algebraic topology for sensor networks

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algebraic topology for sensor networks

• robert ghrist

department of mathematics coordinated
science laboratory university of illinois
urbana-champaign, usa
msri sept 2006
joint work with v. de silva mathematics, pomona
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what is algebraic topology?
topological object
algebraic object
equivalent inputs (e.g., same homotopy type)
yield equivalent outputs (e.g., isomorphic
groups)
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example euler characteristic
simplicial complex
integer
? vertices - edges faces (2-d)
alternating sum of k-simplices
euler characteristic converts a local count of
simplices into a global invariant of the union
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example homology
simplicial complex
Hk k0,1,2... vector spaces (abelian groups)
homology converts local connectivity of simplices
into global invariants of the union
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what does homology mean?
chain complex
build vector spaces Ck with k-simplices as bases
C0 C1 C2 C3
...
boundary map
Hk ker ? / im ?

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what does homology mean?
Hk ker ? / im ?
cycles modulo boundaries
ßi rank(Hk) of independent holes that
k-dimensional probes can detect
ß0 1
ß1 4
ß2 1
ßkgt2 0
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how do you compute homology?
exact sequences
well-stocked toolbox to perform an inductive
computation of homology...
... ? H2(A) ? H2(X) ? H2(X,A) ? H1(A) ? H1(X) ?
H1(X,A) ? H0(A) ? H0(X)
H2(X/A)
H1(X/A)
relative homology
exactness
two-out-of-three aint bad
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sensors sense-ability
goal sense an environment
one/few expensive, strong, global sensors
today
swarms of cheap, weak, local sensors
tomorrow
problem integrate local sensor data into global
features?
thesis homology is the right tool for local ?
global
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an analogy
sensors and simplices each have knowledge only of
their identities and of their local
connectivity...
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the coverage problem
nodes with radially symmetric sensing /
communication
applications
communications
security and surveillance
robotics and navigation
environmental monitoring
goal solve blanket coverage problems
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strategies placement problem
computational geometry
minimal placement ? art gallery problems
everything tends to be np-hard in this business
probabilistic methods
use mobile nodes which sweep domain
try to set up ergodic dynamics...
assumes scarce, controllable nodes...
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strategies interrogating a network
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 control over distribution
homological methods...
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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
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what the network knows...
no assumptions on domain geometry no distances or
angles
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the obvious part...
the communication graph
edges correspond to communication links
computable on hardware level...
...but you dont know the embedding in the plane
and realizing an embedding is np-hard
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the not-as-obvious part...
the (vietoris-) rips complex R
complete the graph to maximal simplicial complex
abstract complex (embedding sold separately)
this allows algebraic topology to usurp network
topology...
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of series and simplices...
given node ids, local communication links
add algebra (chain complex) and topology
(homology)
C0 ? C1 ? C2 ? C3 ? ...
nodes
pairs
triples
quads
algebraic topology converts higher-order network
connectivity into geometric structure...
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a homological coverage criterion
theorem under A1-A4, the radius rc discs cover
the domain if there exists a in H2(R ,F) with
?a?0
intuition
such a generator spans the boundary of the domain
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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
?
H2(R ,F)
H1(F)
0
s
s
?
H2(R2,?D)
H1(?D)
if p lies in D-s(R), then the left map factors
and vanishes
commutativity yields a contradiction
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beware of fake generators
betti numbers are not sufficient for coverage...
this forms an octahedron in the rips complex
generator for H2(R ) but it is not a true
relative class (it has vanishing boundary)
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good news bad
bad news...
not an if only if statement provides a
certificate
not distributed yet
good news...
good software available
chomp
plex
broadly applicable techniques...
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example
computing a generator...
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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
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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
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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
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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.
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time-dependent pursuit / evasion
question is a mobile network secure?
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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
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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
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questions
1. how restrictive is this homological criterion?
2. what is the best way to distribute this
computation?
3. what are the appropriate signal protocols?
4. how can one incorporate uncertainty?
references http//www.math.uiuc.edu/ghrist/
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the second half of this talk...
1. cech!
2. arbitrary dimensions!
3. persistent homology!
4. diagram chasing!
5. software demo!
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recall homological coverage criterion
theorem under A1-A4, the radius rc discs cover
the domain if there exists a in H2(R ,F) with
?a?0
minimal coverage
hole detection repair
barrier coverage
evasion in mobile sweeps
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outline
1. why didnt you just use cech?
2. gee, your proofs were so easy...
3. ok, well, what else can you do?
4. and i suppose youre going to sermonize too?
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oh, the nerve...
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
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how to extract cech data?
unfortunately, the cech complex depends
sensitively upon the exact distances between the
nodes...
the network data alone will never suffice...
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weakening the hypotheses
yesterdays proofs were 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
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fake boundary cycles
homology cant tell that this is not a cover
but notice what happens if we increase rb
slightly
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a persistence theorem
consider nodes in Rd and examine rips complex of
