Efficient Numerical Analysis in Complexity Science

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K. Benedek1,2, B. Hartmann3, G. Ódor2, I. Papp1, M. T. Cirunnay1 (1) Institute of Technical Physics and Materials Science, HUN-REN Centre for Energy Research, P.O. Box 49, H-1525 Budapest, Hungary, (2) Budapest University of Technology and Economics, Műegyetem rkp. 3, H-1111 Budapest, Hungary, (3) Institute of Energy Security and Environmental Safety, HUN-REN Centre for Energy Research, P.O. Box 49, H-1525 Budapest, Hungary benedek.kristof@ek.hun-ren.hu Synchronization is a ubiquitous phenomenon occurring in nature and our everyday life. One such type of system where synchronization is one of the main driving forces is power grids. In our recent studies[1,2], we investigated how synchronization behaves while parametrizing differently the same structural object. Conversely, we examined the impact of maintaining the same parametrization across different structural objects of identical size. As an underlying real-world network, we modeled the Hungarian high-voltage power grid to analyze these effects. One of our key findings is related to the behavior of different synchronization metrics. Several well-known metrics exist in the literature, each with varying sensitivity to system parameters. We compared some of these and evaluated their performance under gradual refinement of model heterogeneity and structural changes. We demonstrate that the universal order parameter proposed by Schröder et al. [3] is particularly useful in distinguishing synchronization differences between heterogeneous models and benchmarking systems with varying inertia. Meanwhile, the classical Kuramoto order parameter more effectively captures differences associated with coupling strength. Further findings reveal that networks with different structures but identical sizes can significantly alter the synchronization "landscape" when subjected to the same parametrization. This is closely related to the Braess paradox, where certain structural reinforcements can counterintuitively reduce overall stability. We find that modifications based on structural and dynamical analysis tend to enhance synchronization more effectively than random or cascade-size-related line additions [2]. These observations highlight the importance of targeted interventions when designing or upgrading power grids. Beyond these structural insights, our study also has implications for the resilience of power grids to perturbations. Understanding how different grid topologies respond to disturbances can inform reinforcement strategies to improve stability and robustness. Future research could extend our framework to include renewable energy sources, which introduce fluctuating power injections and increase the complexity of synchronization dynamics. Additionally, integrating economic feasibility considerations into these reinforcement strategies would enhance the practical applicability of our findings. Our results contribute to a deeper understanding of power grid synchronization by bridging structural and dynamical aspects. By refining models and using appropriate metrics, we aim to provide more accurate tools for assessing stability in real-world power grids. These insights could guide future developments in energy infrastructure planning, ensuring efficiency and resilience in increasingly complex electrical networks.

Sintering of granular powders impacts various industrial applications by forming sintered bridges between particles. When driven by surface tension, these bridges develop sequentially in order of increasing particle size. Identifying the connectivity percolation threshold is vital to predict, prevent, and control sintering. We investigate ranked percolation of granular packs, where particles connect by size. The percolation threshold, defined by the fraction of connected particles, is non-universal and highly sensitive to size dispersion. Conversely, the critical fraction of sintered bridges per particle is a robust estimator for connectivity percolation. Using numerical simulations and a mean-field approach, we elucidate this phenomenon based on the spatial distribution of contacts and discuss its broader implications, including the robustness of embedded networks against targeted attacks.

Transient dynamics and metastability play a crucial role in complex systems, where critical transitions can give rise to extreme events. These transitions often result from intricate, time-dependent interactions between subsystems rather than isolated tipping points. We here introduce the concept of tipping subnetworks -- dynamically evolving structures within networked dynamical systems -- that are involved in orchestrating transitions between metastable states. Employing the complex network approach we identify vertices and edges that not only act as tipping elements on an individual level, but also are intricately interconnected. This allows us to reveal how transiently emerging subnetworks mediate critical transitions. We demonstrate the existence of tipping subnetworks in excitable complex networks and in human epileptic brains. Our findings suggest that tipping subnetworks encapsulate the fundamental mechanisms underlying extreme events, offering novel insights into metastability and transient phenomena across complex systems.

Down syndrome (DS) disease encompasses various physical and neurological features, notably intellectual disability, with cognitive deficits linked to prefrontal cortex (PFC) dysfunction. Here, using spike data from high-density Neuropixel probes and tools from information theory, we examine differences in the communication among the PFC layers in awake trisomic mice, revealing important drops of information encoding in the Prelimbic cortex.

