Papers

We test whether the Kuramoto model quantitatively describes spatial synchronization in regional epidemic waves, using daily COVID-19 incidence for 400 German Kreise (2021–2023) and weekly ILI rates for 12 European countries (ECDC, 2021–2026). Bandpass filtering and Hilbert-transform phase extraction yield high global order parameters ($\langle r\rangle\approx0.87$ for Germany; $\langle r\rangle\approx0.97$ for Europe), yet both are dominated by a common-mode driver rather than pairwise local coupling. Removing the common mode reveals genuine short-range synchronization: an exponentially decaying residual local order with correlation length $\xi=152\pm6$~km in Germany, and a significant neighbor-vs.-non-neighbor contrast (Mann–Whitney $p=0.005$) in the European panel. We interpret this convergent pattern across two diseases and two spatial scales as a universal two-scale structure: a long-range common-mode field superimposed on shorter-range diffusive coupling. Cross-validation against a companion gauge-mediated Doi–Peliti framework shows that district-level effective screening mass $m_R$ is uncorrelated with local~$r$ (null result), the national $(m_R,r)$ phase portrait independently reproduces a previously reported hysteresis loop, and pairwise phase lags quantitatively match fractional-calculus predictions. A frequency–degree correlation provides the structural signature of explosive synchronization; excess synchronization between districts sharing the same Bundesland implicates shared school-holiday calendars and state-level policy as local coupling drivers.

In a mechanistic model of the dawn chorus, Kaye showed that heterogeneous activation thresholds and a shared feedback signal determined by the population's active fraction can produce abrupt collective activation and hysteresis. We extend this mechanism to a network of interacting agents. Each node has a continuous activation level, and a nonnegative row-stochastic matrix determines how node activities contribute to individual feedback. We prove that sufficiently weak feedback yields a unique globally attracting equilibrium. For all feedback strengths, the homogeneous dynamics exactly reproduce Kaye's scalar equation; consequently, network topology does not alter the folds or cusp of the homogeneous branch, and no heterogeneous mode becomes unstable before the homogeneous mode. For equitable partitions, the network admits an exact quotient system in which nodes within a block receive the same aggregate input from every block. When blocks are uncoupled, the quotient reduces to independent copies of Kaye's scalar equation. We show that every assignment of stable scalar equilibria to blocks persists under sufficiently weak interblock coupling, producing a combinatorial family of stable quotient equilibria that lift to stable full-network equilibria and remain under small perturbations that break exact equitability. For two symmetrically coupled blocks, branches with unequal block activities terminate at a pair of symmetry-related cusp bifurcations. Near the onset of bistability, we derive scaling laws for the interblock coupling at which these bifurcations occur, the activity difference between the blocks at bifurcation, and the corresponding shift of the external stimulus from the scalar cusp. Numerical continuation confirms the scaling laws for gamma, logistic, and normal threshold distributions.

Synchronization can be observed in various systems in physics and biology. The choruses of male frogs are known as an example of biological synchronization in which the well-organized temporal structure, i.e., anti-phase synchronization between neighbors, is realized. Given that male frogs produce sounds to advertise their territories to competitors, the dynamics of phases and spatial coordinates should be mutually coupled in the frog choruses. In this study, we examined the spatio-temporal dynamics in the choruses consisting of male Japanese tree frogs and other acoustic animals. First, we carried out playback experiments using actual frogs and observed that male Japanese tree frog synchronized in anti-phase with the stimuli of a similar frequency but did not synchronize with the stimuli of a much different frequency. Second, we modeled the choruses with two species as a system of coupled mobile oscillators and numerically evaluated how the spatio-temporal structure depends on the similarity of call frequencies. Numerical simulations of the model showed that (1) the two-cluster antisynchronization is established in the same species when the distributions of call frequencies are much different between two species and (2) the two-cluster antisynchronization is disturbed when the distributions of call frequencies are similar. These results highlight the occurrence of various spatio-temporal patterns in the proposed model, indicating the importance of repulsive effects with different weights on the variation of the spatio-temporal patterns.