radius e
theorem de silva 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 the persistent homology of rips
complexes yields rigorous conclusions about the
homology of the cover
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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
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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
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hypotheses for relative coverage
nodes can detect the boundary ?D within distance
rf
fence nodes
fence subcomplex of rips
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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
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a persistent homology criterion
otherwise said, coverage is implied by a
persistent relative top-dimensional
homology class
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proof
given a in Hd(R s,Fs) nonzero in Hd(R w,Fw)
consider map s(R s,Fs)?(Rd,E) where ERd-(D-C)
use long exact sequences of the pairs
if is s? nonzero, then proceed as before
otherwise...
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proof
construct auxiliary complexes at midrange radius
rmrsv5
0
diagram chasing does the trick...
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some open questions...
how does one deal with constraints?
how does one deal with asymmetry?
how does one deal with stochasticity?
beyond coverage...
there are numerous problems in sensor networks
for which the perspectives (if not the actual
tools) of topology are efficacious...
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two simple problems in planar topology...
1. separation given a point and a loop, does the
loop enclose it?
2. isolation given a point, find a loop that
encloses it.
these are both simple... but what about a
coordinate-free network version?
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networks, nodes, and coordination
assume nodes in euclidean plane
consider unit disc graph edges correspond to
nodes within unit distance
nodes ? sensors, robots
edges ? communication, coverage, visibility
graph ? summary of proximity data for the entire
system
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whats easy? whats difficult?
if you are given only the abstract unit disc
graph of the network, can you solve the
isolation / separation problems?
in general no
finding some embedding of a general unit disc
graph in the plane is np-hard breu, kirkpatrick
many others...
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...and why do i care about all this?
environmental monitoring
security defense
separation
is an (rfid-tagged) animal within a given area?
isolation
an alarm goes off at node x find a collection
of sensors that are guaranteed to surround x for
further monitoring
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whats the minimal sensing required?
assume that you are given the proximity graph
plus some additional (local) geometric data how
hard is it to solve these (global) problems?
local coordinates of nodes everything is
trivial
angles between the nodes not trivial, but
doable, especially with a dense collection of
nodes...
cf. work of fang, gao, guibas
we will weaken this to use only the order in
which neighbors are arranged...
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formal assumptions
assumption (P) planar
nodes with unique labels represent points in the
euclidean plane
assumption (N) network
the unit disc graph with nodes as vertices and
edges based on unit distance in the plane is
connected and has sufficient diameter
assumption (O) cyclic ordering
each vertex is decorated with the clockwise
cyclic ordering of its immediate neighbors
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what is cyclic ordering of neighbors?
this is weaker than the order type of
goodman-pollack
reasonable data from a primitive omnidirectional
camera
assume (for the moment) no ambiguity in order
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separation intuition
if this were a problem of smooth topology...
...you would then count the number of transverse
intersections mod 2
but not in the network setting --- cannot
guarantee transverse crossings
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separation intuition
adopt a complementary strategy
choose a node guaranteed to lie outside image of
the cycle in the plane
choose paths from these two nodes to the cycle
use cyclic ordering to determine if paths are on
same side of the cycle
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separation details
1. its necessary to assume that the cycle L is
chord-free so that its image in the
euclidean plane is a non-intersecting curve
2. its necessary to assume that x is not within
1-hop of L
3. to find y guaranteed to lie outside of the
cycles image it suffices to go 2L2 / p2
hops away from L (where L of nodes in
cycle)
(why, yes, that is an isoperimetric inequality
thank you very much)
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separation details
4. to determine whether two paths approach L from
the same side, first, orient the cycle
(arbitrarily) and then see how each path
approaches
the ugly way enumerate all approaches and
classify by ordering data
the nice way compute a simple topological index
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separation details
define ordering index Inda(bc,d) -1 (if b c
d at a) 1 (if b d c at a)
-1 cw 1 ccw
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isolation intuition
if this were a problem of smooth topology...
choose a path from the node to infinity and
divide neighborhood into left right sides
find a path which connects the left side to
the right side
complete this to a loop
in the discrete setting...that works!
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so, what have we done?
lipsky, poduri
given any network that satisfies (N), (P), (O)
theorem given any chord-free cycle L and any
node x more than 1-hop away from L, this is a
complete algorithm for determining whether or
not x is surrounded by the image of L in the
plane
theorem given any node x and any integers Rmax gt
Rmin gt 1, this is a complete algorithm to
determine whether there is a cycle within the
Rmax-hop neighborhood of x (but not within the
Rmin-hop neighborhood) whose image surrounds x
and to construct such a cycle when it exists
what if we want to weaken some of the
assumptions?
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dealing with angular uncertainty
coarse sensing demands robustness
goal exploit topological concepts to permit
outrageous slopTM
almost-true theorem the separation problem can
be solved with angular uncertainty up to
p/3
the best way to deal with uncertainty?
ignore assumption (O) altogether. . .
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tri-path algorithm intuition
find three mutually avoiding paths to infinity
this step may fail if the graph is too
sparse...
find a three new paths which connect the three
old ones cyclically
connect pairwise such a loop must surround the
initial node
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concluding scientific postscript
exploiting topological ideas
theres more to computational homology that
computing homology
engineers need excision, exactness,
mayer-vietoris, obstructions,...
what do you do in the zone between sparse dense?
minimality and sensing
what is the minimal sensing capability required
to solve ltXgt?
topologists can contribute to miniaturization /
deployment
the parting quote
topology for networks is more than network
topology