robert ghrist professor of mathematics & electrical/systems engineering the university of pennsylvania topological methods for networks infodynets kickoff : sept 2009
tools for applied mathematics…
differential equations
linear algebra
numerical analysis
novel challenges necessitate novel mathematics
hardware improves...
topology
topological methods for networks…
homological coverage
sensors and simplices each have knowledge only of their identities and of their local connectivity... sensorssimplices homological coverage
the flag complex of a network is the maximal simplicial completion given node id’s, local communication links count nodes & cancel via signal connectivity C 0 ← C 1 ← C 2 ← C 3 ←... [nodes][pairs] [triples][quads] homology converts higher-order network connectivity into global structure... [1-d network] [flag complex] [environment] [H 1 generator]...without coordinates; density assumptions; uniform distributions, etc. networks & complexes
1. compact polygonal domain D in R 2 2. nodes broadcast unique id’s to neighbors 3. coverage regions of a 2-simplex of connected nodes contain the convex hull 4. dedicated fence cycle defines ∂D F coverage assumptions
Theorem [DG]: under above assumptions, the sensor network covers the domain without gaps if there exists [α] in H 2 ( R, F ) with ∂α≠0 F H 2 ( R, F )H1(F)H1(F) H 2 ( R 2,∂D )H 1 ( ∂D ) H 2 ( R 2 -p,∂D ) ∂*∂* ∂*∂* σ * ≈σ*σ* =0 proof: build a commutative diagram of homology groups map σ:( R, F )→(R 2,∂D) convex hulls of simplices if p lies in D-σ( R ), then the left passes through zero commutativity of diagram yields a contradiction intuition: [α] “triangulates” the domain with covered simplices coverage criterion
Not an if & only if statement: provides a certificate The relative condition really is necessary coverage remarks
power conservation via minimal homology generators hole detection & repair via H 1 basis computation distributed (gossip) algorithms for homology computation pursuit/evasion results for time-dependent nodes current results
idea: choose a minimal generator [α] in H 2 ( R, F ) Corollary: [DG] nodes implicated in generator of H 2 ( R, F ) suffice to cover domain question: is the cover redundant? idea: choose a generating set {[α i ]} for H 1 ( R ) where |α i |=N i Theorem: [DG] expanding r c at the nodes α i of to the value ½ r b csc (π/N i ) suffices to cover domain question: how to fix the holes? coverage power
question: is the computation distributable? Jadbabaie & Tahbaz-Salehi C0C0 C1C1 ← ∂ C2C2 ← ∂ C3C3 ← ∂ 0 ← ∂ … ← ∂ ← ∂* ← ← ← ← Laplacian: L = ∂*∂ + ∂∂* Hodge theory: ker( L k ) ≈ H k use dynamics… Egerstedt & Muhammad = - L k c(t) dt dc dynamics of heat flow is globally asymptotically stable iff H k = 0 distributed (“gossip”) algorithms to compute ker L Mrozek et al. distributed algebraic algorithms… bonus: subgradient methods yield sparse generators for homology… Tahbaz-Salehi and Jadbabaie distributed coverage
question: what happens when the sensors (and an evader) are in motion? dynamic coverage
a network is the skeleton of higher order structure… C0C0 C1C1 0 ← ∂ ← ∂ C2C2 C3C3 ← ∂ ← ∂ ← ∂ moral
euler calculus
χ = Σ (-1) k # { k-dimensional cells } k χ = 2 χ = Σ (-1) k rank H k k euler calculus χ (AuB) = χ (A)+ χ (B) – χ (A B) u euler characteristic is a topological invariant of spaces thus: euler measure d χ explicit definition: euler integral ∫ h d χ = ∫ ( Σ c i 1 U i ) d χ = Σ ( ∫ c i 1 U i ) d χ = Σ c i χ (U i )
signal processing ∫ h d χ geometry probability topology networks kashiwara macpherson schapira viro blaschke hadwiger rota chen adler taylor
target detection
a network of simple sensors returns target counts without IDs how many targets are there? = 0= 1= 2= 3= 4 problem
theorem: [baryshnikov-g.] assuming target footprints have uniform χ (U i )=c≠0 # targets = ( 1/c ) ∫ h d χ ∫ h d χ = ∫ ( Σ 1 U i ) d χ = Σ ( ∫ 1 U i d χ ) = Σ χ ( U i ) = c # i “ target space ” “ sensor space ” “ target footprint ” U i for each i “ local count ” h(x) = #{ i : x lies in U i } h trivial proof: ∫ h d χ = ∫ ( Σ 1 U i ) d χ = Σ ( ∫ 1 U i d χ ) = Σ χ ( U i ) = N # i amazingly, one needs no convexity, no leray (“good cover”) condition, etc. this is a purely topological result.
example
numerical analysis for (planar) sampled integrand via alexander duality integration formulae for counting time-dependent waves or moving targets via fubini theorem extensions to real-valued integrands for numerical analysis integral transforms for topological signal processing current results
topological integration theories aggregate data moral F*F* XY CF(X)CF(Y) F pt CF( pt )=Z ∫ d χ
what is the right global tool for this infdynets muri?
sheaf theory
closing credits… research sponsored by professional supportuniversity of pennsylvania a. mitchell darpa (stomp program) primary collaboratorsy. baryshnikov, bell labs v. de silva, pomona acme klein bottleb. mann