Massive jellyfish swarms observed at sea can extend over tens of kilometres and contain millions of individuals, yet the mechanisms governing their formation and large-scale dynamics remain poorly understood. Agent-based models provide a framework for describing this dynamics based on jellyfish responses to ocean currents and environmental cues, but become computationally prohibitive when extended to large populations and spatial scales relevant to ocean circulation. Here we derive a continuous kinetic theory from an active-particle model of jellyfish motion. The resulting Fokker-Planck framework incorporates transport by prescribed currents, stochastic reorientation, direct interactions and stimulated steering, allowing chemical signalling to be represented through a coupled field. We further derive a hydrodynamic closure for large swarms by exploiting the separation between fast orientational and slow spatial dynamics, yielding a reduced density equation suitable for implementation in ocean-current models. This framework provides a route from individual behavioural mechanisms to continuum descriptions of jellyfish populations and establishes a basis for constraining model parameters using observations and in-situ measurements. This approach offers a theoretical foundation for future numerical prediction of large jellyfish swarm formation and evolution in realistic ocean flows.

Higher-order networks exhibit rich critical phenomena that cannot be captured by traditional pairwise models. Here, we develop an analytical dimension-reduction framework that maps higher-order networked dynamics onto an effective low-dimensional system, allowing accurate prediction of tipping boundaries, bistability regions, and the nature of phase transitions. We demonstrate the power of this framework across a range of dynamical processes, revealing distinct effects of higher-order interactions on transition continuity and hysteresis. Furthermore, we find that system resilience exhibits a profound dependence on the alignment between pairwise and higher-order connectivity, with assortative mixing enhancing tipping toward active states. Our findings establish a general theory for understanding the critical transitions in higher-order networks, offering new insights for anticipating and managing systemic risk in complex systems.

We study the stability of collective neutrino oscillations using a distributional approach motivated by the statistical mechanics of Kuramoto synchronization. Treating the ensemble of neutrino flavor polarization vectors in the thermodynamic limit $N\to\infty$, we derive an exact nonlinear Fokker–Planck (continuity) equation for the one-body distribution $F(\vec{\mathbf{S}},\omega,t)$ on the flavor sphere. This equation admits a two-parameter family of azimuthally symmetric stationary solutions, whose stability we analyze by linearizing around them. The resulting eigenvalue condition determines the growth or decay rate of small perturbations from any initial distribution – not merely from a state close to full flavor coherence – thereby going significantly beyond the conventional linear stability analysis of collective modes. In special limits the condition reproduces known synchronization thresholds in the two-beam model, providing a non-trivial check of the framework. We present analytical results for the eigenvalue equation and explore stability phase diagrams for physically relevant frequency distributions.

From protein complexes to electronic circuits, many natural and engineered systems function only if assembled in an exact, reproducible fashion. The structure of each of these systems can be understood as a network, yet network science lacks the mechanisms to consistently reproduce exact topologies, focusing instead on generating network ensembles. We introduce the framework of network design where we encode the local constraints obeyed by a system's building blocks in a design set, and derive the Unigraphical Design Theorem, which determines when these constraints guarantee reproducible assembly into a unique structure, a process we call unigraphical assembly. For systems whose design sets do not specify a unique outcome, we identify guided assembly as a second route to reproducibility, in which temporal ordering decomposes construction into unigraphical steps. Applying these results to 3,618 reproducible systems, including protein complexes, molecules, and robots, we classify those that undergo unigraphical assembly and those that require guided assembly. We further identify a diversity-redundancy boundary that explains how systems trade component variety for structurally interchangeable parts while retaining unique assembly. Finally, we experimentally test the theory using 3D-printed components to re-engineer generative construction sets into systems that assemble unigraphically into prescribed topologies. Network design thus reframes reproducibility as a mathematically testable property of real networks, opening a route to the rational engineering of complex systems.

