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Distributed Estimation of Network Cardinalities
Luleå University of Technology, Department of Computer Science, Electrical and Space Engineering, Signals and Systems.
2017 (English)Licentiate thesis, comprehensive summary (Other academic)Alternative title
Distribuerad skattning av nätverkskardinalitet (Swedish)
Abstract [en]

In distributed applications knowing the topological properties of the underlying communication network may lead to better performing algorithms. For instance, in distributed regression frameworks, knowing the number of active sensors allows to correctly weight prior information against evidence in the data. Moreover, continuously estimating the number of active nodes or communication links corresponds to monitoring the network connectivity and thus to being able to trigger network reconfiguration strategies. It is then meaningful to seek for estimators of the properties of the communication graphs that sense these properties with the smallest possible computational/communications overheads.

Here we consider the problem of distributedly counting the number of agents in a network. This is at the same time a prototypical summation problem and an essential task instrumental to evaluating more complex algebraic expressions such as products and averages which are in turn useful in many distributed control, optimization and estimation problems such as least squares, sensor calibration, vehicle coordination and Kalman filtering.

Being interested in generality, we consider computations in anonymous networks, i.e., in frameworks where agents are not ensured to have unique IDs and the network lacks a centralized authority. This setting implies that the set of distributedly computable functions is limited, that there is no size estimation algorithm with uniformly bounded computational complexity that can provide correct estimates with probability one, and thus that scalable size estimators are non-deterministic functions of the true network size. Natural questions are then: which one is the scheme that leads to topology estimators that are optimal in Mean Squared Error (MSE) terms? And what are the fundamental limitations of information aggregation for topology estimation purposes, i.e., what can be estimated and what not?

Our focus is then to understand how to distributedly estimate cardinalities given devices with bounded resources (e.g., battery/energy constraints, communication bandwidth, etc.) and how considering different assumptions and trade-offs leads to different optimal strategies. We specifically consider the case of peer-to-peer networks where all the participants are required to: i) share the same final result (and thus the same view of the network) and ii) keep the communication and computational complexity at each node uniformly bounded in time.

To this aim, we study four different estimation strategies that consider different tradeoffs between accuracy and convergence speed and characterize their statistical performance in terms of bias and MSE.

Place, publisher, year, edition, pages
Luleå: Luleå University of Technology, 2017.
Series
Licentiate thesis / Luleå University of Technology, ISSN 1402-1757
National Category
Control Engineering
Research subject
Control Engineering
Identifiers
URN: urn:nbn:se:ltu:diva-62460ISBN: 978-91-7583-843-4 (print)ISBN: 978-91-7583-844-1 (electronic)OAI: oai:DiVA.org:ltu-62460DiVA: diva2:1081162
Presentation
2017-05-17, A1545, Luleå, 13:00
Available from: 2017-03-15 Created: 2017-03-13 Last updated: 2017-05-02Bibliographically approved
List of papers
1. Networks cardinality estimation using order statistics
Open this publication in new window or tab >>Networks cardinality estimation using order statistics
2015 (English)In: American Control Conference (ACC), 2015: Chicago, IL, 1-3 July 2015,, Piscataway, NJ: IEEE Communications Society, 2015, 3810-3817 p.Conference paper (Refereed)
Abstract [en]

We consider a network of collaborative peers that aim at distributedly estimating the size of the network they belong to. We assume nodes to be endowed with unique identification numbers (IDs), and we study the performance of size estimators that are based on exchanging these IDs. Motivated by practical scenarios where the time-to-estimate is critical, we specifically address the case where the convergence time of the algorithm, i.e., the number of communications required to achieve the final estimate, is minimal. We thus construct estimators of the network size by exploiting statistical inference concepts on top of the distributed computation of order statistics of the IDs, i.e., of the M biggest IDs available in the network. We then characterize the statistical performance of these estimators from theoretical perspectives and show their effectiveness in practical estimation situations by means of numerical examples.

Place, publisher, year, edition, pages
Piscataway, NJ: IEEE Communications Society, 2015
Series
American Control Conference, ISSN 0743-1619
National Category
Control Engineering
Research subject
Control Engineering; Enabling ICT (AERI); Intelligent industrial processes (AERI)
Identifiers
urn:nbn:se:ltu:diva-32416 (URN)10.1109/ACC.2015.7171924 (DOI)6e948819-84da-4c91-9255-260d3c9ac8d4 (Local ID)978-1-4799-8685-9 (ISBN)6e948819-84da-4c91-9255-260d3c9ac8d4 (Archive number)6e948819-84da-4c91-9255-260d3c9ac8d4 (OAI)
Conference
American Control Conference : 01/07/2015 - 03/07/2015
Note
Validerad; 2016; Nivå 1; 20150327 (damvar)Available from: 2016-09-30 Created: 2016-09-30 Last updated: 2017-03-13Bibliographically approved
2. Network cardinality estimation using max consensus: the case of Bernoulli trials
Open this publication in new window or tab >>Network cardinality estimation using max consensus: the case of Bernoulli trials
2015 (English)In: IEEE 54th Annual Conference on Decision and Control (CDC): Osaka, Japan, 15-18 Dec. 2015, Piscataway, NJ: IEEE Communications Society, 2015, 895-901 p., 7402342Conference paper (Refereed)
Abstract [en]

Interested in scalable topology reconstruction strategies with fast convergence times, we consider network cardinality estimation schemes that use, as their fundamental aggregation mechanism, the computation of bit-wise maxima over strings. We thus discuss how to choose optimally the parameters of the information generation process under frequentist assumptions on the estimand, derive the resulting Maximum Likelihood (ML) estimator, and characterize its statistical performance as a function of the communications and memory requirements. We then numerically compare the bitwise-max based estimator against lexicographic-max based estimators, and derive insights on their relative performances in function of the true cardinality.

