Abstract Spiral waves are a prominent form of spatiotemporal organization in excitable media. In brain networks, these waves are characterized by intermittent events separated by quiescent periods, which are not captured by models that generate sustained spiral activity. Here, we develop a stochastic reaction-diffusion model based on the Barkley framework to examine how transient waves emerge from a stable resting state. The model includes an activator variable, a recovery variable, a slow adaptation variable, and stochastic forcing. Numerical simulations reveal three dynamical regimes: a quiescent regime, a regime with sustained waves, and an intermittent regime where transient events emerge and terminate. In the intermittent regime, a subset of wave events exhibit broad phase coverage, rotational structure, and sustained phase winding. Linear stability analysis shows that the quiescent state is asymptotically stable, indicating that wave initiation does not arise from a local instability but from noise-driven threshold crossings. A first-passage approximation predicts an exponential dependence of initiation rates on noise amplitude, damping, and excitation threshold, in agreement with numerical simulations. Together, these results identify a novel stochastic threshold-crossing regime that combines stable quiescence, noise-driven wave nucleation, and slow adaptive recovery to generate intermittent wave dynamics, including events with transient spiral organization.
Abstract The future behavioural fate of a forced nonlinear system can depend sensitively on how it is forced, and on natural fluctuations within the system. This is especially the case if there is rate-induced tipping, where small changes in the forcing can lead to drastically different eventual behaviours. This sensitivity may be present only for a limited period of time, for example, when the forcing is changing most rapidly. We extend a recently proposed geometric early warning that can measure when a system is in such a sensitive state. We show how to compute the signed distance to an approximate R-tipping threshold, which we refer to as the R-tipping indicator. The R-tipping threshold is a dynamic state that embeds knowledge of the system and future behaviour of the forcing. As an example, we consider early prediction of the final state for a 3-box model of the Atlantic Meridional Overturning Circulation (AMOC) with specified and rapid forcing. For this model, we give a method to approximate the R-tipping indicator and estimate its skill in forecasting the eventual state of the system, even when the forcing is still underway. The skill of the geometric early warning based on the R-tipping indicator compares favourably to thresholds based on state variables or critical slowing down.
Higher-order contagion models show that group interactions can create tipping points: small outbreaks die out, while larger ones persist. In temporal systems, group interactions do not simply exist or not exist; they appear, disappear, and sometimes persist over time. This paper investigates when that persistence is strong enough to matter qualitatively. A minimal SIS contagion model with ordinary pairwise transmission and an additional triadic reinforcement mechanism is studied. The key assumption is that higher-order transmission requires a candidate triangle to be active in two consecutive observation windows. This gives a persistence-dependent reinforcement term derived from an explicit microscopic rule. Under a homogeneous mean-field closure, closed-form expressions are obtained for the bistability boundary, the critical persistence threshold, and the unstable seed-size threshold. The result is an analytical criterion showing when repeated small-group interactions are persistent enough to generate tipping-point epidemic behavior in the model. Finite-population simulations reproduce the persistence-controlled transition qualitatively, while explicit sparse-network simulations reveal a systematic reduction in realized triadic exposure compared to the homogeneous closure.
Critical slowing down (CSD) is a widely used indicator of resilience loss in systems approaching an abrupt transition. In this study, we analyse a paleo-climate run of the Parallel Ice Sheet Model that shows an abrupt collapse of the Baffin Bay ice shelf, investigating whether CSD signals are present in nearby ice sheet basins. We carefully assess the assumptions underlying CSD in the context of this non-idealised timeseries. We show that the assumptions can be considered sufficiently fulfilled, through the presence of stationary noise, an alternative stable state, a timescale separation between response and forcing and the likely existence of a fold-type bifurcation via the melt-elevation feedback. We find a clear and statistically significant increase in variance and autocorrelation in the North West basin, adjacent to the location of the abrupt ice loss, beginning around -9 ka. This is appropriately before the abrupt change and therefore provides a meaningful forewarning of the subsequent ice loss. Statistical analysis, across a wide range of detrending bandwidths and window lengths, confirms that this signal is not an artefact of parameter choices. Further analysis of the forcing aids us in rejecting the null hypothesis that the indicators reflect inherited forcing variability rather than genuine resilience loss. Our results demonstrate that meaningful early warning signals can be recovered from non-idealised model output, even when assumptions are not perfectly met. We argue that CSD-based methods can provide valuable information when analysing non-idealised model timeseries, but need to be applied with careful assumption testing, parameter sensitivity analysis and statistical rigour.
