1. robert ghrist professor of mathematics & electrical/systems engineering the university of pennsylvania topological methods for networks infodynets kickoff : sept 2009

2. tools for applied mathematics…

3. differential equations

4. linear algebra

5. numerical analysis

6. novel challenges necessitate novel mathematics

7. hardware improves...

9. topology

10. topological methods for networks…

11. homological coverage

12. sensors and simplices each have knowledge only of their identities and of their local connectivity... sensorssimplices homological coverage

13. 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

14. 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

15. 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

16. Not an if & only if statement: provides a certificate The relative condition really is necessary coverage remarks

17. 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

18. 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

19. 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

20. question: what happens when the sensors (and an evader) are in motion? dynamic coverage

21. a network is the skeleton of higher order structure… C0C0 C1C1 0 ← ∂ ← ∂ C2C2 C3C3 ← ∂ ← ∂ ← ∂ moral

22. euler calculus

23. 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 )

24. signal processing ∫ h d χ geometry probability topology networks kashiwara macpherson schapira viro blaschke hadwiger rota chen adler taylor

25. target detection

26. a network of simple sensors returns target counts without IDs how many targets are there? = 0= 1= 2= 3= 4 problem

27. 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.

28. example

31. 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

33. topological integration theories aggregate data moral F*F* XY CF(X)CF(Y) F pt CF( pt )=Z ∫ d χ

34. what is the right global tool for this infdynets muri?

35. sheaf theory

36. 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