Phase Transitions and Statistics Gathering for Sensor Networks

1
Phase Transitions and Statistics Gathering for
Sensor Networks Ashish GoelStanford University
Joint work with Sanatan Rai and Bhaskar
Krishnamachari Enachescu, Govindan, and Motwani
http//www.stanford.edu/ashishg
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Sensor Networks

• Sensors often reside on the Euclidean plane
• Whether two sensors can communicate is determined
  largely by their Euclidean distance
• Often, laid out regularly
• Grids, random geometric graphs are reasonable
  models
• Big advantage over general networks in algorithm
  design and analysis
• This talk
• Phase transitions in geometric random graphs
• Statistics gathering over grids

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Geometric Random Graphs

• G(nr) in d-dimensions
• n points uniformly distributed in 0,1d
• Two points are connected if their Euclidean
  distance is less than r
• Sensor networks can often be modeled as G(nr)
  with d2
• Eg. sensors sprinkled from a helicopter over a
  corn field
• The wireless radius corresponds to r
• Question How should n and r be chosen to ensure
  that G(nr) has a desirable property P (eg.
  connectivity, 2-connectivity, large cliques) with
  high probability?

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1
Any other point Y is a neighbor of X with
probability ?r2
Expected degree of X is ? r2 (n-1)
r
X
0
1
5
Thresholds for monotone properties?

• A graph property P is monotone if, for all graphs
  G1(V,E1) and G2(V,E2) such that E1 µ E2,
• G1 satisfies P ) G2 satisfies P
• Informally, addition of edges preserves P
• Examples connectivity, Hamiltonianicity, bounded
  diameter, expansion, degree k, existence of
  minors, k-connectivity .
• Folklore Conjecture All monotone properties have
  sharp thresholds for geometric random graphs
• Also, simulation evidence for several properties
• Krishnamachari, Bejar, Wicker, Pearlman 02
• McColm 03

6

• To quote Bollobás (via Krishnamachari et al)
• One of the main aims of the theory of random
  graphs is to determine when a given property is
  likely to appear.

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Example Connectivity

• Define c(n) such that p c(n)2 log n/n
• Asymptotically, when d2
• G(nc(n)) is disconnected with high probability
• For any e gt 0, G(n (1e)c(n)) is connected whp
• So, c(n) is a sharp threshold for
  connectivity at d2 Gupta and Kumar 98 Penrose
  97
• Similar thresholds exist for all dimensions
• cd(n) ¼ (log n/(nVd))1/d, where Vd is the volume
  of the unit ball in d dimensions
• Average degree ¼ log n at the threshold

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Width and sharp thresholds

• For property P, and 0 lt e lt 1, if there exist two
  functions L(n) and U(n) such that
• PrG(nL(n)) satisfies P e, and
• PrG(nU(n)) satisfies P 1 - e,
• then the e-width we(n) of P is defined as
  U(n)-L(n)
• If we(n) o(1) for all e, then P is said to have
  a sharp threshold

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Example
PrG(nr) satisfies P
1
1-e
e
r
0
Width
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Connections (?) to Bernoulli Random Graphs

• Famous graph family G(np)
• Popular model for general random graphs
• Edges are iid each edge present with probability
  p
• Connectivity threshold is p(n) log n/n
• Average degree exactly the same as that of
  geometric random graphs at their connectivity
  threshold!!
• All monotone properties have e-width O(1/log n)
  for any fixed e in the Bernoulli graph model
• Friedgut and Kalai 96
• Can not be improved beyond O(1/log2 n)
• Almost matched Bourgain and Kalai 97
• Proof relies heavily on independence of edges
• There is no edge independence in geometric random
  graphs gt we need new techniques

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Our results
Goel, Rai, Krishnamachari, Ann App Prob 2005

• The e-width of any monotone property is
• Sharp thresholds in the geometric random graph
  model
• Sharper transition (inverse polynomial width)
  than Bernoulli random graphs (inverse logarithmic
  width)
• There exist monotone properties with width
• Tight for d1, sub-logarithmic gap for dgt1

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Why cd(n)?