Strategic interactions between tumor cells have been hypothesized to arise through evolutionary processes in which cell actions are conditioned on their fitness payoffs. Temporal change in payoffs impacts cell strategies, leading to the evolution of unwanted traits and genetic variation. Consequently, understanding the stochastic game dynamics of tumor metabolism for fluctuating payoffs is an important problem. Here, we study a game-theoretic model of tumor metabolism with fluctuating payoffs. A key feature of this model is the metabolic exploration rate, a quantity that governs metabolic symbiosis in tumor cells and thereby affects tumor progression to metastasis. We find that intrinsic noise induces a transition from a dominance game to a coordination game for an increase in metabolic exploration rate. This transition coincides with a change in the payoffs for the tumor cells at the threshold for critical transitions of the corresponding stochastic system. As a result, our model suggests that variations in payoffs due to intrinsic fluctuations can lead to unexpected changes in tumor cell persistence, transitioning from dominance to coexistence. Additionally, we investigate some candidate early warning signals that we hypothesize may anticipate critical transitions in the stochastic tumor system, guiding therapy to decouple tumor metabolism and thereby prevent cancer proliferation.
We report a series of detailed statistical analyses on the distributions of waiting times pertaining to a diverse set of complex systems, including terrestrial and space weather, sea-level variations, currency trading (for both fiat and cryptocurrencies), and synthetic automotive data. Given a generic time series, we define a waiting time as the shortest time interval needed to find an entry of value of at least $A+\delta$ , with $\delta$ a given threshold, after a certain entry of value $A$ was observed. Going through the entire time series we obtain the complete set of waiting times for a specific value of $\delta$ and can determine their distribution. This distribution can be seen as a dynamic fingerprint of the process to which the time series pertains and is particularly useful to directly compare the dynamics of otherwise very different systems. To this end, we show that the aforementioned distributions have a prominent scale-free character for small values of $\delta$ for all datasets under scrutiny, while for large values of $\delta$ the observed distributions of waiting times converge to a Pareto–Tsallis distribution. We identify the threshold values $\delta$ at which this transition occurs using the goodness-of-fit indicators, and further substantiate these results by analyzing the behavior of the generalized Kullback–Leibler divergence. Our results are robust across all of the considered datasets.
For systems of identical oscillators with all-to-all phase-difference coupling, four classes of phase-locked solutions are guaranteed to exist under mild non-degeneracy conditions on the coupling function. The linear stability of three of these classes-in-phase, two-block, and rotating-block states-has been previously characterized. Here, we complete this analysis by determining the linear stability properties of the remaining class, the double rotating-block solutions. We also construct an explicit coupling function for which these solutions are linearly stable. These results complete the stability classification of guaranteed phase-locked states in this class of Kuramoto-like models and further demonstrate the breadth of stable clustered dynamics possible in globally coupled oscillator systems.
This paper reframes economic complexity metrics as centrality values in the Product Space. Instead of searching for the ‘best’ algorithm to calculate a superior complexity index, I propose utilizing different centrality values in a complementary fashion to explain export diversification and income. The linear economic complexity index, the non-linear country Fitness, and network degree and closeness centralities share a large common component, yet each retains a distinct aspect of a country’s position in the network. As a result, country income, diversification, and growth are each best explained by different centrality metrics. The paper argues that the explanatory power of complexity metrics stems from the connections an export basket holds in the Product Space. The centrality approach can strengthen the network implications of the Product Space for developing countries. Since the current location and opportunities of each developing country differ, a centrality approach can better provide country-specific development targets and policy prescriptions.
