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University of Rome La “Sapienza” INFOCOM Department

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.

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University of Rome La “Sapienza” INFOCOM Department

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  1. 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

  2. Overview • Motivating remarks • Was the problem already solved in some unpublished notes ?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, 2007

  3. Motivating remarks • 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 … it may look like a nightmare for an engineer ! … or maybe is an opportunity to apply for research funds ? Perugia, February 13, 2007

  4. a life cycle much smaller than • average human being lifetime • limited individual reliability and precision design sensor networks as a population of mutually coupled oscillators Idea: Was the problem already solved in some unpublished notes ? Example: heartbeat a single natural pacemaker cell has nevertheless … a population of mutually coupled pacemakers gives rise to a very stable and reliable system Perugia, February 13, 2007

  5. Was the problem already solved in some unpublished notes ? Other examples • fireflies in South East Asia • brain neurons • lasers • muscular contraction in digestive system • menstrual cycle in women living in close contact • cellular mitotic division Perugia, February 13, 2007

  6. Huyghens, 1658 nearby pendula tend to synchronize Kaempfer, 1680 Kuramoto, 1984 South-East Asia fireflies flash simultaneously chemical oscillations, waves and turbulence Was the problem already solved in some unpublished notes ? References a population of globally coupled oscilllators may converge to a unique stable equilibrium under very mild conditions Mirollo, Strogatz 1990 Perugia, February 13, 2007

  7. Fundamental limits in wireless communications: A tiny step towards semantic • 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] Perugia, February 13, 2007

  8. Fundamental limits in wireless communications: A tiny step towards semantic 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 Perugia, February 13, 2007

  9. Basic message: Efficient network design should take into account the goal of the network data-centric and event-driven approaches Fundamental limits in wireless communications: A tiny step towards semantic Besides providing bounds on transport capacity, fundamental scaling laws suggest also strategies to approach the bounds: • Spatial reuse / multihop • Distributed source / channel coding • In-network processing / computing • Hierarchical layering clustering Perugia, February 13, 2007

  10. Hierarchical layering • Low level nodes pre-process the data and take local decisions • High level nodes carry relevant information to control centers Perugia, February 13, 2007

  11. 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, 2007

  12. In-network processing: Distributed consensus algorithms • Motivating problems: • 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) ? Perugia, February 13, 2007

  13. 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) ? Perugia, February 13, 2007

  14. In-network processing: Distributed consensus algorithms 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) pj= transmit power of sensor j dij = distance between nodes i and j hij = channel fading coefficient Perugia, February 13, 2007

  15. In-network processing: Distributed consensus algorithms Basic questions: • 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 ? Perugia, February 13, 2007

  16. Directed graphs: How to model interactions Network topologies weakly connected (WC) digraph, with a two-tree forest strongly connected (SC) digraph quasi strongly connected (QSC) digraph 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, the corresponding strongly connected component is a root SCC (RSCC) Perugia, February 13, 2007

  17. Decentralized decision through self-synchronizationConvergence conditions in the no-delay case • 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 The system is totally democratic: the final consensus depends on all Perugia, February 13, 2007

  18. Decentralized decision through self-synchronizationConvergence conditions in the no-delay case 2. If thecoupling function is linear, the network is not SC and it contains only one directed tree, then all state derivatives converge to the root decision All the nodes obey to the decision taken by the leader (root node) Perugia, February 13, 2007

  19. 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 cluster Cq The network forms K clusters of consensus is the i-th entry of the left eigenvector associated to the smallest eigenvalue of the graph Laplacian consensus depends on topology ! Perugia, February 13, 2007

  20. 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 Perugia, February 13, 2007

  21. Decentralized decision through self-synchronizationConvergence conditions in the nonlinear, non-delayed case 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 All dynamical systems converge to the same value of the state derivative, which is unique, irrespective of their initial conditions Perugia, February 13, 2007

  22. Decentralized decision through self-synchronizationConvergence conditions in the nonlinear, non-delayed case The critical coupling coefficient is lower bounded by depends only on measurement statistics depends only on network topology depends only on coupling function • If the network is not connected, • the oscillators cannot reach a consensus • If coupling is linear, and the network is connected, • the oscillators always reach a consensus • The consensus speed is proportional to Perugia, February 13, 2007

  23. Summary of convergence conditions • 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 Perugia, February 13, 2007

  24. Decentralized decision through self-synchronizationDistributed computing through self-synchronization 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, … Perugia, February 13, 2007

  25. Decentralized decision through self-synchronizationHow to set network parameters to achieve optimal estimates 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) Perugia, February 13, 2007

  26. 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 Perugia, February 13, 2007

  27. Decentralized decision through self-synchronizationHow to choose network parameters to build MAP detector 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 Perugia, February 13, 2007

  28. Numerical examples Linear coupling with random locations and geometry dependent delays Perugia, February 13, 2007

  29. Numerical examples Estimated value +/- standard deviation vs. time two step decentralized delayed decentralized undelayed centralized MLE (BLUE) decentralized - delayed Rayleigh fading channels with distance-dependent variance, random network topology, 40 nodes Observation: with , Perugia, February 13, 2007

  30. Examples: Estimation of vector parameters Example: random Gaussian (6x3) mixing matrices network topology: regular graph with fixed node degree = 4 Average estimation variance vs. number of nodes Estimation variance decreases as 1 / N, even if degree is fixed Performance improves adding nodes, without changing node Tx power (degree) Perugia, February 13, 2007

  31. Examples:Decentralized detection Binary hypothesis testing Signal model: Gaussian random patterns with known variances Optimal centralized rule: Decentralized solution: run and compare with a threshold Perugia, February 13, 2007

  32. Examples:Decentralized detection Example of performance over a random grid: Colors encode detection decisions Eventually, all oscillators end up with the same decision statistic Perugia, February 13, 2007

  33. Decentralized detection Detection probability vs. SNR and number of nodes, for a given Pfa N Degree is four, for any N Performance improves by increasing number of sensors, even if the degree is kept fixed Perugia, February 13, 2007

  34. global consensus systems cluster consensus system Entropy flow: How to monitor self-organization Entropy evolution [Bar-Scu ‘07] Entropy decreases ! Final value depends on how many nodes contribute to final decisions Are we contradicting the second law of thermodynamic ? Perugia, February 13, 2007

  35. The dark side of distributed consensus: iterations Total energy spent for achieving consensus is inversely proportional to [Bar-Scu-Swami ‘07]: There exists an optimal transmit power that minimizes total energy Random spacing is equivalent to uniform spacing Perugia, February 13, 2007

  36. Spatial clustering Using K less than critical value, the network may be forced not to synchronize Example: Noisy temperature field observed by a regular grid of sensors The network tends to form spontaneous clusters Perugia, February 13, 2007

  37. Spatial smoothing or clustering If sensors are distributed over a line, they are sufficiently close to each other and there is linear coupling only between adjacent sensors, i.e., except 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 Perugia, February 13, 2007

  38. Spatial smoothing or clustering Example: Noisy temperature field observed by a regular grid of sensors Initial observation Smoothed phase Perugia, February 13, 2007

  39. Conclusion • 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 Perugia, February 13, 2007

  40. Conclusion self- synchronization may be a beautiful subject to study … Perugia, February 13, 2007

  41. Conclusion … sometimes it doesn’t work Perugia, February 13, 2007

  42. Conclusion … but when it works, it may be rewarding … Perugia, February 13, 2007

  43. Conclusion … sometimes, it may be necessary … sometimes, it may be just for fun thank you ! Perugia, February 13, 2007

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