• Why express results in terms of cd(n)?
• Width gives a sharp additive threshold
• We are typically interested in properties that
  subsume connectivity
• For such properties, an additive threshold in
  terms of cd(n) also corresponds to a
  multiplicative threshold
• The exact sharpness of the multiplicative
  threshold depends on L(n) and on the exact
  additive bounds

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Bottleneck Matchings

• Draw n blue points and n red points uniformly
  and independently from 0,1d
• B, R denotes the set of blue, red points resp.
• A minimum bottleneck matching between R and B is
  a one-one mapping
• fB ! R
• which minimizes
• maxu2 Bf(u)-u2
• The corresponding distance (maxu2 Bf(u)-u2)
  is called the minimum bottleneck distance
• Let Xn denote this minimum bottleneck distance

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Example
Bottleneck distance g
?
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Bottleneck Matchings and Width

• Theorem If PrXn gt g p then the sqrt(p)-width
  of any monotone property is at most 2g
• Implication Can analyze just one quantity, Xn,
  as opposed to all monotone properties (in
  particular, can provide simulation based
  evidence)
• Proof Let P be any monotone property
• Let e sqrt(p)
• Choose L(n) such that PrG(nL(n)) satisfies P
  e
• Define U(n) L(n) 2g
• Draw two random graphs GL and GU (independently)
  from G(nL(n)) and G(nU(n)), resp.
• Let B, R denote the set of points in GL, GU resp.

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Bottleneck Matchings and Width (proof contd.)

• Assume Xn g.
• Let f be the corresponding minimum bottleneck
  matching between R and B.
• For any u,v 2 B
• f(u)-f(v)2 f(u)-u2 u-v2
  f(v)-v2 2g u-v2
• Hence, (u,v) is an edge in GL ) (f(u),f(v)) is an
  edge in GU
• i.e. GL is a subgraph of GU
• By definition, PrXn gt g p
• ) PrGL is not a subgraph of GU p ?2 (1)

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Illustration I Triangle Inequality
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Bottleneck Matchings and Width (proof contd.)

• PrGL is not a subgraph of GU p ?2 (1)
• Let q PrGU does not satisfy P
• P is monotone, PrGL satisfies P e,
• ) PrGL is not a subgraph of GU eq (2)
• Combining (1) and (2), we get eq p i.e. q e
• Therefore, PrGU satisfies ? 1-?
• i.e. the ?-width of ? is at most U(n) L(n) 2?
• Done!

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Illustration II Probability Amplification
GU
GL
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Our Goal now Analyze the bottleneck matching
distance Xn Specifically, we are done if Xn
O(g(n)) with high probability, for some small
g(n)
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Comparison with Bernoulli Random Graphs?

• We are attempting to show something quite strong
• G(nr) is a subgraph of G(nrg) whp, for small g
• Laminar structure
• Corresponding result is NOT true for Bernoulli
  random graphs even for ? ½
• If small bottleneck matchings exist whp, we will
  get stronger thresholds than for Bernoulli random
  graphs

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Existence of Small Bottleneck Matchings

• The bottleneck matching length is
• O(cd(n)) whp for d 3
• Shor and Yukich 1991 we present a simpler
  proof
• O(c2(n) log1/4n) whp for d 2
• Leighton and Shor 1989
• O (sqrt(log(1/e))/sqrt(n)) with probability 1-e
  for d 1
• Our paper (folklore?)
• This gives us the desired widths
• Will omit the proof, focus on implications
• There is a small bottleneck matching between a
  random geometric graph and the grid

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Lower bound examples

• For d1, the property min-degree n/4 has
  width
• W(sqrt(log 1/e)/sqrt(n))
• Basic idea Just the two endpoints on the line
  are interesting for the purpose of finding the
  minimum degree
• For d 2, the property G is a clique has width
  W(1/n1/d)
• Open problems
• Plug the gap in the upper/lower bounds on the
  width for d 2
• Also, all our lower bound examples undergo phase
  transitions at r Q(1). Is there something
  interesting and different in the region where r
  is of the order of the connectivity threshold?