Scaling quantum networks to long distances necessitates the development of advanced quantum repeater (QR) technologies. Depending on the error mitigation mechanisms adopted to suppress loss and errors, QRs are typically classified into memory-based or all-photonic QRs; and each type of QR may be best suited for a specific type of underlying quantum technology, a particular scale of quantum networks, or a specific regime of operational parameters. This paper presents a comparative theoretical and simulation-based analysis of first-generation memory-based trapped-ion QRs and networks, and all-photonic entanglement-based QRs and networks. Our model incorporates a wide range of realistic parameters and noise sources to evaluate performance trade-offs. Major research findings of this study include: (a) the entanglement rate does not necessarily scale with the number of QRs. Although adding repeaters in a chain helps to reduce photon loss in quantum links, it also introduces overheads that can be attributed to imperfect photon collection, inefficient quantum frequency conversion, and limited detector efficiency. The entanglement rate only improves when the distance between QRs exceeds a certain threshold, specifically, when the reduction in photon loss in quantum links gained by adding QRs outweighs the inherent overheads they introduce. (b) Without quantum error correction and entanglement purification, the entanglement fidelity decreases with the number of QRs, regardless of the chain distance. Finally, the theoretical models and simulation software package developed in this study provide a versatile framework for the design and evaluation of future QR technologies, protocols, and architectures.
The notion of internal reliability in dynamical networks describes whether replicas of a particular unit follow the dynamics of the reference unit. Reliability and anti-reliability can be quantified by the transversal Lyapunov exponents. We study phase oscillators coupled via Kuramoto-Sakaguchi-type interactions. Already the simplest solvable system of two oscillators demonstrates nontrivial reliability properties. We present numerical evidence of reliability and anti-reliability in small networks with a uniform distribution of natural frequencies. The dynamics of an ensemble of replicas can be described within the Watanabe-Strogatz theory, which predicts symmetry of the transversal Lyapunov exponents for replica-attractor and replica-repeller.
We hereby correct mistakes in equations (49) and (51) of the original manuscript, Trugenberger (2025 J. Phys.: Complex. 6 042001).
Tipping phenomena represent critical transitions to alternative states, often irreversible, arising from parameter changes over time. Indirect tipping is a special kind of cascading dynamics observed in coupled systems which occurs when a downstream system collapses despite the upstream driver system remaining almost undisturbed in spite of being perturbed externally. We investigate this phenomenon in a seasonally forced, unidirectionally coupled ecological model consisting of two competing toxin-producing phytoplankton and a fish population with an Allee effect. The phytoplankton growth rates vary periodically with a phase lag phi, capturing distinct seasonal niches and both single and multiple bloom cycles. Although the phytoplankton system remains oscillatory and stable for all phi, variations in phase lag can induce tipping in fish population leading to a state of extinction. In particular, synchronous periodic growth of the two driving species ( phi=0) leads to complete tipping, whereas anti-phase variation of growth ( phi=pi) results in partial tipping in fish population. These results demonstrate that seasonal variation of two competing phytoplankton bloom with a phase lag and bloom periodicity critically govern indirect tipping in aquatic ecosystems. Furthermore, we apply the Fourier spectrum-based early warning indicators that reliably signal the impending transition in this periodically forced system.
