Abstract

Robustness to perturbation is a key topic in the study of complex systems occurring across a wide variety of applications from epidemiology to biochemistry. Here, we analyze the eigenspectrum of the Jacobian matrices associated to a general class of networked dynamical systems, which contains information on how perturbations to a stationary state develop over time. We find that stability is always determined by a spectral outlier, but with pronounced differences to the corresponding eigenvector in different regimes. We show that, depending on model details, instability may originate in nodes of anomalously low or high degrees, or may occur everywhere in the network at once. Importantly, the dependence on extremal degrees results in considerable finite-size effects with different scaling depending on the ensemble degree distribution. Our results have potentially useful applications in network monitoring to predict or prevent catastrophic failures, and we validate our analytical findings through applications to epidemic dynamics and gene regulatory systems.

Abstract

Binary tree network, being a subclass of Cayley tree network, is a significant topological structure used for information transfer in a hierarchical sense. In this article, we consider four types of binary tree repeater networks (directed and undirected, asymmetric and symmetric) and obtain the analytical expressions of the average of the maximum teleportation fidelities for each of these binary tree networks. We contribute a methodology for the analytical calculation of path lengths in all considered graph types. Based on these, we have used simple Werner state-based models and are able to identify the parameter ranges for which these networks can show quantum advantage. We also explore the role of maximally entangled states in the network to enhance the quantum advantage. We provide a detailed examination of the large-scale behaviour of these networks, obtaining the limiting value of the average maximum teleportation fidelity as the number of nodes, N, approaches infinity, same as fractal tree. Our findings reveal that the directed symmetric binary tree represents the most advantageous topology for quantum teleportation within this context. From the context of quantum repeater networks, this work makes a significant advancement in the process of identifying resourceful tree networks for distributed quantum teleportation, i.e. teleportation between all possible sources and targets.

Abstract

Cluster synchronization in multilayer networks of phase oscillators with phase-lag poses significant challenges due to the destabilizing effects of delayed interactions. Leveraging the Sakaguchi-Kuramoto model, this study addresses these challenges by systematically exploring the role of natural frequency distributions in sustaining cluster synchronization under high phase-lag conditions. We focus on four distributions: uniform (uni-uni), partially degree-correlated (deg-uni, uni-deg), and fully degree-correlated (deg-deg), where oscillators' intrinsic frequencies align with their network connectivity. Through numerical and analytical investigations, we demonstrate that the deg-deg distribution where both layers employ degree-matched frequencies remarkably enhances synchronization stability, outperforming other configurations. We analyze two distinct network architectures: one composed entirely of nontrivial clusters and another combining trivial and nontrivial clusters. Results reveal that structural heterogeneity encoded in the deg-deg coupling counteracts phase-lag-induced desynchronization, enabling robust cluster synchronization even at large phase-lag values. Stability is rigorously validated via transverse Lyapunov exponents (TLEs), which confirm that deg-deg networks exhibit broader synchronization regimes compared to uniform or partially correlated systems. These findings provide critical insights into the interplay between topological heterogeneity and dynamical resilience, offering a framework for designing robust multilayer systems from delay-tolerant power grids to adaptive biological networks, where synchronization under phase-lag is paramount.

Abstract

Understanding and forecasting precipitation events in the Arctic maritime environments, such as Bear Island and Ny-Ålesund, is crucial for assessing climate risk and developing early warning systems in vulnerable marine regions. This study proposes a probabilistic machine learning framework for modeling and predicting the dynamics and severity of precipitation. We begin by analyzing the scale-dependent relationships between precipitation and key atmospheric drivers (e.g., temperature, relative humidity, cloud cover, and air pressure) using wavelet coherence, which captures localized dependencies across time and frequency domains. To assess joint causal influences, we employ synergistic-unique-redundant decomposition, which quantifies the impact of interaction effects among each variable on future precipitation dynamics. These insights inform the development of data-driven forecasting models that incorporate both historical precipitation and causal climate drivers. To account for uncertainty, we employ the conformal prediction method, which enables the generation of calibrated non-parametric prediction intervals. Our results underscore the importance of utilizing a comprehensive framework that combines causal analysis with probabilistic forecasting to enhance the reliability and interpretability of precipitation predictions in Arctic marine environments.

