On a generalization of power algorithms over max-plus algebra

Fahim, Kistosil; Subiono,; Woude, Jacob Der

doi:10.1007/s10626-016-0235-4

Cited by 8 publications

(8 citation statements)

References 9 publications

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“…which is the same with (12). Notice that the equation above is well defined since k + 1 ≥ p max , or equivalently k + 1 − p max ≥ 0.…”

Section: Modeling Of Public Transportation Networkmentioning

confidence: 91%

“…Proposition 3.5 implies the max-plus eigenvalue and max-plus eigenvectors of z 1 (k+1) in (13) coincide with the max-plus eigenvalue and max-plus eigenvectors in (12). Since the model in (12) has a unique max-plus eigenvalue, z 1 (k + 1) in (13) also has a unique max-plus eigenvalue.…”

Section: Modeling Of Public Transportation Networkmentioning

confidence: 95%

“…This can be done by replacing the terms "max-plus eigenvalue" and "the corresponding max-plus eigenvector" by "entry of cycle-time vector" and "a state in the periodic regime," respectively. The interested reader is referred to [12] for the details.…”

Section: Spectral and Generalized Eigenvalue Problemsmentioning

confidence: 99%

See 2 more Smart Citations

Generalized public transportation scheduling using max-plus algebra

Subiono¹,

Fahim²,

Adzkiya³

2018

Kybernetika

Self Cite

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“…which is the same with (12). Notice that the equation above is well defined since k + 1 ≥ p max , or equivalently k + 1 − p max ≥ 0.…”

Section: Modeling Of Public Transportation Networkmentioning

confidence: 91%

Section: Modeling Of Public Transportation Networkmentioning

confidence: 95%

See 1 more Smart Citation

Generalized public transportation scheduling using max-plus algebra

Subiono¹,

Fahim²,

Adzkiya³

2018

Kybernetika

Self Cite

View full text Add to dashboard Cite

“…Note that the concept of an eigenspace can be extended for the case where the timing dependency matrix A is not irreducible. There is a little more work involved, but the periodic regime is guaranteed [7]. Thus, Theorem 3.3 is also true for a network that is not strongly connected.…”

Section: The Eigenspace Of a Ismentioning

confidence: 97%

A new threshold model based on tropical mathematics reveals network backbones

Patel

2021

Preprint

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Maxmin-ω is a new threshold model, where each node in a network waits for the arrival of states from a fraction ω of neighborhood nodes before processing its own state, and subsequently transmitting it to downstream nodes. Repeating this sequence of events leads to periodic behavior. We show that maxmin-ω reduces to a smaller, simpler, system represented by tropical mathematics, which forges a useful link between the nodal state update times and circuits in the network. Thus, the behavior of the system can be analyzed directly from the smaller network structure and is computationally faster. We further show that these reduced networks: (i) are not unique; they are dependent on the initialisation time, (ii) tend towards periodic orbits of networks -"attractor networks." In light of these features, we vary the initial condition and ω to obtain statistics on types of attractor networks. The results suggest the case ω = 0.5 to give the most stable system. We propose that the most prominent attractor networks of nodes and edges may be seen as a 'backbone' of the original network. We subsequently provide an algorithm to construct attractor networks, the main result being that they must necessarily contain circuits that are deemed critical and that can be identified without running the maxmin-ω system. Finally, we apply this work to the C. elegans neuron network, giving insight into its function and similar networks. This novel area of application for tropical mathematics adds to its theory, and provides a new way to identify important edges and circuits of a network.

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“…It has been shown in [13,Theorem 3.11] that if the cycle-time vector of A exists for at least one initial vector then it exists for any initial vector. Instead of computing the limit as in Definition 4, the cycle-time vector can be generated using a procedure [11,Algorithm 31].…”

Section: Cycle-time Vectormentioning

confidence: 99%

Computation of the Transient in Max-Plus Linear Systems via SMT-Solving

Abate¹,

Cimatti²,

Micheli³

et al. 2020

Preprint

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This paper proposes a new approach, grounded in Satisfiability Modulo Theories (SMT), to study the transient of a Max-Plus Linear (MPL) system, that is the number of steps leading to its periodic regime. Differently from state-of-the-art techniques, our approach allows the analysis of periodic behaviors for subsets of initial states, as well as the characterization of sets of initial states exhibiting the same specific periodic behavior and transient. Our experiments show that the proposed technique dramatically outperforms state-of-the-art methods based on max-plus algebra computations for systems of large dimensions.

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On a generalization of power algorithms over max-plus algebra

Cited by 8 publications

References 9 publications

Generalized public transportation scheduling using max-plus algebra

Generalized public transportation scheduling using max-plus algebra

A new threshold model based on tropical mathematics reveals network backbones

Computation of the Transient in Max-Plus Linear Systems via SMT-Solving

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