Antifragility refers to the counterintuitive phenomenon whereby a system improves its global performance in response to damage. Although antifragile behavior has been reported in a variety of complex systems, the structural and dynamical mechanisms underlying this response remain poorly understood. In this work, we investigate antifragility in the synchronization dynamics of identical oscillators on networks, focusing on the effect of damage defined as the reduction in the weight of an edge connecting pairs of nodes in the network. Using a linearized description of the Kuramoto model with normalized couplings, we introduce a spectral measure of antifragility based on the response of the second eigenvalue of the normalized Laplacian. This framework enables an analytical and numerical characterization of how local reductions in coupling strength can accelerate global synchronization. Through first-order perturbation theory, we show that the sign and magnitude of the response are controlled by the structure of the slowest relaxation mode. In networks with modular organization, this mechanism leads to a systematic distinction between intra- and inter-community links: damage to intra-community edges is typically antifragile, whereas damage to inter-community edges is fragile. These predictions are validated by numerical simulations on deterministic, random, and real-world networks with community structure. Overall, our results establish a direct connection between spectral properties, modular topology, and antifragility in synchronization processes.
Identifying key nodes is crucial in applications like epidemic control and information dissemination. Many conventional methods rely primarily on local or global topological properties, overlooking the role of intermediate higher-order structures, such as network motifs, particularly in directed networks where these patterns can reveal nuanced functional roles in propagation. To address this, we propose the Roles within Motifs (RM) method for identifying key nodes in directed networks. In this framework, roles within motifs are defined as $recipient$ or $disseminator$ , and a node’s propensity to act as a $disseminator$ within motifs is quantified to capture its local higher-order structural information. Experiments on real-world networks show that RM significantly outperforms eight benchmark methods, with a 17.08% reduction in average ranking deviation in single-source propagation. Moreover, it attains near-optimal performance in multi-source propagation, demonstrating robust efficacy across tested conditions. Furthermore, the method has low computational complexity, making it suitable for large-scale network analysis.
The Atlantic Meridional Overturning Circulation (AMOC) is a climate-relevant ocean current system responsible for the meridional heat transport in the Atlantic. The AMOC strength is affected by a meridional density difference, where the density in the northern North Atlantic is controlled by an advective and a convective feedback. Here, we introduce and study a conceptual mathematical model of the variability of the AMOC strength by representing these feedbacks as delayed terms in a scalar delay differential equation (DDE) for the salinity in the northern North Atlantic. After scaling and without external input, this DDE has the associated delay times tau and sigma as its only parameters. We perform a numerical bifurcation analysis of this deceptively simple-looking DDE AMOC model with the continuation software package DDE-BifTool. We find and characterize intricate dynamical regimes, including those exhibiting complicated oscillations associated with homoclinic connections. These results are presented as bifurcation diagrams in the (tau,sigma)-plane, where we identify a codimension-two Belyakov transition as an organizing center for nearby complicated dynamics. Moreover, we present a detailed analysis of different attractor regions in the (tau,sigma)-plane, which we identify by computing the (strong) unstable manifold of a physically relevant equilibrium. As a general picture, we find that attractor regions repeat as the values of either sigma or tau increase, including in physically relevant regions of these two (scaled) delay times. In this way, we clarify where different types of dynamics - such as periodic orbits of different periods, invariant tori, and chaotic dynamics - can be observed, and how they emerge or disappear.
This study introduces the Large Network Generator, an algorithm capable of creating undirected graphs with three main characteristics of real-world networks: long-tailed degree distribution, short distances between nodes (small-world phenomenon), and large clustering coefficients. The main idea is to add one node to the network at each step, perform random walks on the existing nodes, and select a subset of marked nodes to connect to the new node. Additionally, with an adjustable probability, edges may be created between the marked nodes themselves. Key advantages of our algorithm are its simplicity, efficiency, and flexibility in creating networks with different characteristics without using global information about network topology. The parameters can be adjusted to generate networks with specific characteristics, producing a wide range of values for each parameter, including high clustering (up to C & horbar;approximate to 0.7), markedly short average distances (as low as L & horbar;approximate to 2.8 for N=200000), and a strong presence of hubs. Additionally, the algorithm's near-linear time complexity allows the creation of networks with one million nodes in less than 60 s. The implementation of our algorithm is publicly available on a GitHub repository.
