Information flows, graphs and their guessing numbers

Riis, Søren

doi:10.1109/wiopt.2006.1666516

Cited by 41 publications

(113 citation statements)

References 14 publications

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“…The optimality of this bound is a central problem in network coding: the so called binary network coding problem consists in deciding if φ ′ (G) = 2 τ [23,10]. Concerning lower bounds, let us mention the following "folklore" lower bound…”

mentioning

confidence: 99%

Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks

Aracena¹,

Richard²,

Salinas³

2017

SIAM J. Discrete Math.

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Given a digraph G, a lot of attention has been deserven on the maximum number φ(G) of fixed points in a Boolean network f : {0, 1} n → {0, 1} n with G as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the feedback bound φ(G) ≤ 2 τ , where τ is the minimum size of a feedback vertex set of G. In this paper, we study the maximum number φ m (G) of fixed points in a monotone Boolean network with interaction graph G. We establish new upper and lower bounds on φ m (G) that depends on the cycle structure of G. In addition to τ , the involved parameters are the maximum number ν of vertex-disjoint cycles, and the maximum number ν * of vertex-disjoint cycles verifying some additional technical conditions. We improve the feedback bound 2 τ by proving that φ m (G) is at most the largest sub-lattice of {0, 1} τ without chain of size ν + 2, and without another forbidden pattern described by two disjoint antichains of size ν * + 1. Then, we prove two optimal lower bounds: φ m (G) ≥ ν + 1 and φ m (G) ≥ 2 ν * . As a consequence, we get the following characterization: φ m (G) = 2 τ if and only if ν * = τ . As another consequence, we get that if c is the maximum length of a chordless cycle of G then 2cν . Finally, with the techniques introduced, we establish an upper bound on the number of fixed points of any Boolean network according to its signed interaction graph.

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mentioning

confidence: 99%

Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks

Aracena¹,

Richard²,

Salinas³

2017

SIAM J. Discrete Math.

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“…This network coding instance is solvable over A if all the demands of the sinks can be satisfied at the same time. We assume the network instance is given in its circuit representation, where each vertex represents a distinct coding function and hence the same message flows every edge coming out of the same vertex [15]; again this loses no generality. This circuit representation has r source nodes, r sink nodes, and m intermediate nodes.…”

Section: Guessing Game and Guessing Numbermentioning

confidence: 99%

“…By merging each source with its corresponding sink node into one vertex, we form the digraph D on n = r +m vertices. In general, we have g(D, A) ≤ r for all A and the original network coding instance is solvable over A if and only if g(D, A) = r [15]. Note that the protocol on the digraph is equivalent to the coding and decoding functions on the original network.…”

Section: Guessing Game and Guessing Numbermentioning

confidence: 99%

“…, 10}. Therefore, after removing the vertices 6 to 10, we are left with the undirected cycle of length 5,C 5 , which is not solvable [15].…”

Section: B Application To Removing Useless Verticesmentioning

confidence: 99%

“…In [13] it was proved that an instance of network coding with r sources and r sinks on an acyclic network (referred to as a multiple unicast network) is solvable over a given alphabet if and only if the guessing number of a related digraph is equal to r. Moreover, it is proved in [14], [15] that any network coding instance can be reduced into a multiple unicast network. Therefore, the guessing number is a direct criterion on the solvability of network coding.…”

Section: Introductionmentioning

confidence: 99%

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Closure Solvability for Network Coding and Secret Sharing

Gadouleau

2013

IEEE Trans. Inform. Theory

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Network coding is a new technique to transmit data through a network by letting the intermediate nodes combine the packets they receive. Given a network, the network coding solvability problem decides whether all the packets requested by the destinations can be transmitted. In this paper, we introduce a new approach to this problem. We define a closure operator on a digraph closely related to the network coding instance and we show that the constraints for network coding can all be expressed according to that closure operator. Thus, a solution for the network coding problem is equivalent to a so-called solution of the closure operator. We can then define the closure solvability problem in general, which surprisingly reduces to finding secret-sharing matroids when the closure operator is a matroid. Based on this reformulation, we can easily prove that any multiple unicast where each node receives at least as many arcs as there are sources is solvable by linear functions. We also give an alternative proof that any nontrivial multiple unicast with two source-receiver pairs is always solvable over all sufficiently large alphabets. Based on singular properties of the closure operator, we are able to generalise the way in which networks can be split into two distinct parts; we also provide a new way of identifying and removing useless nodes in a network. We also introduce the concept of network sharing, where one solvable network can be used to accommodate another solvable network coding instance. Finally, the guessing graph approach to network coding solvability is generalised to any closure operator, which yields bounds on the amount of information that can be transmitted through a network.

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On the possible values of the entropy of undirected graphs

Gadouleau

2017

Journal of Graph Theory

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The entropy of a digraph is a fundamental measure which relates network coding, information theory, and fixed points of finite dynamical systems. In this paper, we focus on the entropy of undirected graphs. We prove that for any integer k the number of possible values of the entropy of an undirected graph up to k is finite. We also determine all the possible values for the entropy of an undirected graph up to the value of four.

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Information flows, graphs and their guessing numbers

Cited by 41 publications

References 14 publications

Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks

Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks

Closure Solvability for Network Coding and Secret Sharing

On the possible values of the entropy of undirected graphs

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