Large-language models (LLMs) are rapidly becoming part of human culture, reshaping how information is produced, transmitted, and used. Here we propose that their diffusion can be understood through a viral analogy, with LLM use spreading through populations, becoming embedded in cognitive and cultural practices. We model transitions among uncoupled, coupled, and persistently dependent users, and show that the interplay between social transmission, recovery, and collective reinforcement can generate tipping points and technological lock-in. A central consequence is the possibility of runaway dynamics: once a critical threshold is crossed, small increases in adoption can trigger rapid population-level shifts toward persistent dependence, with abrupt losses in cognitive competence. The same framework, however, identifies conditions for cognitive immunization, based on reducing transmission and facilitating reversibility. Our results highlight how LLM adoption may involve nonlinear collective transitions with important consequences for cognitive autonomy.

Clusters that merge and split have no unique genealogy. Tracking the host size of a conserved constituent defines $q_*$ by $\langle(S_{t+\tau}/S_t)^{q_*}\rangle=1$, with $q_*\ne0$. Across coupled map networks, coagulation-fragmentation dynamics, and Vicsek systems, $q_*\simeq\alpha-2$ without fitting, where $c(s)\propto s^{-\alpha}$ on the same size window. Radius scans locate the known Vicsek interaction radius and recover an independently specified clustering scale in experimental active rods. Local scale covariance yields the relation, while a tent map counterexample shows when reducing the stationary balance at each parent size to one moment fails.

This chapter reconstructs the Hopfield network as a physical theory of memory rather than merely an early neural-network algorithm. It begins with the problem as it stood before 1982-threshold logic, Hebbian association, correlation memories, and recurrent binary networks-and isolates what Hopfield's synthesis added: a dynamical definition of content-addressable memory, a symmetric recurrent architecture with a Lyapunov function, a Hebbian embedding of patterns in its couplings, and a physical account of basins, robustness, and graceful degradation. The binary and graded-response energy functions are derived in full, together with the signal-crosstalk decomposition governing pattern stability, the mean-field theory of retrieval at extensive load, and the zero-temperature retrieval spinodal at (alpha 0.138) established by Amit, Gutfreund, and Sompolinsky. The energy-based program is then followed through analog optimization networks, polynomial dense associative memories, exponential interactions, and modern continuous Hopfield updates, including the precise conditions under which the update becomes scaled dot-product attention. Throughout, capacity claims are tied to their disorder ensemble, scaling limit, and success criterion, showing why numerically different storage limits need not conflict. A closing assessment distinguishes established results from surviving principles, assumption-bound limitations, and open problems, treating the Hopfield network as an effective theory whose symmetry, locality, and point-neuron assumptions delimit its biological reach. Fixed-seed numerical experiments expose the mechanisms discussed but do not substitute for analytical results.

We study how network connectivity and heterogeneous phase-delays shape the spatiotemporal dynamics of finite oscillator networks. Phase-delays can destabilize global synchronization and promote phase-locked patterns, including states with uniform phase gradients and more complex combinations of these modes. Yet, how connectivity and phase-delays jointly determine which states the network selects remains unclear. Here, we show that the spectrum of a composite matrix, which combines connectivity and phase-delays, governs not only the linear stability of the network's collective states but also their basin sizes. This, in turn, enables analytical estimates of basin size of phase-locked states for individual networks from connectivity and phase-delays alone. Applying this framework to nonlocal and global networks, including cases with random phase-delays, we uncover multistability and strong asymmetries in basin sizes, revealing chiral dynamics that conventional stability analysis cannot detect.

A population of $N$ identical pulse-coupled oscillators ultimately settles into one of two outcomes: full synchrony or a state of co-existing synchronized clusters. We show that which outcome occurs is controlled by the prime factorization of $N$. At the critical charging curve — linear, the boundary between the synchronizing and clustering regimes — the synchronization basin acquires exact arithmetic structure. The synchronization probability is $\Psync=A_{N,1}/N^N$ for all $N$, where $A_{N,1}$ satisfies an exact recurrence relation. For prime $N$, $A_{N,1}=N^N-1$ giving the closed form $\Psync=1-1/N^N$; for composite $N$, the observed asymptotic scaling is $1-\Psync\sim C_m N^{-(m-1)}$, where $m$ is the smallest prime divisor. The result adds a new member to the atlas of exotic basin geometries: alongside fractal, riddled, and tentacled basins, we now have a basin that is arithmetic.