Place, publisher, year, edition, pages
Piscataway, NJ: IEEE Communications Society, 2015
Series
I E E E Conference on Decision and Control. Proceedings, ISSN 0743-1546
National Category
Control Engineering
Research subject
Control Engineering
Identifiers
urn:nbn:se:ltu:diva-34143 (URN)10.1109/CDC.2015.7402342 (DOI)000381554501031 ()842e74f7-bfaf-4be6-9fb4-bfba308dc423 (Local ID)978-1-4799-7884-7 (ISBN)842e74f7-bfaf-4be6-9fb4-bfba308dc423 (Archive number)842e74f7-bfaf-4be6-9fb4-bfba308dc423 (OAI)
Conference
IEEE Conference of Decision and Control : 16/12/2015 - 18/12/2015
Note

Godkänd; 2015; 20150725 (ricluc)

Available from: 2016-09-30 Created: 2016-09-30 Last updated: 2017-03-13Bibliographically approved
3. Average consensus via max consensus
Open this publication in new window or tab >>Average consensus via max consensus
2015 (English)Article in journal (Refereed) Published
Abstract [en]

Since intuition states that it is simple and fast to compute maxima over networks, we aim at understanding the limits of computing averages over networks through computing maxima. We thus build on top of max-consensus based networks’ cardinality estimation protocols a novel estimation strategy that infers averages through computing maxima of opportunely and locally generated random initial conditions. We motivate the max-consensus strategy explaining why it satisfies practical requirements, we characterize completely its statistical properties, and we analyze when and under which conditions it performs favorably against classical linear consensus strategies in static Cayley graphs

Keyword
Information technology - Automatic control, Informationsteknik - Reglerteknik
National Category
Control Engineering
Research subject
Control Engineering
Identifiers
urn:nbn:se:ltu:diva-30245 (URN)10.1016/ifacol.2015.10.307 (DOI)2-s2.0-84992521919 (Scopus ID)402a528a-8fc5-488f-8f91-576376428abd (Local ID)402a528a-8fc5-488f-8f91-576376428abd (Archive number)402a528a-8fc5-488f-8f91-576376428abd (OAI)
Conference
IFAC Workshop on Distributed Estimation and Control in Networked Systems : 10/09/2015 - 11/09/2015
Note

Konferensartikel i tidskrift

Available from: 2016-09-30 Created: 2016-09-30 Last updated: 2017-03-13Bibliographically approved
4. A tight bound on the Bernoulli trials network size estimator
Open this publication in new window or tab >>A tight bound on the Bernoulli trials network size estimator
2016 (English)In: 2016 IEEE 55th Conference on Decision and Control, CDC 2016, Piscataway, NJ: Institute of Electrical and Electronics Engineers (IEEE), 2016, 3474-3480 p., 7798790Conference paper (Refereed)
Abstract [en]

We consider the problem of finding exact statistical characterizations of the Bernoulli trials network size estimator, a simple algorithm for distributedly counting the number of agents in an anonymous communication network for which the probability of committing estimation errors scales down exponentially with the amount of information exchanged by the agents. The estimator works by cascading a local, randomized, voting step (i.e., the i.i.d. generation of some Bernoulli trials) with an average consensus on these votes. We derive a tight upper bound on the probability that this strategy leads to an incorrect estimate, and refine the offline procedure for selecting the Bernoulli trials success rate.

Place, publisher, year, edition, pages
Piscataway, NJ: Institute of Electrical and Electronics Engineers (IEEE), 2016
Series
IEEE Conference on Decision and Control, E-ISSN 0743-1546
National Category
Control Engineering
Research subject
Control Engineering
Identifiers
urn:nbn:se:ltu:diva-37510 (URN)10.1109/CDC.2016.7798790 (DOI)000400048103103 ()2-s2.0-85010720911 (Scopus ID)b90d2ca1-9f38-48bc-b98a-6e0211ccf71f (Local ID)9781509018376 (ISBN)b90d2ca1-9f38-48bc-b98a-6e0211ccf71f (Archive number)b90d2ca1-9f38-48bc-b98a-6e0211ccf71f (OAI)
Conference
55th IEEE Conference on Decision and Control, CDC 2016, Las Vegas, United States, 12-14 December 2016
Available from: 2016-10-03 Created: 2016-10-03 Last updated: 2017-06-02Bibliographically approved

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