quantifying coverage in mobile sensor networks using ... › ... › 03 ›...

Homology for sensor networks Zigzag Persistence Hop distance filtration Tracking representative cycles

Quantifying Coverage in Mobile SensorNetworks using Zigzag Persistent Homology

Jennifer Gamble

SAMSI (LDHD) Workshop - Topological Data Analysis

Department of Electrical and Computer EngineeringNorth Carolina State University

Joint work with Hamid Krim and Harish Chintakunta

Feb 7, 2014

Jennifer Gamble NC State University, ECE


Homology for sensor networks

Zigzag Persistence

Hop distance filtration

Tracking representative cycles



Sensor network model

I Sensors deployedover area of interest

I Each can sensewithin a coveragedisk of radius r

I Nodes withindistance 2r cancommunicate

I Represent coverageregion with asimplicial complex



I Simplicial complex is the 2-skeleton of the Rips complexdefined by the communication graph

I i.e.) whenever three nodes can communicate pairwise, theyform a triangle in the complex

I Complex stored as binary adjacency matrix, indicatingwhich sensors can communicate

simplicialhomology=⇒ “5 holes”

I Does the homology of the complex match the homology ofthe coverage region?



Missed holes



Options for dealing with missed holes:1. (de Silva and Ghrist, 2006) Set communication radius to

2√3r , instead of 2r , to guarantee no such missed holesI Allows statements about global coverage guarantees

2. Consider small missed holes to be unimportantI Lowers rate of false alarms (holes in simplicial complex not

truly present in coverage region)I Worst case: sensors on hexagonal lattice ∼ 7% area missedI Typical case: missed holes cover � 1% of total area



Dynamic sensor network

⇓ · · ·

⇓ · · ·⇒ Zigzag ⇒

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Persistence



Regular persistent homology

I Takes a nested family of spaces

∅ = X0 ⊆ X1 ⊆ X2 ⊆ . . . ⊆ Xn

I Tracks the associated sequence of homology spaces

0→ Hk(X1)→ Hk(X2)→ . . .→ Hk(Xn)

induced by the inclusion maps between successive Xi ’s




Zigzag Persistence





Zigzag Persistence

I Still tracks homology classes over sequence of spacesX1, . . . ,Xn , but no longer assumes sequence is nested

I Inclusion maps can go “backward” or “forward”, so insteadof

X1 → X2 → X3 → . . .→ Xn

we now have

X1 ← X2 → X3 → . . .← Xn



Zigzag Persistence

I In particular, can be used to analyze a sequence of spacesby mapping through their union

X1 X2 X3↘ ↙ ↘ ↙ · · ·

X1⋃

X2 X2⋃

X3

I Induces sequence in homology

Hk(X1) Hk(X2) Hk(X3)↘ ↙ ↘ ↙ · · ·Hk(X1 ∪X2) Hk(X2 ∪X3)



Tracking coverage holes

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Tracking coverage holes

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Output of lifetimes of homology classes

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Quantifying coverage properties

I Do the lifetimes of homology classes describe ‘lifetimes’ oftime-varying coverage holes?

I What is a time-varying coverage hole?I Holes moving in a network can split or merge. Geometric

definition of which is the ‘same hole’, is not clear.I Evasion path: route through space and time that an

intruder could take, while remaining undetectedI (Adams and Carlsson, 2013) Evasion path 6⇔ Long bar




I Correspondences between barcode and coverage holes:I An evasion path exists over interval [b, d] =⇒ There exists a

bar containing [b, d]I At each time point, the number of bars present equals the

number of holes in the complexI If a hole is born at b, remains isolated, and is killed at d,

then the pair [b, d] will be in the barcode, corresponding toit

I In general, longer bars and more bars mean worse coverage



Comparing mobility patterns

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Hol

e nu

mbe

r

Brownian motion, n = 200, E[deg] = 15

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ole

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I Barcode gives us descriptor of time-varying network, whichis related to its coverage over time

I Can we infer geometric information from the adjacencymatrix?

I Number of holes tells us nothing about their sizeI Similarly, long bar may or may not correspond to a large

hole




I Using only adjacency matrix information, want getestimates of the sizes of the holes

I Only estimate of size available is hop-length of shortestcycle surrounding a hole

I Hop distance filtration




Zigzag Persistence






1 hop (original complex)




2 hops




3 hops




Hop-length of shortest Persistence of hole incycle surrounding hole hop distance filtration

4, 5, 6 17, 8, 9 2

10, 11, 12 3...

...3k + 1, 3k + 2, 3k + 3 k

I At what point a given cycle becomes trivial in the hopdistance filtration depends on the size of the largest hole itsurrounds




I Compute hop distance filtration at each time pointI Obtain size estimates for the holes present at that timeI Want to link the set of sizes at time i to the bars at time i,

for each time point iI Need geometrically-meaningful choice of representative

cycle for each bar, perhaps time-varying




Zigzag Persistence





Representative cycles - regular persistent homology

I Algorithm for regular persistent homology returns set ofbirth-death intervals

{[bj , dj ] | j = 1, . . . ,m}

as well as a specific representative cycle cj for each interval[bj , dj ]

I The cycle cj consists of the same set of simplices for eachsimplicial complex in bj , . . . , dj




I The representative cycles {cj | j = 1, . . . ,m } given bypersistent homology are compatible with the intervaldecomposition {[bj , dj ] | j = 1, . . . ,m} in the followingsense:

1. cj remains non-trivial over the interval [bj , dj ], becominghomologous to zero after dj

2. The set of representative cycles corresponding to intervalsalive at time i have corresponding homology classes whichform a basis for the homology at time i

3. The homology class corresponding to cj at time i maps intothe homology class corresponding to cj at time i + 1



Representative cycles




I Relax condition 3, and let representative cycles vary overtime

I Choices of homology bases that satisfy 1 and 2 will still becompatible with the barcode intervals

I Idea: Make ‘canonical’ choice of basis at death time, andpropagate backwards through time for as long as possible

I Boundary of simplex which kills homology class acts ascanonical representative cycle



Representative cycles - Adaptive

I Store representative cycles in Wi , and update byproceeding backwards through time

I Two cases:1. If σ−

i (i.e. death): Wi−1 = [Wi ∂σi ]2. If σ+

i (i.e. birth): Perform change of basis



Adaptive representative cycles



Adaptive representative cycles - Zigzag setting

I Four cases:1. Birth (by removal): append ∂σ2. Death (by removal): change of basis3. Birth (by addition): append cycle born in zigzag algorithm4. Death (by addition): reduce matrix storing representative

cycles, remove trivial oneI No canonical choice available for choices 3 and 4I Algorithm still performs well in many cases




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Expanding failure region




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Conclusions

I Barcodes obtained from zigzag persistent homology can beused to quantify coverage in dynamic sensor networks

I Only requires adjacency matrix at each time pointI Bars are related to coverage holes (although not one-to-one)

I Choice of adaptive representative cycles allowsgeometrically informed homology basis to be chosen ateach time point

I Hop distance filtration can give coarse size estimates ofholes

I Visualize using barcode thickened by size information



Thank you.

email: [email protected]


quantifying coverage in mobile sensor networks using ... › ... › 03 ›...

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