Ensuring a sustainable energy supply requires maintaining power-grid stability. Rotor dynamics are governed by the swing equation, which takes the form of a second-order Kuramoto model with a correlation between power and total coupling strength. Yet the microscopic mechanisms that nucleate and propagate synchronized clusters remain poorly understood. Using a minimal model motivated by empirical grid data and the \Adhoc potential method, we reveal two distinct seed cluster types and propagation pathways: a population-driven seed cluster at the center of the power distribution propagating to its tails by rotor accretion, and, for a symmetric distribution, coupling-driven seed clusters at its tails propagating inward to the center by cluster merger. These differences generate distinct order-parameter patterns, while inertia controls whether the seed clusters persist with distinct angular velocities. The same propagation pathways also govern recovery following external disturbances. We further confirm that the same selection–persistence rule holds in data-derived annealed representations of European power grids. Therefore, our results can inform strategies for sustaining stable power-grid operation. More generally, our pathway-based framework reframes synchronization by emphasizing the dynamics of cluster formation and recovery rather than relying solely on static criteria.

Many natural and engineered systems are large complex networks of interacting components, and external perturbations drive them along different dynamical paths. Identifying which perturbation occurred matters for diagnosis, control, and prediction. Yet often times only a few components can be jointly monitored. Which components should be monitored? Experimental practice usually favors placing reporters at the most sensitive sites, where perturbations produce the largest effects. Using a simple dynamical model for complex systems with heterogeneous connectivity, we ask how sparse reporter panels should be chosen to classify shocks from partial trajectories when repeated trials only approximately reproduce an ideal initial condition. Once that reproduction is imperfect, sensitivity ranked panels fall far short of optimal, and the shortfall grows with the noise. We find that the best panels mix two kinds of reporters. A promiscuous reporter responds to most shocks, so it separates them mainly by degree, and degree fluctuates from trial to trial. A dormant reporter responds to only a few shocks but strongly, and its answers do not scatter as much between trials. As noise grows, the cost of losing a dormant reporter rises to meet the cost of losing a promiscuous one. Panels of either kind alone classify worse than the mixture, and no property of the members collected individually explains the ordering. Most of all, we find that only a minuscule number of reporters are needed on a panel to accurately identify which shock hit the system. We implement an efficient algorithm to assemble such a panel. Together these results provide a low cost practical design principle for monitoring large complex dynamical systems.

Closed-loop human-AI systems generate high-dimensional behavioural trajectories whose collective dynamics remain obscure. Using 297,915 learners' adaptive-tutoring histories, we define semantic order variables before model fitting and test them in user-disjoint cohorts. The state exhibits reproducible basin-like flow and operationally defined, state-heterogeneous metastable-like kinetics. A construction-matched null distinguishes normalised-memory relaxation from a reproducible excess field. A four-term conditional mechanism recovers population drift (r = 0.946; learner-bootstrap 95% CI, 0.935-0.955). Predictive event-level self-supervised learning recovers the state and learned-plane flow; null-referenced corrections retain directional, partial-amplitude excess-field structure without full calibration. Shuffled-order training reverses learned-plane flow on ordered trajectories; support-alignment randomisation selectively reduces inward transport. Both axes remain linearly accessible without state supervision. Without cross-model fitting, the models share leading population drift (r = 0.866; learner-bootstrap 95% CI, 0.857-0.875) and persistence ordering; residual directions remain model-specific. These results identify an externally anchored leading-order effective field linking empirical dynamics, an interpretable mechanism and neural computation.

