university of rome la “sapienza” infocom department
DESCRIPTION
BIO-INSPIRED SENSOR NETWORK DESIGN: DISTRIBUTED DECISION THROUGH SELF-SYNCHRONIZATION Sergio Barbarossa. University of Rome La “Sapienza” INFOCOM Department. Ack’s: WINSOC project (IST-FP6) and ARL/ERO - R&D 9989-CE-01 Collaborators: G. Scutari, L. Pescosolido. Overview. - PowerPoint PPT PresentationTRANSCRIPT
1Perugia, February 13, 2007
University of Rome La “Sapienza”INFOCOM Department
BIO-INSPIRED SENSOR NETWORK DESIGN:
DISTRIBUTED DECISION THROUGH SELF-SYNCHRONIZATION
Sergio Barbarossa
Ack’s: WINSOC project (IST-FP6) and ARL/ERO - R&D 9989-CE-01 Collaborators: G. Scutari, L. Pescosolido
Perugia, February 13, 20072
Univ. of Rome “La Sapienza”
Overview
• Motivating remarks
• Was the problem already solved in some unpublished notes ?
• Fundamental limits in wireless communications: A tiny step towards semantic
• In-network processing: Distributed consensus algorithms
• Directed graphs: How to model interactions
• Decentralized decision through self-synchronization
• Entropy flow: How to monitor self-organization
• What is the price for self-organization ?
• Conclusion
Perugia, February 13, 20073
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• Fundamental motivating application: Sensor networks
• Requirements: High reliability, small energy consumption, economy of scale, adaptive MAC / routing capabilities, energy scavenging
• Criticalities: Energy consumption, survivability, vulnerability to node failures, sleep modes or intentional attacks, congestion around sink nodes, scalability
• Resources: very inexpensive, simple, unreliable nodes, with very limited energy supply and simple MAC / routing mechanisms
Motivating remarks
… it may look like a nightmare for an engineer !
… or maybe is an opportunity to apply for research funds ?
Perugia, February 13, 20074
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Was the problem already solved in some unpublished notes ?
Example: heartbeat
a single natural pacemaker cell has
• a life cycle much smaller than average human being lifetime
• limited individual reliability and precision
a population of mutually coupled pacemakers gives rise to a very stable and reliable system
nevertheless …
design sensor networks as a population of mutually coupled oscillators Idea:
Perugia, February 13, 20075
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Was the problem already solved in some unpublished notes ?
Other examples
• fireflies in South East Asia
• brain neurons
• lasers
• menstrual cycle in women living in close contact
• muscular contraction in digestive system
• cellular mitotic division
Perugia, February 13, 20076
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References
Was the problem already solved in some unpublished notes ?
Huyghens, 1658 nearby pendula tend to synchronize
Kaempfer, 1680 South-East Asia fireflies flash simultaneously
Kuramoto, 1984 chemical oscillations, waves and turbulence
Mirollo, Strogatz 1990a population of globally coupled
oscilllators may converge to a unique stable
equilibrium under very mild conditions
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• Capacity of one-to-one link [Shannon ‘48]
• Transport capacity of many one-to-one wireless links [Gupta, Kumar, ‘00], [Xie, Kumar , ‘04]
• Transport capacity of many-to-one wireless links [Duarte-Melo, Liu, ‘03]
Fundamental limits in wireless communications: A tiny step towards semantic
logC
n n
1
Cn
1
Perugia, February 13, 20078
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Scaling laws for wireless sensor networks
Goal of a sensor network: Compute a function of the measurements
collected by N sensors
Data-centric view: is invariant to any permutation of the collected data
what is important is the value of the collected data, not which sensor has collected which measurement
transport capacity scales as
( , ,..., )Nf x x x1 2 , ,..., Nx x x1 2
( , ,..., )Nf x x x1 2
logC
n
1
Fundamental limits in wireless communications: A tiny step towards semantic
Perugia, February 13, 20079
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Besides providing bounds on transport capacity, fundamental scalinglaws suggest also strategies to approach the bounds:
• Spatial reuse / multihop
• Distributed source / channel coding
• In-network processing / computing
• Hierarchical layering clustering
Fundamental limits in wireless communications: A tiny step towards semantic
Basic message: Efficient network design should take into account the goal of the network data-centric and event-driven approaches
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Hierarchical layering
environment
lower-level nodes
higher-level nodes
controlnodes
- Low level nodes pre-process the data and take local decisions
- High level nodes carry relevant information to control centers
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In-network processing: Distributed consensus algorithms
References:
• Eisenberg, Gale, “Consensus of subjective probabilities: The pari-mutuel method”, 1959
• DeGroot, “Reaching a consensus”, 1974
• Borkar, Varaiya, “Asymptotic agreement in distributed estimation”, 1982
• J. N. Tsitsiklis, “Problems in decentralized decisions making and computation,” 1982
• Olfati-Saber and Murray, “Consensus Protocols for Networks of Dynamic Agents”, 2003
• Jadbabaie, Lin, and Morse, “Coordination of groups of mobile autonomous agents using nearest neighbour rules”, 2003
• Xiao, Boyd, and Lall, “A scheme for robust distributed sensor fusion based on average consensus,” 2005
• Barbarossa, Scutari “Decentralized ML estimation through nonlinearly coupled osc. ”, 2005
Perugia, February 13, 200712
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In-network processing: Distributed consensus algorithms
Motivating problems:
1. Distributed estimation
Given the vector measurements gathered by N nodes
can we achieve the globally optimal (ML or BLUE) estimate
using a totally decentralized approach (without a fusion center) ?