Drawing on the understanding of the logistic map, we propose a simple predator-prey model where predators and prey adapt to each other, leading to the co evolution of the system. The special dynamics observed in periodic windows contribute to the coexistence of multiple time scales, adding to the complexity of the system. Typical dynamics in ecosystems, such as the persistence and coexistence of population cycles and chaotic behaviors, the emergence of super-long transients, regime shifts, and the quantifying of resilience, are encapsulated within this single model. The simplicity of our model allows for detailed analysis, reinforcing its potential as a conceptual tool for understanding ecosystems deeply.

Imagine a set of devices connected electrically to each other in a network, as for example a power-grid. A short-circuit in one of the nodes may cause damage also to other nodes directly connected to it, making them unusable. Analogously, a physical attack on a site of a technological network may not only destroy it, but also disrupt activity in neighboring sites. To model these systems where the inactivation of some elements may damage their neighboring active ones we define a neighbor-induced damage percolation (NIDP) process. We present an exact solution for the size of the giant usable component (GUC) and the giant damaged component (GDC) in uncorrelated random graphs. We show that, even for strongly heterogeneous distributions, the GUC always appears at a finite threshold and its formation is characterized by homogeneous mean-field percolation critical exponents. The threshold is a nonmonotonic function of connectivity: robustness is maximized by networks with finite optimal average degree. We also show that, if the average degree is large enough, a damaged phase appears, characterized by the existence of a GDC, bounded by two distinct percolation transitions. The birth and the dismantling of the GDC are characterized by standard percolation critical exponents in networks, except for the dismantling in scale-free networks where new critical exponents are found. Extensive numerical simulations on regular lattices in D = 2 show that the existence of a GDC depends not only on the spatial dimension but also on the lattice coordination number, thus exhibiting qualitative differences between different lattice topologies in the same spatial dimension.

Background: In Kreuz et al., J Neurosci Methods 381, 109703 (2022) two methods were proposed that perform latency correction, i.e., optimize the spike time alignment of sparse neuronal spike trains with well defined global spiking events. The first one based on direct shifts is fast but uses only partial latency information, while the other one makes use of the full information but relies on the computationally costly simulated annealing. Both methods reach their limits and can become unreliable when successive global events are not sufficiently separated or even overlap. New Method: Here we propose an iterative scheme that combines the advantages of the two original methods by using in each step as much of the latency information as possible and by employing a very fast extrapolation direct shift method instead of the much slower simulated annealing. Results: We illustrate the effectiveness and the improved performance, measured in terms of the relative shift error, of the new iterative scheme not only on simulated data with known ground truths but also on single-unit recordings from two medial superior olive neurons of a gerbil. Comparison with Existing Method(s): The iterative scheme outperforms the existing approaches on both the simulated and the experimental data. Due to its low computational demands, and in contrast to simulated annealing, it can also be applied to very large datasets. Conclusions: The new method generalizes and improves on the original method both in terms of accuracy and speed. Importantly, it is the only method that allows to disentangle global events with overlap.

Non-equilibrium systems can be identified by the violation of the fluctuation-dissipation relation (as shown in references [1, 2]). Recently, the fluctuation-dissipation violation in neural systems has been studied in references [3, 4], where its connection to the asymmetry of interactions was demonstrated using both empirical human neuroimaging data and whole-brain models. These systems often display dynamical scaling, even if the stationary state is very far from being critical [5]. Criticality, which often emerges at continuous second-order phase transitions, is a common phenomenon in nature and plays a crucial role in enhancing the adaptability and functionality of complex systems. We investigate the distance from equilibrium using the Shinomoto-Kuramoto by measuring the magnitude of fluctuation-dissipation violation as the consequence of different levels of edge weight anisotropies. We achieve this by solving the phase oscillator equations on the raw, homeostatic weighted and random inhibitory edge variants of a fully grown fruit-fly [6] connectome. Our investigations were performed at close to synchronization transition critical points. By measuring the auto-correlations and the auto-response functions for small perturbations we calculate the fluctuation-dissipation ratios for different scenarios at different anisotropy levels. Evidence shows that these ratios deviate from constant values hat could otherwise be associated with an equilibrium temperature. This deviation follows the degree of anisotropies observed across different scenarios. [1] L. F. Cugliandolo, D. S. Dean, and J. Kurchan, Phys. Rev. Lett. 79, 2168 (1997), URL https://link.aps.org/doi/10.1103/PhysRevLett.79.2168. [2] U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, Physics Reports 461, 111 (2008), ISSN 0370-1573. [3] G. Deco, C. W. Lynn, Y. Sanz Perl, and M. L. Kringel bach, Phys. Rev. E 108, 064410 (2023), URL https://link.aps.org/doi/10.1103/PhysRevE.108.064410. [4] J. M. Monti, Y. S. Perl, E. Tagliazucchi, M. Kringelbach, and G. Deco, bioRxiv (2024), URL https://doi.org/10.1101/2024.04.04.588056. [5] M. Henkel and M. Pleimling, Non-Equilibrium Phase Transitions: Volume 2: Ageing and Dynamical Scaling Far from Equilibrium, Theoretical and Mathematical Physics (Springer Netherlands, 2011), ISBN 9789048128693, URL https://books.google.hu/books?id=AiofeEteLVcC. [6] Sven Dorkenwald, Arie Matsliah, Amy R Sterling, et al. Neuronal wiring diagram of an adult brain. bioRxiv, 2023.