Physical learning lets a trainable material or network use its own physical response to carry error signals, reducing the need for a separately programmed backward computation. We ask what determines whether such a system follows conventional gradient descent or evolves along a genuinely different learning trajectory. Our canonical model is a directed layered transport network in which every node redistributes a fixed amount of flow, so learning preserves positivity and total mass. In this model, conservation constrains only the allowable learning directions. Within the matched response class studied here, adjoint matching gives the physical output response a symmetric form. Non-negative mode-wise feedback then produces a reciprocal closed-loop response and a reweighted gradient flow. Adding an antisymmetric boundary component makes the closed-loop response rotational: the learning path can turn while the error driving that update still decreases at that moment. Turning is not automatically beneficial. Its finite-step effect is set by local curvature, and its accumulated effect also depends on step selection and on the new states visited along the path. Numerical consistency checks reproduce the exact response structure, predict the sign of the local effect across new network families, and show how trajectory drift can negate a local advantage. These results separate the roles of conservation, reciprocity, and nonreciprocity in physical learning.

Large language models (LLMs) are increasingly adopted as closed-loop controllers in physical multi-agent systems, yet their emergent collective dynamics remain incompletely characterised. We deploy 22 LLM agents as direct, real-time target-speed controllers (per 0.5 s cycle, with IDM as collision-avoidance clamp) on a 230 m ring road under the Sugiyama 2008 paradigm, reproducing human-like stop-and-go waves. Six matched controls spanning stochasticity (white noise, OU noise, temperature), population variance, and dynamical instability (delay, OV model) are systematically excluded. The surviving phenomenon, termed Sustained Heterogeneity (SH), is the persistent, approximately temperature-insensitive (approx. 8 percent across a 6x T sweep), per-cycle divergence in LLM-chosen target-speed adjustments, propagating through a three-stage cascade of drift, gap erosion, and nonlinear braking. Across four traffic densities, the critical LLM penetration fraction p_c decreases monotonically from no transition at density 43.5 veh/km to p_c approx 0.23 at density 95.7 veh/km, consistent with an initiation-threshold model governed by trigger distance, stochasticity, and fleet size. Chain-of-thought analysis of 39,600 decisions across three seeds shows agents engage in multi-factor safety reasoning, yet systematic divergence persists, implying stability must be enforced at the dynamics layer. This is the first study to identify a previously uncharacterised collective mechanism in LLM-controlled traffic and map a density-dependent phase boundary p_c(rho).

Synchronization is conventionally understood as the emergence of a phase-locked collective state among interacting dynamical units. However, extending this notion to systems whose dynamical variables are defined on higher-dimensional simplices introduces fundamentally new constraints arising from the topology of the underlying simplicial complex. In the simplicial Kuramoto model, nontrivial topological cycles give rise to a higher dimensional harmonic subspace that is unaffected by the coupling and can therefore drift indefinitely, preventing a globally synchronized state. To resolve this we employ Hodge decomposition on simplicial Kuramoto dynamics and investigated the components. Despite this topological obstruction, we show that the simplicial Kuramoto model admits fixed-point states in the exact and coexact sectors of the Hodge decomposition above critical coupling strengths, which we derive analytically for both sectors. This decomposition provides a generalized notion of synchronization in which the non-harmonic components converge to fixed states. We further investigate the robustness of these fixed-point states to external perturbations. Excluding the drifting and non-interacting harmonic component, the perturbation response is governed by the spectrum of weighted Laplacian and recovers a Kirchhoff index dependence analogous to standard Kuramoto case. In contrast, the harmonic sector is non-dissipative, and consequently the fragility of the system increases with the dimension of this topological subspace. We characterize this effect numerically using triangulated tori with varying first Betti number and find a superlinear scaling relation between system fragility and the dimension of the harmonic sector.

