Evasion Paths in Mobile Sensor Networks
Henry Adams and Gunnar Carlsson, IMA and Stanford University
The Evasion Problem
Zigzag Persistence
Suppose ball-shaped sensors wander in a bounded
d
and contractible domain D ⊂ R over time interval
I = [0, 1]. Assume immobile sensors cover the boundary ∂D. The sensors don’t know their locations but do
measure their time-varying C̆ech complex (the nerve of
the sensor balls). An evasion path exists if it is possible for a wandering intruder to avoid the sensors. Using
only the time-varying C̆ech complex, can we determine
if an evasion path exists?
Theorem [Ada13]: If there is no full-length interval in the (d − 1)-dimensional zigzag persistent
homology of the covered region, then no evasion path exists.
What Minimal Sensing Capabilities
Might We Add?
Theorem [Ada13]: In a planar sensor network
that remains connected, the time-varying alpha
complex and the cyclic orderings of the edges about
each sensor determine if an evasion path exists.
4CechRips.nb
CechRips.nb3
Cech simplicial complex
Cech simplicial complex
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Filtration parameter
t
Open Question: Do the time-varying C̆ech complex
and the cyclic orderings of the edges about each sensor
also suffice?
Filtration parameter
0.186
t
0.186
What is the Space of Evasion Paths?
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What information do we need about covered region
X and its fibrewise embedding in spacetime D × I to
c
describe the space of evasion paths (sections I → X )?
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CechRips.nb
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Cech simplicial complex
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Filtration parameter
Sensor balls
t
0.186
Connectivity graph
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The zigzag persistence barcodes on the right are incorrect. Why?
This criterion allows for streaming computation.
C̆ech complex
Dependence on the Embedding
Theorem of de Silva and Ghrist
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a.nb
Rips data1, t, 0, 1
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Rips!data3y, t, 0, 1"
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Rips!data5y, t, 0, 1"
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Rips!data6, t, 0, 1"
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Rips data7, t, 0, 1
Rips simplicial complex
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Filtration parameter
t0.358
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References
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a.nb
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This theorem is not sharp. Can you see why?
Rips data2, t, 0, 1
Rips simplicial complex
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Rips!data1, t, 0, 1"
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Rips data2, t, 0, 1
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Rips!data3n, t, 0, 1"
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a.nb
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Rips!data4n, t, 0, 1"
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a.nb
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Rips!data6, t, 0, 1"
a.nb
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a.nb
Rips simplicial complex
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Filtration parameter
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Filtration parameter
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An evasion path exists.
t0.358
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Filtration parameter
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t0.358
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No evasion path exists because the intruder cannot travel backwards in time.
a.nb
Filtration parameter
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Rips data7, t, 0, 1
Rips simplicial complex
I
I
a.nb
Rips simplicial complex
t
No evasion
path exists.
a.nb
Rips simplicial complex
Filtration parameter
D
a.nb 107
Rips simplicial complex
Filtration parameter
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D
a.nb 8 5 a.nb
Rips simplicial complex
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A relative
2-cycle.
a.nb
Neither the time-varying C̆ech complex nor the fibrewise homotopy type of the covered region determine
if an evasion path exists. Consider the two sensor networks below: their time-varying C̆ech complexes are
equal and their covered regions are fibrewise homotopy equivalent, but the top network contains an evasion
path while the bottom one does not.
In[422]:=
Theorem [dG06]: If there is a relative 2-cycle
in the region of spacetime covered by sensors with
boundary wrapping around ∂D ×I, then no evasion
path exists.
6
We apply a homotopy spectral sequence for function
complexes between diagrams of spaces [DZ87], which
in this setting has input depending on unstable invariants of the uncovered region, and which converges to
information about the homotopy groups of the space
of evasion paths. It remains to obtain these unstable
invariants from embedding invariants of the covered
region, and one idea is to use the tools of embedding
calculus in a fibrewise setting.
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[Ada13] Henry Adams, Evasion paths in mobile sensor
networks, PhD Thesis, Stanford University, 2013.
arXiv:1308.3536.
[Cd10] Gunnar Carlsson and Vin de Silva, Zigzag persistence,
Foundations of Computational Mathematics 10 (2010),
367–405.
[dG06] Vin de Silva and Robert Ghrist, Coordinate-free
coverage in sensor networks with controlled
boundaries via homology, International Journal of
Robotics Research 25 (2006), 1205–1222.
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[DZ87] Emmanuel Dror Farjoun and Alexander Zabrodsky, The
homotopy spectral sequence for equivariant function
complexes, Algebraic Topology Barcelona, Lecture
Notes in Mathematics, vol. 1298, Springer, 1987,
pp. 54–81.