arXiv subjects

Michiel Rollier

Publications and source records attributed to Michiel Rollier.

8 recordsLinked to original sources

arXiv2026

Exact Lyapunov spectra of affine cellular automata and the parity rule on networks

The Lyapunov exponent quantifies the sensitivity of a dynamical system to perturbations, and the full Lyapunov spectrum extends this to every orthogonal direction in tangent space. For cellular automata the spectrum is almost always approximated numerically, and the approximation is delicate. We show that the affine rules, those whose update is a XOR of a subset of the inputs together with a constant, admit an exact Lyapunov spectrum. An affine rule has a configuration-independent Boolean Jacobian, so the spectrum reduces to the logarithms of the singular values of a single constant matrix, with no simulation and no limit involved. Two cases carry a closed form. For an affine cellular automaton on a periodic lattice the Jacobian is a multilevel circulant matrix, and the spectrum is the discrete Fourier transform of the rule's gradient stencil, valid in any spatial dimension. For the parity rule on an arbitrary graph the Jacobian is the adjacency matrix itself, so the Lyapunov spectrum is the logarithm of the absolute adjacency spectrum, and the maximal exponent is the logarithm of the spectral radius. The long-time amplitude of a single-site perturbation then scales with the eigenvector centrality of the seeded node. Reading the periodic lattice as the Cayley graph of an abelian group unifies the two cases. Because they are exact, the affine spectra also serve as benchmarks: they reveal numerical artefacts in previously reported spectra and turn the informal correspondence between spectral radius and dynamical sensitivity into an exact identity.

nlin.CG↗

arXiv2025

Essential metrics for Life on graphs

We present a strong theoretical foundation that frames a well-defined family of outer-totalistic network automaton models as a topological generalisation of binary outer-totalistic cellular automata, of which the Game of Life is one notable particular case. These "Life-like network automata" are quantitatively described by expressing their genotype (the mean field curve and Derrida curve) and phenotype (the evolution of the state and defect averages). After demonstrating that the genotype and phenotype are correlated, we illustrate the utility of these essential metrics by tackling the firing squad synchronisation problem in a bottom-up fashion, with results that exceed a 90% success rate.

nlin.CG↗

arXiv2024

Efficient Simulation of Non-uniform Cellular Automata with a Convolutional Neural Network

Cellular automata (CAs) and convolutional neural networks (CNNs) are closely related due to the local nature of information processing. The connection between these topics is beneficial to both related fields, for conceptual as well as practical reasons. Our contribution solidifies this connection in the case of non-uniform CAs (nuCAs), simulating a global update in the architecture of the Python package TensorFlow. Additionally, we demonstrate how the highly optimised out-of-the-box multiprocessing in TensorFlow offers interesting computational benefits, especially when simulating large numbers of nuCAs with many cells.

nlin.CG↗

arXiv2024

Convolutional Neural Networks for Automated Cellular Automaton Classification

The emergent dynamics in spacetime diagrams of cellular automata (CAs) is often organised by means of a number of behavioural classes. Whilst classification of elementary CAs is feasible and well-studied, non-elementary CAs are generally too diverse and numerous to exhaustively classify manually. In this chapter we treat the spacetime diagram as a digital image, and implement simple computer vision techniques to perform an automated classification of elementary cellular automata into the five Li-Packard classes. In particular, we present a supervised learning task to a convolutional neural network, in such a way that it may be generalised to non-elementary CAs. If we want to do so, we must divert the algorithm's focus away from the underlying 'microscopic' local updates. We first show that previously developed deep learning approaches have in fact been trained to identify the local update rule, rather than directly focus on the mesoscopic patterns that are associated with the particular behavioural classes. By means of a well-argued neural network design, as well as a number of data augmentation techniques, we then present a convolutional neural network that performs nearly perfectly at identifying the behavioural class, without necessarily first identifying the underlying microscopic dynamics.

nlin.CG↗

arXiv2024

A Comprehensive Taxonomy of Cellular Automata

Cellular automata (CAs) are fully-discrete dynamical models that have received much attention due to the fact that their relatively simple setup can nonetheless express highly complex phenomena. Despite the model's theoretical maturity and abundant computational power, the current lack of a complete survey on the 'taxonomy' of various families of CAs impedes efficient and interdisciplinary progress. This review paper mitigates that deficiency; it provides a methodical overview of five important CA 'families': asynchronous, stochastic, multi-state, extended-neighbourhood, and non-uniform CAs. These five CA families are subsequently presented from four angles. First, a rigorous mathematical definition is given. Second, we map prominent variations within each CA family, as such highlighting mathematical equivalences with types from other families. Third, we discuss the genotype and phenotype of these CA types by means of mathematical tools, indicating when established tools break down. Fourth, we conclude each section with a brief overview of applications related to information theory and mathematical modelling.