Anticipating the onset of collective synchronization is important in many networked systems, yet observing every oscillator is often impractical. We investigate whether synchronization transitions can be detected from a small set of monitored, or sentinel, nodes. Using a stochastic Kuramoto model on networks, we numerically compare three early warning signals: the local order parameter, its temporal variance, and the variance of individual oscillator phases after removing their mean rotational trends. We also compare sentinel-selection strategies based on node dynamics, degree, and random sampling. We show that, under partial observation, the local order parameter and the variance of detrended phases provide substantially stronger warning signals than the variance of the local order parameter. Selecting nodes according to their dynamical behavior near the transition consistently improves performance over other sentinel-selection methods. With only $\lfloor\ln N\rfloor$ dynamically selected sentinels, two success indicators approach the performance obtained by observing all $N$ nodes. These results demonstrate that synchronization transitions can be anticipated from sparse observations when the warning signal and monitored nodes are chosen appropriately.

Collective behaviour in living systems is usually modelled as the outcome of a direct social drive: agents are rewarded, or hard-wired, to align with or approach their neighbours. Here we show that aggregation can instead emerge from an indirect objective. We let reinforcement learning foragers, initially performing a random walk, optimize their dynamics from a purely individual reward for finding replenishable targets, while perceiving only their conspecifics and never the targets themselves. As the visual range grows, the agents undergo a sharp crossover from an environment-tuned individual search to a scale-agnostic collective one, and this crossover coincides with the onset of spatial aggregation. Thus a collective phase arises as a by-product of optimal foraging, without any direct reward for grouping. A minimal analytical first-passage model reproduces the transition as a crossover between the two search strategies. Our results identify indirect, resource-driven reward as a generic route to emergent collective phenomena.

How do organisms navigate when the signals guiding them are noisy? Variance adaptation, the rescaling of sensitivity to noise, is common in sensory systems, but its role in navigation is unexplored. We introduce a minimal active Brownian particle whose run-and-tumble policy follows from an optimality principle. Variance adaptation emerges as part of this policy. Adaptation keeps chemotactic drift finite as noise grows, while a non-adaptive particle's collapses exponentially. Adaptation also carries a cost, degrading performance in quiet environments and requiring a tuned adaptation sensitivity.

Low-dimensional reduction theories such as the Ott-Antonsen ansatz have played a crucial role in the study of populations of coupled oscillators. Their application, however, has largely been restricted to systems with frequency distributions of rational-function form, such as the Cauchy distribution. For such distributions, the residue theorem allows the dynamics of the global order parameters to be closed in terms of a finite number of poles, thereby yielding a finite-dimensional system of ordinary differential equations. Rational frequency distributions, however, generally have heavy tails and only finitely many well-defined moments, and therefore may not always be realistic as frequency distributions. In this paper, we develop an approximate low-dimensional reduction theory based on perturbation theory for weakly heterogeneous populations of phase oscillators with a Gaussian frequency distribution. We construct the theory not only for first-harmonic coupling, for which the Ott-Antonsen ansatz applies, but also for populations of phase oscillators with multi-harmonic coupling. The effectiveness of the proposed low-dimensional reductions is demonstrated through both theoretical analysis and numerical simulations.

We demonstrate a physical mechanism for causal information filtering in a physical reservoir computing (PRC) by exploiting asymmetric coupling anisotropy. Using a network of coupled Duffing oscillators, we show that the directionality of internal coupling induces a spatial gradient in the effective potential, establishing a deterministic upstream-to-downstream information flow. This anisotropy allows for the selective amplification of semantic drifts, triggering a macroscopic saddle-node bifurcation as a physical interlock before global computational failure. Through spatiotemporal analysis of a 50-node system under traveling wave inputs, we confirm that local phase transitions effectively purge anomalous information while preserving the computational integrity of the remaining nodes. The results suggest that the intrinsic causality of the reservoir's topology provides a robust framework for autonomous reliability and fault-tolerant physical intelligence.