Multiplicative (or parametric) noise is ubiquitous in real world systems, often playing a decisive role in state transitions. We provide a simple framework for the study of multiplicative noise, making use of scale parameters. We show that for a large class of stochastic differential equations increasing the multiplicative noise intensity surprisingly causes the mass of the stationary probability distribution to become increasingly concentrated around the minima of the multiplicative noise term, whilst under quite general conditions exhibiting a kind of intermittent burst like jumps between these minima. Our framework relies on first term expansions, which become more accurate for larger noise intensities. In this work we show that the full width half maximum (FWHM) in addition to the maximum is appropriate for quantifying the stationary probability distribution (instead of the mean and variance, which are often undefined). We show how the mean and FWHM can be obtained numerically far more efficiently than the moments. As a first application we consider how multiplicative noise may be beneficial for synchronisation of flagella of eukaryotes. As a second application we consider the role of such noise on tipping points in a double well potential.

In this research we introduce a mathematical framework to assess the impact of damage, defined as the reduction of weight in a specific link, on identical oscillator systems governed by the Kuramoto model and coupled through weighted networks. We analyze how weight modifications in a single link affect the system when its global function is to achieve the synchronization of coupled oscillators starting from random initial phases. We introduce different measures that allow the identification of cases where damage enhances synchronization (antifragile response), deteriorates it (fragile response), or has no significant impact. Using numerical simulations of the Kuramoto model in graphics processing units (GPUs), we investigate the effects of damage on network links where antifragility emerges. Our analysis includes lollipop graphs of varying sizes and a comprehensive evaluation and all the edges of 109 non-isomorphic graphs with six nodes. The approach is general and can be applied to study antifragility in other oscillator systems with different coupling mechanisms, offering a pathway for the quantitative exploration of antifragility in diverse complex systems.

We study the dynamic behavior in a vertical-cavity surface-emitting laser (VCSEL) subject to orthogonal optical injection through the computation of Lyapunov exponents and isospikes for a wide range of intervals in the plane of the injection parameters, i.e., the frequency detuning vs. injection strength plane. Our thorough numerical experiments on this plane constitute a deep quantitative analysis of the different bifurcation scenarios leading to polarization switching (PS). Firstly, we obtain similar results for the linearly polarized (LP) intensities for the different PS scenarios, especially when the injection strength is increased. It allows us to determine the parameter values that will be used for further analysis of the bifurcation scenarios in the parameter space. Analysis of different phase diagrams enables us to show multistability in the system and identify in the parameter planes several regions such as the predominantly chaotic lobe ones inside them are embedded some mainly regular structures such as spirals, rings, tricorns, shrimp networks, ``eye(s) of chaos'' and chiral and nonchiral distribution of periodicities characterized by sequences of quint points. We emphasize two routes to chaos, namely period-doubling and quint-point-based bifurcations.

Assessing complexity of time series and finding quantities that can be used as early warning signals of critical transitions is of primary importance in nonlinear science. Here we contribute to Markov chain-based methods to calculate properties of the general Rényi entropy spectrum of symbolic trajectories. In addition to the Sinai-Kolmogorov entropy $S$ (whose value is at $q=1$ on the spectrum), one can approximate the derivative of the spectrum $\Lambda$ at $q=1$ (here called the Lyapunov measure), which can change drastically near chaos-to-periodic transitions [1], where the dynamics before the transition is intermittent. These measures can be efficiently computed numerically from the transition probabilities between the symbols, after the original time series is transformed into a symbolic one. We test the performance of the proposed metrics using several symbolization schemes (phase space partitions, ordinal patterns), with relation to added noise and in cases where the bifurcation parameter is allowed to slide. We present applications to physiological data such as heart rhythm ECG in a sliding-window analysis setting to assess whether transitions to/from fibrillation are noise-induced or bifurcation-induced with intermittent dynamics. [1] Sándor, B., Rusu, A., Károly, D., Ercsey-Ravasz, M., Lázár I.Zs., Measuring dynamical phase transitions in time series, 2024, https://arxiv.org/abs/2407.13452