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Implications Mixing Time

• Fastest mixing Markov chain defined on G(nr) has
  mixing time Q(r-2 log n) for large enough r
  Boyd, Ghosh, Prabhakar, Shah 05
• Alternate proof
• GRID(nr) n points are laid on a grid in 0,12
  and two points are connected if they are within
  distance r.
• Fastest mixing time of GRID(nr) Q(r-2log n)
  Trivial
• G(nr) is a super-graph of GRID(nr-d) and a
  sub-graph of GRID(nrd) whp for small enough d
  Our result
• ) Fastest mixing time of G(nr) is Q(r-2log n)
  whp

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Implications Spectra

• Our techniques can be extended to show that the
  spectrum of random geometric graphs converges to
  the spectrum of the grid graph. Rai 05

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Implications Coverage

• Coverage Any point in the unit square must be
  within a distance r from one of the n sensors
• Known there is a sharp threshold in r
  Shakkottai, Srikant, Shroff 04
• Coverage is NOT a graph property, so it does not
  fall within our framework
• But the laminar structure in our proof implies a
  sharp threshold for coverage as well (weaker than
  the sharpest known)
• Fixed density, increasing area Muthukrishnan,
  Pandurangan

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Conclusions

• Monotone properties in G(nr) have sharp
  thresholds
• Much sharper than for Bernoulli Random Graphs
• Much stronger too Random geometric graphs
  exhibit a laminar structure
• Useful for recovering several known
  results/proving new ones
• Randomness is often a red-herring since the grid
  often yields tight upper/lower bounds
• Rule of thumb Grids are nearly as good a model
  for sensor networks as random geometric graphs
• No useful analogue in general networks
  (deterministic expanders are no easier to analyze
  than Bernoulli random graphs)
• Open problem Does laminarity imply anything
  about throughput (via separators)?

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Statistics Gathering

• Assume sensors are laid out on an n n grid
• Each sensor has 4 neighbors
• Each sensor knows its (x,y)-coordinates
• A processing agent (sink) at (0,0)

• Vast literature model based, gossip based, ODI,
  throughput maximization, coding
• This talk
• Answering box queries Goel, unpublished
• Aggregating spatially correlated data Enachescu,
  Goel, Govindan, Motwani, 2004

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Answering Box Queries

• The sink can issue queries for the aggregate
  statistics in any box
• Will focus on statistics that can be estimated
  from a random sample of size K
• Examples mean, median, quantiles, standard
  deviations

• Some naive strategies
• All sensors send their value to the sink every
  time unit
• No query-time cost, very high pre-processing
  cost (n per node)
• The sink samples K sensors from the box at query
  time
• No preprocessing cost, cost Kn at query time

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Notation K-uniform Samples

• A K-uniform sample of S is a subset of S obtained
  by choosing each element with probability K/S,
  independently of all other elements
• K-uniform samples are equally good for estimation
  purposes

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Our Solution Hyperbolic Gossip

• Pre-processing Every time unit, sensor (i,j)
  chooses z(i,j) uniformly at random from 0,1)
• Sensor (i,j) sends its value to all sensors (x,y)
  satisfying
• z(i,j) K/(x-iy-j) hyperbolic
  region
• Probability that (x,y) hears from (i,j)
  K/(x-iy-j)
• H(i,j) set of sensors contacted by (i,j)
• H(i,j) is connected, and has expected size O(K
  log2n)
• Pre-processing cost O(K log2n) per sensor
  (sharply concentrated)
• Claim Every sensor can now derive K-uniform
  samples from every box that contains this sensor
• Querying becomes trivial sink can query any
  sensor in this box
• Side-benefit Each sensor has a rough map of the
  entire sensor-field

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Deriving K-uniform Samples

• Suppose sensor (x,y) needs to derive K-uniform
  samples from box B
• Sub-sample For every node (i,j) in B which has
  sent its value to (x,y), add the value to the
  sample with probability x-iy-j/B
• Probability that (x,y) hears from (i,j)
  K/(x-iy-j)
• Probability that the value from (i,j) is in the
  sample
• (x-iy-j/B)(K/(x-iy-j))
• K/B

(x,y)
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Aggregation in Sensor Networks

• Highly constrained in power (and hence
  communication)
• Data gathering Each sensor (source) sends its
  data to a central agent (the sink)
• In-network aggregation (coding) can result in
  great power savings due to correlations in data
  or the nature of the query
• Example spatially correlated data
• Aggregation gains are unpredictable
• Basic Question What is the best routing
  structure to aggregate data on?
• Does it depend on the correlation?