Mixed-mode oscillations (MMOs), characterized by the alternation of small-amplitude oscillations (SAOs) and large-amplitude oscillations (LAOs), and bursting oscillations are common forms of complex oscillatory dynamics observed in systems with multiple timescales and have been widely studied across scientific disciplines. Mixed-mode bursting oscillations (MMBOs) combine features of MMOs and bursting, with LAOs organized into burst events. Most existing studies treat MMBOs as two-timescale phenomena, identifying distinct geometric mechanisms depending on how the timescales are grouped. In this work, we use a three-timescale biophysical cortical neuronal oscillator model to demonstrate that a three-timescale implementation of geometric singular perturbation theory (GSPT) provides stronger predictive insight into MMBO dynamics. This perspective unifies mechanisms previously identified from fast-slow analysis, while revealing the canard-delayed-Hopf (CDH) singularity as an organizing center for MMBOs near the singular limit. We perform a detailed bifurcation analysis of the full eight-dimensional model to determine how MMBOs are organized along families of isolas. We then combine GSPT with full-system bifurcation analysis to show how tuning the relative timescales induces transitions among MMOs, MMBOs, and bursting dynamics. Our results highlight the importance of studying MMBOs from a three-timescale perspective and demonstrate that combining GSPT with full-system bifurcation analysis can reveal organizing structures and mechanisms that are not apparent from either approach alone.

In the Derrida–Higgs model of sympatric speciation, a sexually reproducing population with finite binary genomes can mate only when the genetic overlap between two individuals exceeds a threshold $q_{\min}$. Depending on the genome length $L$, the population either remains genetically connected or fragments into reproductively isolated species. A central problem is therefore to predict the critical genome length $L_c$ at which fragmentation begins. At finite $L$, fluctuations broaden the overlap distribution and allow genetically distant regions of the population to remain connected. A previously proposed transient variance criterion captures this effect, but its evaluation requires numerical iteration of coupled moment equations. Here we first correct the unrestricted moment equations by removing a previously implicit assumption and then derive an explicit closed form expression for $L_c$. The resulting formula shows that the critical genome length is determined, at the time the mean overlap reaches $q_{\min}$, by the competition between deterministic separation from the unrestricted equilibrium and the transient genealogical variance of the overlap distribution. This expression permits a systematic asymptotic analysis. When the deterministic contribution dominates, $L_c$ becomes essentially independent of the population size $M$ and scales as $\mu^{-2}$. When the transient genealogical variance dominates, $L_c$ grows as $M^{3/2}$ or as $\sqrt{M}/\mu$, depending on how $M$ and the mutation rate $\mu$ jointly vary. Simulations support all predicted behaviors. Our results identify transient genealogical variance as the principal mechanism linking finite genome fluctuations to the onset of reproductive fragmentation and provide a practical analytical prediction for the critical genome length.

In oscillator networks sustained by finite resources, synchronization can depend on both the total resource and its spatial distribution. We study a duplex system whose activity layer consists of chaotic Hindmarsh–Rose oscillators and whose transport layer redistributes a conserved resource through a degree-biased Markov process. The stationary resource field is characterized analytically, and its existence, uniqueness, and convergence are established. By embedding this field into a local adaptive feedback law, the available resource is converted into node-dependent dissipation, for which Lyapunov analysis guarantees convergence to the synchronization manifold. Numerical results show that topology bias reorganizes the transient route to synchronization. Weak bias produces an almost collective contraction, whereas intermediate bias creates a hub-initiated recruitment hierarchy that extends toward middle-degree and peripheral nodes. Under stronger bias, the degree hierarchy becomes more pronounced while peripheral recruitment slows because adaptive dissipation is concentrated on structurally privileged nodes. Across the explored parameter range, this localization–coverage tradeoff is accompanied by a non-monotonic synchronization response at fixed total resource. The largest Lyapunov exponent remains positive after synchronization and the correlation dimension changes only modestly, consistent with suppression of transverse deviations while chaotic motion is retained on the synchronization manifold.

The Kuramoto model provides a paradigmatic framework for studying synchronization of interacting oscillators, and has been generalized to arbitrary dimensions to describe swarms, flocks and multi-dimensional opinion dynamics. Yet, existing formulations neglect inertia, a key mechanism known to enhance information propagation and collective responsiveness. Here, we introduce and analyze an inertial Kuramoto model in arbitrary dimensions. We show that inertia fundamentally alters the nature of the synchronization transition, inducing a crossover from continuous to discontinuous behavior, with the onset of hysteresis depending explicitly on both inertia and dimensionality parity. Our analytical theory is supported by extensive numerical simulations. These results establish inertia as a crucial ingredient of high-dimensional collective dynamics and reveal a novel universal structure in synchronization phenomena.

