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MAE 298, Lecture 8April 27, 2006
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“Spectral Methods, Sensor Nets andSelf-organization”
Last time: spectral methods, eigen-spectrum
• If two distinct graphs have the same eigen-spectrum, they arelikely isomorphic (esp for large graphs).
• Eigenvalues: degeneracy of λ = 1 tells us how manydisconnected components in the graph.
Summary: spectral methods, measures
• Mixing time (time to forget where the walk started)
• Relaxation time (related to mixing time, gives bounds)
• Cover time (time to occupy each node)
• Spectral gap: the largest mixing time, tmax = −1/ ln(λ2)
– the larger tmax the longer it takes for a random walk to coverthe graph.
– the larger tmax the more accurately a graph can bepartitioned into two pieces.
Applications: Wireless sensor networks
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• Start with isolated sensor distributed at random.
• Is there a local way to build up global connectivity?
• Locality — why?
Locality
1. Locality ∼ distributed
2. Adapt quickly to changing environment
3. Minimal growth in overhead with increasing system size
4. “Self-organizing”
“self-organization”
• Not quantitatively defined.
• (Wikipedia:) Self-organization is a process in which the internalorganization of a system , normally an open system , increasesin complexity without being guided or managed by an outsidesource. Self-organizing systems typically (though not always)display emergent properties.
Beaconing
A geometric graph problem
One idea — percolation
Call the graph describing connectivity of nodes: GR
Is this a local algorithm?
(How to determine Rc?)
How to determine Rc?
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R = 1.362
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R = 2.724
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R = 4.904
Keep increasing until only one eigenvalue λ = 1
Percolation
Why is it bad?
• Farthest away node sets operating power for all
• Need to communicate this value Rc (critical operating range)
• Assumes wireless footprint a uniform disk
Why is in good?
• Guarantees full global connectivity. In the asymptotic limit(N → ∞) know how Rc scales with N . So for large N canuse theoretical estimate rather than λ = 1 construction.
• Want small range R to conserve power and also reduceinterference. Percolation is a “sweet spot” (full connectivity without too much interference).
Refining percolation graph GR
(also called the “unit disk graph”)
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RNG & GGRelative Neighbor Graph (RNG)! An edge (u, v) exist if !w, d(u,v) " max(d(u, w), d(w, v)
Gabriel graph (GG)! An edge (u, v) exist, if no other
vertex w is present within the circle
2 2 2, : ( , ) ( , ) ( , )w u v d u v d u w d w v! # " $
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Planarized Graph (Cont’d)
Original Unit Disk Graph
GG RNG
• Preserve connectivity of GR, but sparser (and also planar).
• There is an R-package that computes the disk, Gabriel andrelative neighbor graph given an input set of coordinates(http://rss.acs.unt.edu/Rdoc/library/spdep/html/graphneigh.html)
Planar disk graphs =⇒ greedy routing
• What is greedy forward routing?
• Packets are discarded if there is no neighbor which is nearerto the destination node than the current node; otherwise,packets are forwarded to the neighbor which is nearest to thedestination node.
• Each node needs to know the locations of itself, its 1-hopneighbors and destination node.
• Pros: easy implement
• Cons: deliverability (stuck in local voids)
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Examples
uw4
v
w1 w2 w3
w5 w6w6 is a local
minimum w.r.t. v
Theory versus realityWireless footprints
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!k!!
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Monotonicity
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Uniform-disk model Experimental[Ganesan, et. al., 2002]
Adaptive Power Topology control(Local algorithms to build connectivity)
• Percolation (Common power) neglects natural clustering.– Too much power consumption and unnecessary interference.– Misses certain paths which could optimize traffic.
• How to build up a connected network using only localinformation?– Moreover, want to avoid uniform disk requirement
Adaptive Power Topology control
• Adaptive power topology control (APTC)[Wattenhofer, Li, Bahl, and Wang. Infocom 2001]
[D’Souza, Ramanathan, and Temple Lang. Infocom 2003]
Each node individually increases power until it has a neighbor inevery θ sector around it:
Call the graph describing connectivity of nodes: Gθ
Sample topology
Topology control:
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τ[max] = 2920 to
Percolation:
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R = 5.164
τ[min] = 183 to
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R = 4.727
τ[med] = 459 to
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R = 4.524
τ[max] = 2590 to
Beyond the uniform disk model
[D’Souza, Galvin, Moore, Randall, IPSN 2006 ]
• Can use a local geometric θ-constraint to ensure full networkconnectivity, independent of wireless footprint!.
• Requires constraints on boundary nodes. (Carefully deployboundary nodes so can communicate, or else have hard-wired boundary channel; then interior nodes can be scatteredhaphazardly).
Proof overview
Theorem 1. If Gθ satisfies the θ-constraint at every internalnode with θ < π and all of the boundary nodes are known tobe connected, then Gθ is fully connected.
Proof: We need only show every internal node v has a path inGθ to some node on the boundary.
Comparison to percolation scheme
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R = 7.513
τ[min] = 109 to
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τ[med] = 604 to
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R = 5.315
τ[max] = 5314 to
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How to quantify “better”?
• Need performance metrics
• Direct measures: energy consumption
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5000 10000 15000
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Common power scheme, Pfixed
Ada
ptiv
e po
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sch
eme,
Pad
pOverall power requirements,
common versus adaptive power(assuming P falls off as R2.5)
∆Padp ∆Pfixed = 0.41
Power Control:Cross-Layer Design Issues
• Physical Layer– Power control affects quality of signal
• Link Layer– Power control affects number of clients sharing channel
• Network Layer– Power control affects topology/routing
• Transport Layer– Power control changes interference, which causes
congestion• Application/OS Layer
– Power control affects energy consumption
The “protocol stack”
In general, networks layered
• Social networks
• → Email networks
• → Data networks
• → Protocol networks
• → Physical networks
Approximating interference
State transition matrix:
Mii = (ki − 1)/ki, for diagonal elements.
Mij = 1/(ki − 1)ki, if an edge exists between i and j.
Network self-discovery time(Using mixing time as a proxy)
τ = −1/ ln(λ2)
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Timescale, τ1 = 5314 to
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Timescale, τ2 = 1092 to
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Timescale, τ3 = 157 to
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− 1 ln(λ)
Den
sity
0 1000 2000 3000 4000 5000
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000.
0010
0.00
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0030
− 1 ln(λ), compared to exponential density of Common Power Adaptive power
− 1 ln(λ)
Den
sity
0 1000 2000 3000 4000 5000
0.00
000.
0010
0.00
200.
0030
Common power− 1 ln(λ) and exponential density
Comparison of timescales
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Scatterplot of timescalescommon versus adaptive power
Regimes for routing
Two timescales:
1. tinfo = −1/ ln(λ2)
2. tnetwork: time for network topology to change.
Routing:
• If tnetwork � tinfo, network essentially static during packetrouting, so build up routing information.
• If tnetwork � tinfo, any info on the network topology will beimmediately obsolete, so no routing strategy.
Is there a sharp threshold? Or even any way to bound thesebehaviors? (What routing protocols work best in which regimes?
Other pressing issues: Sensor networks
• Deployed networks: optimal sensor placement
• Gossip algorithms: spreading shared information quicklythrough local exchanges.
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