homological methods for sensor networks

1
homological methods for sensor networks

• robert ghrist
• department of mathematics
• university of illinois
• urbana-champaign, usa

pacm colloquium 2005
joint work with v. de silva pomona
(special thanks a. muhammad gatech, d.
lipsky uiuc)
2
sensors sense-ability
goal sense an environment
one/few expensive, strong, global sensors
today
tomorrow
swarm of cheap, weak, local sensors
problem integrate local sensor data into global
features?
thesis topology is the right tool for local ?
global
3
an analogy
homology converts local combinatorial data into
global, algebraic topological data for spaces
4
what is homology?
simplicial complex X
Hk(X) k0,1,2...
homology converts local combinatorial data into
global, algebraic topological data for spaces
5
the coverage problem
nodes with radially symmetric sensing /
communication
applications
communications
security and surveillance
robotics and navigation
goal solve blanket coverage problems
6
strategies
computational geometry
li et al. huang et al. meguerdichian et al zhang
et al..
advantages
good algorithms rigorous
drawbacks
requires fine sensor data
probabilistic methods
koskinen et al. liu et al. xue et al.
advantages
no coordinates, localization
drawbacks
requires uniform distribution
homological methods...
7
sensor network assumptions
A1 nodes broadcast unique id to anyone within
radius rb
A2 symmetric coverage domains of radius rcgt rb
/v3
A3 compact polygonal domain D in R2
A4 special fence nodes define ?D
8
what the network knows...
there are no assumptions on the geometry of the
domain other than dimension (2)
9
sensors and simplices
classical tool the cech complex of a cover (aka,
nerve)
k-simplices correspond to depth k overlaps
the cech theorem the cech complex has the
homotopy type of the cover
10
how to extract cech data?
unfortunately, the cech complex depends
sensitively upon the exact distances between the
nodes...
what can we discern from the network data?
11
the obvious part...
the communication graph
edges correspond to communication links
computable on hardware level...
...but you dont know the embedding in the plane
12
the not-so-obvious part...
the vietoris-rips complex R
complete the graph to maximal simplicial complex
construction comes from geometric group
theory...
fence subcomplex F corresponding to ?D
13
a homological coverage criterion
theorem under A1-A4, the radius rc discs cover
the domain if there exists a generator of H2(R
,F) with nonvanishing boundary
intuition
a generator looks like a patchwork of
triangles which spans the boundary of the domain
14
beware of fake generators
betti numbers ranks of homology groups are not
sufficient for coverage...
this forms an octahedron in the rips complex
generator for H2 but it has vanishing boundary
15
proof
given a in H2(R ,F) with ?a nonzero
consider map s(R ,F)?(R2,?D) convex hulls of
simplices
use long exact sequences of the pairs
d
H2(R ,F)
H1(F)
s
s
d
H2(R2,?D)
H1(?D)
if p lies in D-s(R), then the left map factors
and thus vanishes
commutativity yields a contradiction
16
good news bad
bad news...
not an if only if statement provides a
certificate
not distributed yet
good news...
good software available
chomp mischaikow al. georgia tech
plex de silva, zomorodian, al. stanford
broadly applicable techniques...
17
generators power conservation
question is the cover redundant?
idea choose a minimal generator a in H2(R ,F)
scholium nodes implicated in generator of H2(R
,F) suffice to cover domain
18
hole detection and repair
question how to fix the holes?
idea choose a generating set ai for H1(R )
where aiNi
theorem expanding rc at the nodes ai of to the
value ½ rb csc (p/Ni) suffices to cover domain
19
domains with arbitrary topology
question what if the domain has holes?
assumption know which fence nodes are outer
theorem coverage implied by a generator of H2(R
,F) with nonzero outer boundary
20
barrier coverage in 3-d
question can you block an intruder?
given tunnel of form D x (-8,8) with fence
nodes fixed at ?D x 0 and all nodes having
(3-d) radially symmetric broadcast and coverage
regions
theorem it is impossible for a curve to stretch
from -8 to 8 avoiding detection if there exists
a in H2(R,F) with ?a?0 .
21
time-dependent pursuit / evasion
question is a mobile network secure?
22
time-dependent pursuit / evasion
question is a mobile network secure?
given a sequence of updates to network graph (not
too coarse keep boundary fixed)
can an evader avoid detection?
construct rips complexes Ri, i1...N
23
time-dependent pursuit / evasion
maximal common subcomplex
R i ?R i1
generated by labeled vertices
build amalgamated complex G(R i)
theorem there are no evaders in the mobile
network if there exists a in H2(G(R i),F) with
?a?0 on F
works even if nodes go off-line / on-line
can formulate a distributed update model
24
weakening the hypothses
these proofs are all quite simple...
the reasons controlled boundary nodes 2-d
can one forego the precise control over fence
nodes?
introduce fence radius rf which determines
fence subcomplex of rips
25
fake boundary cycles
homology cant tell that this is not a cover
but notice what happens if we increase rb
slightly
26
a persistence theorem
consider nodes in Rd and examine rips complex of
radius e
theorem the inclusion iR e?R e factors through
the cech complex Ce whenever and this is the
smallest ratio for which this holds
it follows from functoriality and the cech
theorem that any persistent homology class of
rips complexes yields rigorous conclusions about
the homology of the cover
27
a persistence theorem
consider nodes in Rd and examine rips complex of
radius e
theorem the inclusion iR e?R e factors through
the cech complex Ce whenever and this is the
smallest ratio for which this holds
Ce
28
hypotheses for relative coverage
nodes in compact domain D in Rd
nodes have unique id numbers which they broadcast
nodes can detect signal and distinguish between
strong signal xi - xj rs
weak signal rs lt xi - xj rw
nodes have a covering domain of radius rc
29
hypotheses for relative coverage
nodes can detect the boundary ?D within distance
rf
fence nodes
fence subcomplex of rips
30
hypotheses for relative coverage
assume D not too wrinkled (injectivity radii
bounds)
assume D is not pinched D-C is connected
C collar of radius rf rsv2
31
a persistent homology criterion
otherwise said, coverage is implied by a
persistent relative homology class
32
proof
given a in H2(R s,Fs) nonzero in H2(R w,Fw)
consider map s(R s,Fs)?(R2,E) where ER2-(D-C)
use long exact sequences of the pairs
if is sd nonzero, then proceed as before
otherwise...
33
proof
construct auxiliary complexes at midrange radius
rmrsv5
0
diagram chasing does the trick...
34
synthetic degree computation
the difficulty is computing the degree of ?a
idea set up a network-theoretic transversality
theory
build criteria for regular values of the
projection map from the collar to the boundary
this avoids the dual-radius construction of the
persistent homology criterion
software implementation d. lipsky
35
on the horizon...
distributed coverage criteria
target tracking in coordinate - free networks
target isolation
beacon navigation and mapping
36
summary
homological criterion for coverage
gives a certificate
computable
applies to a wide variety of problems
topology as a toolbox
turning dumb dust into smart dust