nlin.CG↗

arXiv2022

A Stochastic Mobility-Driven Spatially Explicit SEIQRD COVID-19 Model with VOCs, Seasonality, and Vaccines

In this work, we extend our previously developed compartmental SEIQRD model for SARS-CoV-2 in Belgium. We introduce SARS-CoV-2 variants of concern, vaccines, and seasonality in our model, as their addition has proven necessary for modelling SARS-CoV-2 transmission dynamics during the 2020-2021 COVID-19 pandemic in Belgium. The model is geographically stratified into eleven spatial patches (provinces), and a telecommunication dataset provided by Belgium's biggest operator is used to incorporate interprovincial mobility. We calibrate the model using the daily number of hospitalisations in each province and serological data. We find the model adequately describes these data, but the addition of interprovincial mobility was not necessary to obtain an accurate description of the 2020-2021 SARS-CoV-2 pandemic in Belgium. We further demonstrate how our model can be used to help policymakers decide on the optimal timing of the release of social restrictions. We find that adding spatial heterogeneity by geographically stratifying the model results in more uncertain model projections as compared to an equivalent nation-level model, which has both communicative advantages and disadvantages. We finally discuss the impact of imposing local mobility or social contact restrictions to contain an epidemic in a given province and find that lowering social contact is a more effective strategy than lowering mobility.

physics.soc-ph↗

arXiv2022

Mobility and the spatial spread of SARS-CoV-2 in Belgium

We analyse and mutually compare time series of COVID-19-related data and mobility data across Belgium's 43 arrondissements (NUTS 3). In this way, we reach three conclusions. First, we could detect a decrease in mobility during high-incidence stages of the pandemic. This is expressed as a significant change in the average amount of time spent outside one's home arrondissement, investigated over five distinct periods, and in more detail using an inter-arrondissement ``connectivity index'' (CI). Second, we analyse spatio-temporal COVID-19-related hospitalisation time series, after smoothing them using a generalise additive mixed model (GAMM). We confirm that some arrondissements are ahead of others and morphologically dissimilar to others, in terms of epidemiological progression. The tools used to quantify this are time-lagged cross-correlation (TLCC) and dynamic time warping (DTW), respectively. Third, we demonstrate that an arrondissement's CI with one of the three identified first-outbreak arrondissements is correlated to a significant local excess mortality some five to six weeks after the first outbreak. More generally, we couple results leading to the first and second conclusion, in order to demonstrate an overall correlation between CI values on the one hand, and TLCC and DTW values on the other. We conclude that there is a strong correlation between physical movement of people and viral spread in the early stage of the SARS-CoV-2 epidemic in Belgium, though its strength weakens as the virus spreads

physics.soc-ph↗

arXiv2018

A morphology-independent data analysis method for detecting and characterizing gravitational wave echoes

The ability to directly detect gravitational waves has enabled us to empirically probe the nature of ultra-compact relativistic objects. Several alternatives to the black holes of classical general relativity have been proposed which do not have a horizon, in which case a newly formed object (e.g. as a result of binary merger) may emit echoes: bursts of gravitational radiation with varying amplitude and duration, but arriving at regular time intervals. Unlike in previous template-based approaches, we present a morphology-independent search method to find echoes in the data from gravitational wave detectors, based on a decomposition of the signal in terms of generalized wavelets consisting of multiple sine-Gaussians. The ability of the method to discriminate between echoes and instrumental noise is assessed by inserting into the noise two different signals: a train of sine-Gaussians, and an echoing signal from an extreme mass-ratio inspiral of a particle into a Schwarzschild vacuum spacetime, with reflective boundary conditions close to the horizon. We find that both types of signals are detectable for plausible signal-to-noise ratios in existing detectors and their near-future upgrades. Finally, we show how the algorithm can provide a characterization of the echoes in terms of the time between successive bursts, and damping and widening from one echo to the next.

gr-qc↗