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Distributed Detection with Unreliable Data Sources

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Secure Networked Inference with Unreliable Data Sources

Abstract

In this chapter, we discuss networked detection problems for several practical network architectures such as parallel, multi-hop, and fully autonomous ad hoc networks. Following the taxonomy presented earlier, fundamental limits of networked inference in the presence of Byzantine attacks are first presented. Next, design of optimal countermeasures using the insights provided by the fundamental limits is discussed from a network designer’s perspective.

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Notes

  1. 1.

    It has been shown that the use of identical thresholds is asymptotically optimal [31].

  2. 2.

    These expressions are valid under the assumption that \(\alpha <0.5\). Later in Sect. 3.1.2, we will discuss a generalization of results for any arbitrary \(\alpha \).

  3. 3.

    Condition \(\alpha < \min \{(0.5-P_f),(1-(m/P_d))\}\), where \(m=\frac{N}{2N-2}>0.5\), suggests that as N tends to infinity, \(m=\dfrac{N}{2N-2}\) tends to 0.5. When \(P_d\) tends to 1 and \(P_f\) tends to 0, the condition \(\alpha < \min \{(0.5-P_f),(1-(m/P_d))\}\) simplifies to \(\alpha <0.5\).

  4. 4.

    Note that \(K^*\) might not be an integer.

  5. 5.

    Minimization is performed over attack strategies \(P_{0,1}\) and \(P_{1,0}\).

  6. 6.

    Note that when \(\alpha >0.5\), all flipping probabilities p which satisfy \(\alpha p=0.5\) will correspond to the best strategy.

  7. 7.

    Node i at level \(k'\) covers all its children at levels \(k'+1\le k\le K\) and the node i itself and, therefore, the total number of covered nodes by \(B_{k'}\), Byzantine at level \(k'\), is \(\dfrac{B_{k'}}{N_{k'}}.\sum _{i=k'}^K N_{i}\).

  8. 8.

    IEEE 802.16j mandates tree forwarding and IEEE 802.11s standardizes a tree-based routing protocol. Note that IEEE 802.16j and IEEE 802.11s are standard protocols for tree-based networks.

  9. 9.

    It was shown in [14] that by minimizing/maximizing the fraction of covered nodes, the FC can maximize/minimize the KLD. Using this fact, the authors considered fraction of covered nodes in lieu of the KLD in their analysis.

  10. 10.

    For a regular tree, intermediate nodes at different levels are allowed to have different degrees, i.e., number of children.

  11. 11.

    In practice, one possible way to achieve this is by using the buffer-less TDMA MAC protocol, in which, distinct non-overlapping time slots are assigned (scheduled) to the nodes for communication. One practical example of such a scheme is given in [29].

  12. 12.

    Note that under the conditional independence assumption, the optimal decision rule at the local sensor is a likelihood-ratio test [32].

  13. 13.

    Node i at level \(k'\) covers (or can alter the decisions of) all its children at levels \(k'+1\) to K and itself. In other words, the total number of covered nodes is equivalent to the total number of corrupted paths (i.e., paths containing a Byzantine node) in the network.

  14. 14.

    In practice, parameters such as threshold \(\lambda \) and consensus time step \(\varepsilon \) are set off-line based on well-known techniques [23].

  15. 15.

    The authors in [18] also come up with a robust distributed weighted average consensus algorithm which allows detection of consensus disruption attacks while mitigating the effect of data falsification attacks.

  16. 16.

    In the equal gain combining scheme, all the nodes (including Byzantines) are given the same weight.

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Authors and Affiliations

  1. IBM Research—Thomas J. Watson Research, Yorktown Heights, NY, USA

    Aditya Vempaty

  2. Department of Computing Applications and Research , Lawrence Livermore National Laboratory, Livermore, CA, USA

    Bhavya Kailkhura

  3. Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY, USA

    Pramod K. Varshney

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  1. Aditya Vempaty
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  3. Pramod K. Varshney
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Correspondence to Aditya Vempaty .

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Vempaty, A., Kailkhura, B., Varshney, P.K. (2018). Distributed Detection with Unreliable Data Sources. In: Secure Networked Inference with Unreliable Data Sources. Springer, Singapore. https://doi.org/10.1007/978-981-13-2312-6_3

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  • DOI: https://doi.org/10.1007/978-981-13-2312-6_3

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