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The Spatial Correlation Model

• Imagine each sensor is a camera with range k
• Can take a picture of any point within a 2k 2k
  square centered at the sensor
• Let Ak(s) denote the area imaged by sensor s
• Total amount of information obtained from set of
  sensors S ?s2 S Ak(s)
• What is the right value of k?
• Would like a single aggregation tree that is good
  for all k

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Rationale

• If an algorithm does well for all camera ranges
  k, it also does well when the aggregation
  function is a linear superposition of different
  values of k
• For example, consider events of different
  intensities (fireflies, camp-fires, forest fires,
  volcanic eruptions)
• Different events can be sensed within different
  ranges
• For any sets of intensities and relative
  frequencies, there is a superposition of
  different values of k which measures exactly the
  information in a set of sensors S
• The right model for spatial correlation given
  simultaneous optimization?

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A Useful Transformation

• Instead of focusing on the information captured
  by a sensor, focus instead on the area in a 11
  square
• We call this a value
• A value is sensed by all sensors within a 2k2k
  square centered at the value

The value for k2
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Cost Model

• Find a routing tree (by choosing a parent for
  each node) to send the information from all nodes
  to the processing node.
• Cost of an edge Number of values transmitted
  across the edge
• Cost of the tree Total cost of all the edges
• We want to minimize this transmission cost
  simultaneously for all k, assuming perfect
  aggregation.
• Assume the nodes must send all data values.

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Collision Time

• Consider two adjacent paths in a routing tree.
• Collision time the number of steps before the
  lower path meets the upper path.

Example collision time 3
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Collision Times and Simultaneous Optimization

• Theorem An aggregation tree with average
  expected collision time O(N1/2) gives a constant
  factor approximation to the optimum aggregation
  trees for all k

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A Randomized Algorithm

• Each node chooses one of its two downstream (i.e.
  closer to the origin) neighbors as its parent
• Results in a shortest path tree rooted at the
  sink
• All data flows on this shortest path tree
• Choice of parent Node at position (x,y) chooses
  the node at position (x-1,y) with probability
  x/(xy) and the node at position (x,y-1) with
  probability y/(xy)
• Different nodes choose independently
• Interesting Property The path from a sensor to
  the sink is a shortest path chosen uniformly at
  random from all such shortest paths

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The Randomized Algorithm contd.

• Intuition Consider A (j-1,j1) and B
  (j1,j-1)
• A is above the diagonal, so has a slight
  preference for going down
• B is below the diagonal, so has a slight
  preference for going left
• But if A goes down and B goes left, they meet!
• This centrist tendency introduces a small but
  sufficient bias so that the average expected
  collision time is O(N1/2) as opposed to O(N)
• Enachescu, Goel, Govindan, Motwani
  Theoretical Comput Sci 2006
• Proof omitted
• Implies that the randomized algorithm is an O(1)
  approximation for all correlation parameters k
• Much stronger than what is known for general
  graphs Goel, Estrin

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Conclusions (Aggregating Correlated Data)

• Simple routing techniques work well for a wide
  range of aggregation functions
• Rule of thumb Do not couple coding and routing
  in a sensor network
• Supported by Pattem et al Beferull-Lozano et
  al Goel, Estrin
• Open problems
• Extend our results to other correlation models
• Example Gaussian sources, spatial covariance
  matrix Scaglione, Servetto
• Conjecture Our model of spatial correlations
  captures Gaussian sources
• Making Spatio-temporal maps of sensor readings
  with varying and unknown spatio-temporal
  correlation
• Sampling of correlated data?
• Gossip based protocols for computing statistics
  of correlated data? (along the lines of Kempe et
  al., Boyd et al)