Random%20Walks%20in%20WSN

1
Random Walks in WSN

1. Efficient and Robust Query Processing in Dynamic
   Environments using Random Walk Techniques, Chen
   Avin, Carlos Brito, IPSN 2004
2. Random Walks on Sensor Networks, Luisa Lima, Joao
   Barros, WiOpt 2007, Limassol, Cyprus, April,
   2007.

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Random Walks for Query Processing

• WSN features a highly dynamic environment
• State-based solutions for information extraction
  (clusters, spanning trees) introduce single
  points of failure, like clusterheads or roots of
  spanning trees
• Increased failure rate of nodes requires
  sophisticated failure recovery mechanism
  (increasing the overall complexity)
• This leads to severe impact on the overall
  performance of the network (in terms of energy
  efficiency or bandwidth wasted)

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Random Walks for Query Processing

• Justifying random walks
• No single point of failure for the method to
  operate (all nodes are equally unimportant at all
  times)
• Only a connected neighbor required to keep the
  packet moving
• Simple process of visiting nodes of graph G in
  some sequential random order
• When token arrives at node v, information in
  token is updated with local info stored at node v
• High redundancy in network
• No necessity to consult every node in the network
• Introducing Partial Cover Time (PCT)

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Network Cover Time

• Partial Cover Time, PCT
• Expected number of steps required by a random
  walk to visit a constant fraction of the nodes
  (50, 80, 99)
• Cover Time, C
• Expected number of steps by a random walk to
  visit all nodes in the network (starting from an
  arbitrary node)
• Given graph G(V,E) and two arbitrary nodes i, j
  in G
• hij expected number of steps to move from i to j
• hmax, hmin the max./min. over all ordered pairs
  of nodes in G

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Network Cover Time

• Known results for C
• Best case graphs (dense, highly connected graphs,
  such as the complete graph or d-regular graph
  with dgtn/2 or the hypercube) C O(nlog(n))
• Worst case graphs (when connectivity decreases
  and bottlenecks exist in the graph) C O(n3)
• Upper Bound for PCT (proof in the paper)
• For 0c1 let PCT(c) be the expected time to
  cover nodes of a graph G
• It is shown for PCT that reducing Matthews bound
  by an order of log(n), so it becomes linear in
  hmax

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Network Cover Time

• Comments on PCT
• For graphs where hmax n, the PCT linear in n
• Complete graph, star graph and hypercube are such
  graphs
• Covering x of nodes in Hypercube (d-regular
  graph with dlog(n)) is O(n)
• Cover Time C in Hypercube is O(nlog(n))
• For the grid (d-regular graph with d4) hmax
  nlog(n) so PCT O(nlog(n ))
• Sensor nets are almost d-regular graphs that
  lie between the two (hypercube and grid)

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Efficiency of PCT

• Experiments on grid, Hypercube, random graphs
• Random graphs generated as random geometric graph
  of n nodes on a unit grid with connectivity of
  max. range R
• N 4096 nodes
• R 0.04 units
• Figure depicts efficiency of PCT for
• Grid
• Random network with R 0.035 (aver. node degree
  15)
• Random network with R 0.04 (aver. node degree
  19)
• Hypercube of degree 12

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Efficiency of PCT
O(n)
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Quality of PCT

• Measure the quality of 80 cover
• Metric How far on average are the uncovered
  nodes from the ones that have been covered

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Quality of PCT

• In random graph
• About 90 of uncovered nodes are at most 2 hops
  away from a covered node
• In the grid
• About 60 of uncovered nodes are at most 2 hops
  away from a covered node

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Load Balancing

• Random walk is uncontrolled process
• Some nodes may be visited more than once
  (backtracking, walk in cycles)
• Those nodes spend more energy than others
• RW process is a Markov Chain
• For long walks the process reaches stationary
  distribution p
• Distribution p p1, , pn, where pi is given
  by pi di / 2m (di number of neighbors of node i
  and m number of edges in the network)
• If the graph is d-regular, then stationary
  distribution is uniform distribution

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Load Balancing

• In our case, issuing short walks due to PCT
• No guarantee that some nodes will be much more
  often visited than others
• Histogram presents number of visits to each node
  in 80 cover random walk
• The walk had 13100 steps

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Load Balancing
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Latency

• Random Walk is a sequential process
• Latency proportional to number of steps required
  to accomplish task
• If steps O(n) the applicability of the process
  is limited for large networks
• Possible idea
• Divide the network into regions
• Perform random walks in parallel on each of them
• Our approach? To accelerate RW process??

15
Random Walks on Sensor Networks

• Exploitation of mobile nodes for data gathering
  purposes
• Two types of sensors
• Static sensor nodes with limited connectivity
  (taking local values)
• Mobile, patrol node circulating area and
  collecting data
• Suffices mobile node to enter transmission range
  of static node for its data to be collected
• Node coverage describing effectiveness of
  different mobile data gathering strategies

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Random Walks on Sensor Networks

• Sensor network modelled as random geometric graph
  (n nodes, uniformly random placement, nodes
  within distance ? are connected)
• DEFINITION (Node Coverage)
• A sensor is collected if area defined by its
  transmission radius is visited at least once!
• Node coverage ?(t) expected number of distinct
  sensor nodes collected until time t
• Goal of patrol node To gather as much data as
  possible within a time frame / To maximize ?(t)

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Random Walks on Sensor Networks

• Individual sensor positions are unknown
• Mobile node performs random walk on unit square
  lattice
• Sensors within its transmission radius are
  queried
• Ratio between the transmission radius and step
  size (lattice size) needs to be carefully fixed

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Node Coverage for unconstrained random walk
Mathematical characterization of node coverage in
terms of the following entities
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Node Coverage for unconstrained random walk

• Let ES(t) be the support of the walk at time t
• Average number of sites not visited until time t
  is
• N - ES(t) N (cN) s
• N number of possible sites in finite lattice
• c 1.8456
• s time scaling factor (t sp-1?ln2cN)

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Node Coverage for unconstrained random walk

• Theorem
• Bounds for expected node coverage E? of
  unconstrained random walk until time t

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Node Coverage for unconstrained random walk
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Node Coverage for unconstrained random walk

• Node coverage curves have a steep start
• Node coverage curves stagnate as time progresses
• For increasing number of visited sites the mobile
  node is likely to spend more and more time in
  those (visited sites) instead of the not yet
  visited ones