Discrete Mathematics & Theoretical Computer Science |

- 1 Network Dynamics and Simulation Science Laboratory
- 2 Department of Computer Sciences [Blacksburg]
- 3 Department of Mathematics [Blacksburg]

This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of standard threshold functions and have separate threshold values for the transitions $0 \rightarrow $1 (up-threshold) and $1 \rightarrow 0$ (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation. When the difference $\Delta$ of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for $\Delta \geq 2$ they may have long periodic orbits. The limiting case of $\Delta = 2$ is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graphs.

Source: HAL:hal-01196142v1

Volume: DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems

Section: Proceedings

Published on: January 1, 2011

Imported on: January 31, 2017

Keywords: bifurcation,bi-threshold,threshold,Boolean networks,graph dynamical systems,synchronous,asynchronous,sequential dynamical systems,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS],[NLIN.NLIN-CG] Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

Funding:

- Source : OpenAIRE Graph
*Collaborative Research: NECO: A Market-Driven Approach to Dynamic Spectrum Sharing*; Funder: National Science Foundation; Code: 0831633*CAREER: Cross-layer optimization in Cognitive Radio Networks in the Physical interference model based on SINR constraints: Algorithmic Foundations*; Funder: National Science Foundation; Code: 0845700*Collaborative Research: Modeling Interaction Between Individual Behavior, Social Networks And Public Policy To Support Public Health Epidemiology.*; Funder: National Science Foundation; Code: 0729441*Collaborative Research: NeTS-NBD: An Integrated Approach to Computing Capacity and Developing Efficient Cross-Layer Protocols for Wireless Networks*; Funder: National Science Foundation; Code: 0626964*Synthetic Information Systems for Better Informing Public Health Policymakers*; Funder: National Institutes of Health; Code: 5U01GM070694-13*SDCI NMI New: From Desktops to Clouds -- A Middleware for Next Generation Network Science*; Funder: National Science Foundation; Code: 1032677*NetSE: Large: Collaborative Research: Contagion in large socio-communication networks*; Funder: National Science Foundation; Code: 1011769*Tracking the West Antarctic Rift Flank*; Funder: National Science Foundation; Code: 0003957*Collaborative Research: Coupled Models of Diffusion and Individual Behavior Over Extremely Large Social Networks*; Funder: National Science Foundation; Code: 0904844

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