, 1, 2,...,i i i i N y A ξ v
-1N N1 1
i=1 i=1
ˆ T Ti i i i i i
ξ A R A A R y
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In-network processing: Distributed consensus algorithms
Motivating problems:
2. Multiple hypothesis testing
Denoting by the conditional pdf of the observation vector ,
conditioned to the hypothesis Hk , with a priori known probability P(Hk) ,
assuming that different sensors collect conditionally independent measurements,
can we derive the minimum error rate (MAP) test
using a totally decentralized approach (without a fusion center) ?
( / )i i kp y H iy
1ˆ arg max ( / ) ( ) arg max ( / ) ( )Nm k k i i i k kk k
p P p P y yH H H H H
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Proposed approach [Bar ‘05]: Each sensor takes an initial estimate (decision) as a function of its measurement and starts evolving as follows:
where
running estimate of each sensor
nonlinear, odd, increasing function
global coupling gain
local coupling attenuation (ci > 0)
( )f
ic
ix t
K
In-network processing: Distributed consensus algorithms
ij jia apj = transmit power of sensor j
dij = distance between nodes i and j
hij = channel fading coefficient
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1. Can we design local functions , attenuation coefficients to guarantee that each sensor state (derivative) converges towards globally optimal sufficient statistic ?
2. What is the impact of nonlinear coupling ?
3. What is the impact of delays and (fading) channel coefficients ?
4. Which are the convergence conditions ?
Basic questions:
In-network processing: Distributed consensus algorithms
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Network topologies
Directed graphs: How to model interactions
strongly connected(SC) digraph
quasi strongly connected(QSC) digraph
weakly connected(WC) digraph, witha two-tree forest
Every strongly connected component (SCC) can be substituted by a single node of the so called condensation digraph
If a node of the condensation digraph is a root node, thecorresponding strongly connected component is a root SCC (RSCC)
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1. If the coupling function is linear, the network is SC, then all state derivatives
converge to the same function of the measurements (global consensus)
This state is globally asymptotically stable
Decentralized decision through self-synchronizationConvergence conditions in the no-delay case
The system is totally democratic: the final consensus depends on all
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2. If the coupling function is linear, the network is not SC and it contains only one directed tree, then all state derivatives converge to the root decision
Decentralized decision through self-synchronizationConvergence conditions in the no-delay case
All the nodes obey to the decision taken by the leader (root node)
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Decentralized decision through self-synchronizationConvergence conditions in the no-delay case
3. If the coupling function is linear, the network is not SC and it contains a forest of K RSCC components, then the state derivatives converge to a linear combination of the root decisions
The network forms K clusters of consensus
cluster Cq
is the i-th entry of the left eigenvector associated to the smallest eigenvalue of the graph Laplacian consensus depends on topology !
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Decentralized decision through self-synchronizationConvergence conditions in the delayed case
4. If the coupling function is linear, the delays are not negligible, the network contains a forest of K RSCC components, then the state derivatives converge to a linear combination of the root decisions
Consensus (global or local) depends on channel coefficients and delays
Nevertheless, a two step iterative algorithm is sufficient to remove any bias, without knowing or estimating the channels
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5. If coupling function is nonlinear (odd, monotonically increasing), the graph is
non directed and connected, and K > Kc, then all state derivatives converge
to a global consensus
The synchronized state is globally asymptotically stable
Decentralized decision through self-synchronizationConvergence conditions in the nonlinear, non-delayed case
All dynamical systems converge to the same value of the state derivative,
which is unique, irrespective of their initial conditions
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The critical coupling coefficient is lower bounded by
depends only on network topology
depends only on measurement statistics
depends only on coupling function
• If the network is not connected,
the oscillators cannot reach a consensus
2 0A L
• If coupling is linear, and the network is connected,
the oscillators always reach a consensus
• The consensus speed is proportional to
maxf
2K L
Decentralized decision through self-synchronizationConvergence conditions in the nonlinear, non-delayed case
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• If coupling function is linear, necessary and sufficient condition for reaching a
consensus is that the digraph associated to the network is QSC
• The final consensus depends on the topology (# of root nodes), but it does not
depend on the channel coefficients
• If coupling function is nonlinear (odd, monotonically increasing), global
consensus is achieved if network is QSC and coupling is sufficiently strong
• In the presence of delays, consensus depends on channel coefficients and
delays, but it is possible to remove the bias by running the consensus
algorithm twice, without the need to estimate neither channel coefficients nor
delays
Summary of convergence conditions
Perugia, February 13, 200724
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Any function of the collected measurements that can be expressed in the form
with ci >0, can be computed with a totally distributed strategy based on
self-synchronization
Examples:
parameter estimation, detection of known waveforms in noise, detection of Gaussian process in Gaussian noise, belief propagation, …
Decentralized decision through self-synchronizationDistributed computing through self-synchronization