Cooperation is ubiquitous in both natural and human societies, yet its evolutionary basis remains a major challenge. A long-standing puzzle is whether having more information leads to better decision-making and thus a higher level of cooperation. To address this question, we adopt a recently developed reinforcement learning framework in which individuals learn through trial and error to maximize cumulative rewards - a paradigm that has successfully explained diverse emergent patterns in human behaviors. Specifically, we equip a structured population with the Q-learning algorithm and systematically vary the size of the interactive neighborhood, which serves as a proxy for perceived information. Interestingly, we observe a non-monotonic relationship between cooperation prevalence and neighborhood size in both two-dimensional square lattices and Barabasi-Albert scale-free networks. This inverted U-shaped dependence reveals that an optimal amount of information exists, yielding the highest level of cooperation. Mechanistic analyses show that a moderate neighborhood size enables individuals to strike an optimal balance between information sufficiency and decision-making tractability. This balance allows them to detect reciprocal opportunities while avoiding the deterioration of decision quality due to information overload. Our findings challenge everyday intuition, suggesting that a proper amount of information - not more - is optimal for the emergence of cooperation.

We investigate the joint effects of stochastic forcing and quenched structural heterogeneity on stochastic resonance (SR) and liquid-state computation in a small-world network of excitable FitzHugh–Nagumo neurons. Heterogeneity is introduced separately through coupling strengths and time delays drawn from Gaussian, bimodal, and shifted-exponential distributions. Weak periodic forcing resolves the stochastic-resonance landscape, whereas weak aperiodic driving probes noise-assisted signal encoding and Liquid State Machine (LSM) forecasting. Heterogeneity is not generically beneficial, but reorganizes the resonance landscape and computational performance in a distribution-dependent manner. Gaussian coupling disorder broadens the strong-response region, whereas bimodal disorder produces a more pronounced enhancement and shifts resonance toward weaker noise. Shifted-exponential coupling disorder behaves differently: increasing its scale broadens and shifts the coupling distribution toward less responsive regions. Time-delay heterogeneity can enhance or suppress SR depending on how the delay distribution samples the structured delay-response landscape. Under aperiodic forcing, stronger input–output coherence accompanies lower LSM prediction error, showing that SR can improve LSM performance and that suitable coupling heterogeneity can further enhance this computational benefit; Gaussian and bimodal coupling disorder shift the optimum toward weaker noise and reduce the noise-optimized root-mean-square prediction error, with the largest reduction obtained for the bimodal case. These results identify stochastic forcing and quenched heterogeneity as coupled control parameters and show that the computational enhancement of LSMs depends on disorder structure rather than magnitude alone.

Conventional electrogastrography (EGG) analyses summarize gastric slow-wave activity using dominant frequency, power, or phase-based metrics, without explicitly modeling amplitude-dependent frequency dynamics. We developed a statistical nonlinear-dynamics framework that represents the gastric-band analytic EGG signal as a stochastic Stuart-Landau oscillator and estimate local effective phase-shear, $\beta_{\mathrm{eff}}$, a normalized measure of amplitude-frequency coupling. In overnight recordings from 60 healthy participants, effective phase-shear was consistently negative, generalized to held-out halves of each recording, and was abolished by amplitude surrogates that disrupted amplitude-phase alignment while preserving amplitude-series structure. We then applied the measure to fasted and fed recordings from 16 healthy participants and 24 participants with gastric dysfunction. Phase-shear magnitude differed between groups and classified dysfunction comparable to the strongest multielectrode phase-based traveling-wave measures. These findings establish amplitude-frequency coupling as a dynamical feature of gastric-band EGG that complements conventional spectral and spatial phase metrics, while motivating future multichannel observation models to determine its relationship to slow-wave propagation and gastric function.