1 Choose landmarks
• Choose the first landmark randomly
• Choose the next landmark from the remaining data
  points so as to maximize the minimum distance to
  all previously chosen landmarks
• Step2 Add simplices
• 0-simplices are the landmarks
• A collection of landmarks forms a simplex a0 a1
  ... ak if there is a data point, x, with a0, a1,
  ..., ak among its k nearest landmarks.
• Generalization Add some wiggle room to step 2
• Introduce a threshold, T 0
• For data point xi, let mi be the distance to its
  2nd closest landmark
• Include a b if max(d(a,xi), d(b,xi)) mi T

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Naked Mole Rat Experiment

• Fit a colony of naked mole rats with motes that
  track their encounters
• Collect the motes and analyze the data using
  these methods
• Profit!!

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The Disco

• A disco is crowded and chaotic

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The Disco Movie Phenomenon

• A disco is crowded and chaotic
• But when John Travolta starts dancing everybody
  gets out of the way

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Room Area Cross-section

• Look at the data before the crowd separates
• It should produce a contractible blob (eventually)

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Make Some Deductions
e1
e2
node A encounter node B at time t1, coordinates
(x1,y1)
node B encounter node C at time t2, coordinates
(x2,y2)
Conclusion
If t1- t2 is small, then the points where the
encounters took place must be close t1-t2 lt T
gt dist((x1,y1),(x2,y2)) lt D
A
C
B
B
This is transitive, up to a point
e3
Conclusion
node C encounter node D at time t3, coordinates
(x3,y3)
If t1- t2t2- t3 is small, then the points
where the encounters e1 and e3 took place must be
close t1-t2t2-t3 lt T gt
dist((x1,y1),(x3,y3)) lt D
D
C