A Survey on Hypergraph Products

Hellmuth, Marc; Ostermeier, Lydia; Stadler, Peter F.

doi:10.1007/s11786-012-0109-6

Cited by 21 publications

(23 citation statements)

References 45 publications

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“…, e n ∈ E(T C ), and that every Cartesian product e 1 × · · · × e n of elements of E(T C ) is an element of E(T n C ). This means that H(T n C ) is the square product of H(T C ) (see [23]). For the uncertain wiretap channel from Fig.…”

Section: Appendix a Uncertain Wiretap Channels: Proofs And Further DImentioning

confidence: 99%

Secure Estimation and Zero-Error Secrecy Capacity

Wiese

Oechtering

Johansson

et al. 2019

IEEE Trans. Automat. Contr.

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We study the problem of securely estimating the states of an unstable dynamical system subject to nonstochastic disturbances. The estimator obtains all its information through an uncertain channel which is subject to nonstochastic disturbances as well, and an eavesdropper obtains a disturbed version of the channel inputs through a second uncertain channel. An encoder observes and block-encodes the states in such a way that, upon sending the generated codeword, the estimator's error is bounded and such that a security criterion is satisfied ensuring that the eavesdropper obtains as little state information as possible. Two security criteria are considered and discussed with the help of a numerical example. A sufficient condition on the uncertain wiretap channel, i.e., the pair formed by the uncertain channel from encoder to estimator and the uncertain channel from encoder to eavesdropper, is derived which ensures that a bounded estimation error and security are achieved. This condition is also shown to be necessary for a subclass of uncertain wiretap channels. To formulate the condition, the zero-error secrecy capacity of uncertain wiretap channels is introduced, i.e., the maximal rate at which data can be transmitted from the encoder to the estimator in such a way that the eavesdropper is unable to reconstruct the transmitted data. Lastly, the zero-error secrecy capacity of uncertain wiretap channels is studied.

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Section: Appendix a Uncertain Wiretap Channels: Proofs And Further DImentioning

confidence: 99%

Secure Estimation and Zero-Error Secrecy Capacity

Wiese

Oechtering

Johansson

et al. 2019

IEEE Trans. Automat. Contr.

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“…It is easy to see that H(T n C ) is the n-fold square product H(T C ) n , cf. [6]. For any hypergraph H with vertex set A and hyperedge set E ⊂ 2 A , the square product H 2 of H with itself is defined as follows: The vertex set of H 2 is A 2 and the hyperedge set equals E 2 := {e × e ′ : e, e ′ ∈ E}.…”

Section: Uncertain Wiretap Channelsmentioning

confidence: 99%

Uncertain wiretap channels and secure estimation

Wiese

Johansson

Oechtering

et al. 2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Uncertain wiretap channels are introduced. Their zero-error secrecy capacity is defined. If the sensor-estimator channel is perfect, it is also calculated. Further properties are discussed. The problem of estimating a dynamical system with nonstochastic disturbances is studied where the sensor is connected to the estimator and an eavesdropper via an uncertain wiretap channel. The estimator should obtain a uniformly bounded estimation error whereas the eavesdropper's error should tend to infinity. It is proved that the system can be estimated securely if the zero-error capacity of the sensor-estimator channel is strictly larger than the logarithm of the system's unstable pole and the zero-error secrecy capacity of the uncertain wiretap channel is positive.All authors are with the

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“…In hypergraph terminology, the k-graph with vertex set Y ×V and edge set Y E is the anti-lexicographic product of the empty k-graph on vertex set Y and the kgraph (V, E) (cf. [5]). Observation 6.…”

Section: Liftings Of Hypergraphs and Wreath Product Actionmentioning

confidence: 99%

Monotone Properties of k-Uniform Hypergraphs are Weakly Evasive

Black

2015

Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science

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A boolean function in n variables is weakly evasive if its decision-tree complexity is Ω(n). By k-graphs we mean kuniform hypergraphs. A k-graph property on v vertices is a boolean function on n = v k variables corresponding to the k-subsets of a v-set that is invariant under the v! permutations of the v-set (isomorphisms of k-graphs).Rivest and Vuillemin (1976) proved that all non-constant monotone graph properties (k = 2) are weakly evasive, confirming a conjecture of Aanderaa and Rosenberg (1973). Kulkarni, Qiao, and Sun (2013) proved the analogous result for 3-graphs. We extend these results to k-graphs for every fixed k. From this, we show that monotone boolean functions invariant under the action of a large primitive group are weakly evasive.While KQS (2013) employ the powerful topological approach of Kahn, Saks, and Sturtevant (1984) combined with heavy number theory, our argument is elementary and selfcontained (modulo some basic group theory). Inspired by the outline of the KQS approach, we formalize the general framework of "orbit augmentation sequences" of sets with group actions. We show that a parameter of such sequences, called the "spacing," is a lower bound on the decision-tree complexity for any nontrivial monotone property that is Γ-invariant for all groups Γ involved in the orbit augmentation sequence, assuming all those groups are p-groups. We develop operations on such sequences such as composition and direct product which will provide helpful machinery for our applications. We apply this general technique to k-graphs via certain liftings of k-graphs with wreath product action of p-groups.

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A Survey on Hypergraph Products

Cited by 21 publications

References 45 publications

Secure Estimation and Zero-Error Secrecy Capacity

Secure Estimation and Zero-Error Secrecy Capacity

Uncertain wiretap channels and secure estimation

Monotone Properties of k-Uniform Hypergraphs are Weakly Evasive

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