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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 d>n/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 0≤c≤1 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 nComplete graph, star graph and hypercube are such graphs

• Covering x% of nodes in Hypercube (d-regular graph with d=log(n)) is O(n)

• Cover Time C in Hypercube is O(nlog(n))• For the grid (d-regular graph with d=4) 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 π

– Distribution π = {π1, …, πn}, where πi is given by πi = 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??

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Random Walks on Sensor Networks

• Exploitation of mobile nodes for data gathering purposes

• Two types of sensors:1. Static sensor nodes with limited connectivity (taking local values)

2. 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 γ are connected)

• DEFINITION (Node Coverage)

A sensor is collected if area defined by its transmission radius is visited at least once!

Node coverage η(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 η(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 E[S(t)] be the support of the walk at time t• Average number of sites not visited until time t is

N - E[S(t)] = N (cN) –σ

N: number of possible sites in finite lattice

c: 1.8456

σ: time scaling factor (t = σπ-1Νln2cN)

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Node Coverage for unconstrained random walk

Theorem

Bounds for expected node coverage E[η] 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