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Scalar observation model
where u is the unknown parameter and
Distributed estimation: run
In the non Gaussian, case, this estimator coincides with the Best Linear Unbiased Estimator (BLUE)
Decentralized decision through self-synchronizationHow to set network parameters to achieve optimal estimates
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Decentralized decision through self-synchronizationEstimation of vector parameters
Vector model
Strategy: Initialize each node with local ML estimate and run
All nodes tend to the global ML estimate, without sending neither the observations,nor the mixing matrices, nor the covariance matrices to any fusion center
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MAP test may be achieved by letting the network evolve with
and ci = 1, for as many times as the number of hypotheses
At convergence, each node applies the function
to the consensus value achieved under hypothesis k
The asymptotic consensus value, in the k-th iteration, is proportional to the argument of the MAP detector
Decentralized decision through self-synchronizationHow to choose network parameters to build MAP detector
( / ) log ( / ) ( )i i k i i k kg p Py yH H H
* *( ) exp( )k ku
*k
1 ( / ) ( )Ni i i k kp P y H H
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Linear coupling with random locations and geometry dependent delays
Numerical examples
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.580
85
90
95
100
105
110
time
mµ̂
§¾̂ µ
Estimated value +/- standard deviation vs. time
centralized MLE (BLUE)
decentralized - delayed
two step decentralized delayed
decentralized undelayed
Rayleigh fading channels with distance-dependent variance, random network topology, 40 nodes
Observation: with ,
Numerical examples
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Example: random Gaussian (6x3) mixing matrices network topology: regular graph with fixed node degree = 4
101
102
10-4
10-3
10-2
Number of Sensors
Estim
ation
var
iance
Estimation variance decreases as 1 / N, even if degree is fixed
Performance improves adding nodes, without changing node Tx power (degree)
Average estimation variance vs. number of nodes
Examples: Estimation of vector parameters
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Binary hypothesis testing
Signal model: Gaussian random patterns with known variances
Optimal centralized rule:
Decentralized solution: run
and compare with a threshold
Examples: Decentralized detection
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Examples: Decentralized detection
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Example of performance over a random grid: Colors encode detection decisions
Eventually, all oscillators end up with the same decision statistic
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Detection probability vs. SNR and number of nodes, for a given Pfa
-10 -5 0 5 10 15 20 25 3010
-3
10-2
10-1
100
Peak SNR (dB)
Det
ectio
n P
roba
bilit
y
Detection Probability @ PFA = 10-3 , =2 , = 100.5
N = 4N = 16N = 36N = 64N = 100
Performance improves by increasing number of sensors, even if the degree is kept fixed
Decentralized detection
Degree is four,for any N
N
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Entropy evolution [Bar-Scu ‘07]
Entropy flow: How to monitor self-organization
Are we contradicting thesecond law of thermodynamic ?
0 100 200 300 400 500 600 70050
60
70
80
90
100
110
120
time
Ent
ropy
cluster consensus system
global consensus systems
Entropy decreases !
Final value depends on how many nodes contribute to final decisions
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Total energy spent for achieving consensus is inversely proportional to [Bar-Scu-Swami ‘07]:
The dark side of distributed consensus: iterations
There exists an optimal transmit power that minimizes total energy
Random spacing is equivalent to uniform spacing
10 11 12 13 14 15 16 17 18 19 2010
1
102
103
Transmit power at each node
Tota
l ene
rgy
random grid
uniform grid
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Using K less than critical value, the network may be forced not tosynchronize
Example: Noisy temperature field observed by a regular grid of sensors
Spatial clustering
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The network tends to form spontaneous clusters
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If sensors are distributed over a line, they are sufficiently close to eachother and there is linear coupling only between adjacent sensors, i.e., except
Spatial smoothing or clustering
state evolution follows diffusion equation, triggered by initial observation
information propagates as a heat diffusion process
smoothing against observation noise is a result of diffusion
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Spatial smoothing or clustering
Example: Noisy temperature field observed by a regular grid of sensors
Initial observation Smoothed phase
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• Consensus through self-synchronization proves to be a very versatile strategy matched to the data-centric characteristic of WSN
• The approach allows for an easy implementation of radio transceivers
• Robustness and fault tolerance can be achieved through distributed coding / processing / computing / communicating …
• Great potentials for economy of scale and miniaturization
• Information can propagate as a diffusion wave percolating through the network in analog form
• Switching behavior from global consensus to local clustering or smoothing is possible using the same basic mechanism
Conclusion
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self- synchronization may be a beautiful subject to study …
Conclusion
Perugia, February 13, 200741
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… sometimes it doesn’t work
Conclusion
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… but when it works, it may be rewarding …
Conclusion
Perugia, February 13, 200743
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… sometimes, it may be just for fun… sometimes, it may be necessary
Conclusion
thank you !