On the fiftieth anniversary of the Kuramoto model, synchronization remains a central paradigm for collective behavior in complex oscillator systems. While phase locking, frequency synchronization, and order-parameter coherence are widely used to characterize synchronization, their precise dynamical relationship has remained unclear, especially in finite-dimensional systems beyond mean-field limits. In this work, we show that, in the fully connected Kuramoto model, full phase locking, phase locking, frequency synchronization, and order-parameter synchronization are dynamically equivalent. Moreover, we demonstrate that the equivalence between phase locking and frequency synchronization is governed by a topology-independent mechanism and persists for generic symmetric coupling structures. These results provide a unified dynamical framework for synchronization and clarify the structural role of the Kuramoto order parameter.
Network science is a powerful framework allowing to model complex systems, it is capable to describe and take into account the intricate web of connections existing among the constituting basic element of the system. Recently scholars have brought to the fore the relevance of higher-order networks, namely structures allowing to encode for many-body interaction, differently from the pairwise case handled by networks. This novel research field opens new avenues of research with applications ranging from neurosciences to social sciences; there is thus a need for generative models of higher-order network capable to reproduce features present in empirical data. In this work we present a model for growing simplicial complex rooted on a preferential attachment process acting dimension-wise, i.e., returning a power law distribution for the generalized degree of simplexes of different dimension.
The Atlantic Meridional Overturning Circulation (AMOC) is a key climate tipping element, regulating the meridional transport of heat and freshwater. Its future evolution depends on the interaction between external forcings, such as greenhouse gas increases, and internal climate variability. Given the limitations of deterministic predictions, we apply the trajectory-adaptive multilevel splitting algorithm within the intermediate-complexity model PlaSim-large scale geostrophic to estimate AMOC transition probabilities under different CO2 concentration levels and Shared Socioeconomic Pathway (SSP) scenarios. Results show a non-zero (1%-7.5%) probability of a strong AMOC weakening within 150 years for CO2 levels between 500 and 600 ppm. While the transition is exceptionally unlikely within the 21st century under SSP5-8.5, it becomes unlikely (20%) by 2150 and very likely (95%) by 2200. Notably, the model excludes Greenland meltwater input, suggesting these probabilities may be underestimated. These findings highlight the importance of probabilistic approaches and the value of rare-event techniques in assessing AMOC stability and potential climate tipping points.
As tipping points are extremely difficult to predict using an initial value modelling approach, forewarning of bifurcation tipping points instead often depends on the analysis of observational data, using approaches that aim to detect reducing system resilience. The most commonly used early warning indicators (EWIs) rely on the phenomenon of critical slowing down, which is the tendency for fluctuations of a state variable to get larger (increased variance) and longer lived (increased temporal autocorrelation), as the bifurcation is approached. This is measurable in low dimensional systems that remain close to a quasi-equilibrium state in the run-up to the bifurcation. However, in systems that are subject to rapid changes in external forcing, such as the contemporary Earth system, this condition is unlikely to be met and EWIs become less reliable. In addition, temporal EWIs require long observational time series. For these reasons, it makes sense to consider spatial EWIs that can make use of the spatial detail resolved by present-day observations, especially from remote sensing. In this paper, we explore the conditions under which spatial and temporal EWIs will each be reliable, using a simple spatially coupled model with a fold bifurcation. In the weak coupling limit, we find spatial EWIs give reliable warning of the bifurcation, even if the external forcing is changing rapidly. However, in the limit of strong spatial coupling, spatial early warning disappears. Under these conditions, we find temporal EWIs give reliable warning as long as the external forcing is not changing quickly compared to the characteristic timescale of the system. We conclude that spatial early warnings will be especially useful in systems that have relatively weak spatial coupling, but which are also subject to rapidly changing external forcing (e.g. forests under contemporary climate change).