Abstract

Understanding the collective behavior of dynamical systems is essential for explaining various emergent phenomena in natural and engineered settings. A key step in this process is formulating an appropriate mathematical description of the individual systems and network of systems. In this context, a range of physical systems is considered here, including the classical pendula, superconducting Josephson junctions, power grids, and various others. Despite the diversity of the systems in terms of physical structure and their application domains, they exhibit strikingly similar dynamical features, namely, phase dynamics governed by inertia and damping, and in their response to external forcing. This observation creates interest and motivates a search for a unified theoretical framework capable of capturing the fundamentals of their dynamical behaviors exhibited across the systems. This review critically examines the up-to-date research activities on the dynamics of the second-order phase oscillator, henceforth claimed here as a universality class by its own merits as a simple nonlinear dynamical model representing a broad class of physical systems. It offers a common mathematical framework to develop a comprehensive understanding, from a general perspective, that bridges, the theoretical and experimental observations of pendulum motion, Josephson junctions, and power grids and their collective behaviors. While each of these systems has been discussed in disparate physical contexts, their underlying mathematical structures reveal strong commonalities. In particular, we highlight the importance of analyzing these systems through the lens of nonlinear phase dynamics to uncover their shared mechanisms and system-specific variety of behaviors as well. This survey mainly focuses on some specific interrelated themes: (i) collective phenomena and emergent synchronization; (ii) the role of heterogeneity in terms of system parameters and effects of noise on the emergent dynamics; (iii) multi-stability and complex transient regimes; (iv) the integration of machine learning for model discovery, control, and prediction; and (v) the broader applicability of phase oscillator models across diverse domains beyond the canonical systems considered here. By systematically comparing the dynamical behaviors of the varied physical systems within a cohesive mathematical framework of second-order phase oscillators, this review seeks for the universal and distinctive features of nonlinear dynamics of the three systems, their collective behaviors such as emergent synchrony, partial synchrony, or chimera states, and specifically explains real-life phenomena, and crowd synchrony that may lead to a collapse of a footbridge and the failure of a power grid. Besides our main emphasis on these systems, brief notes have been added on other systems where this second-order phase model explains their dynamical properties. A broad synthesis on the topic will not only deepen our theoretical understanding but also suggest any design and control of complex dynamical systems in both natural and engineered settings.

Abstract

We investigate the effect of partial order parameter adaptation in the form of general functions on the synchronization behavior of coupled Kuramoto oscillators on top of random hypergraph models. The interactions between the oscillators are considered pairwise and triangular. Using the Ott-Antonsen ansatz, we obtain a set of self-consistent equations of the order parameter that describe the synchronization patterns. A broad diversity of synchronization transitions is observed as a result of the interplay between the partial adaptation approach, generalized adaptation functions, and coupling strengths. The system specifically shows a double-jump transition under a power law form of the adaptation function. A polynomial form of the adaptation function leads to the emergence of an intermediate synchronization state for specific combinations of one negative and one positive coefficient. Moreover, the synchronization transition may become continuous or explosive when the pairwise coupling strength varies. This type of synchronization behavior is further found by using a Gaussian adaptation function.

Abstract

Complex biological networks, encompassing metabolic pathways, gene regulatory systems, and protein-protein interaction networks, often exhibit scale-free structures characterized by heavy-tailed degree distributions. However, empirical studies reveal significant deviations from ideal power-law behavior, underscoring the need for more flexible and accurate probabilistic models. In this work, we propose the Compounded Burr (CBurr) distribution, a novel four-parameter family derived by compounding the Burr distribution with a discrete mixing process. This model is specifically designed to capture both the body and tail behavior of real-world network degree distributions with applications to biological networks. We rigorously derive its statistical properties, including moments, hazard and risk functions, and tail behavior, and develop an efficient maximum likelihood estimation framework. The CBurr model demonstrates broad applicability to networks with complex connectivity patterns, particularly in biological, social, and technological domains. Extensive experiments on large-scale biological network datasets show that CBurr consistently outperforms classical power-law, lognormal, and other heavy-tailed models across the full degree spectrum. By providing a statistically grounded and interpretable framework, the CBurr model enhances our ability to characterize the structural heterogeneity of biological networks.

Abstract

We show that the average of the maximum teleportation fidelities between all pairs of nodes in a large quantum
repeater network is a measure of the resourcefulness of the network as a whole. It can be used to rank networks
of arbitrary topologies, with and without loops. For demonstration purposes, we use simple Werner state-based
models to characterize some fundamental topologies (the star, the chain, some trees, and also the ring and the
complete graph) with this measure in three (semi)realistic scenarios. Most of our results are analytic and apply to
arbitrary network sizes. We identify the parameter ranges where these networks can achieve quantum advantages
and show the large-N behaviors.

Abstract

Besides global attention on extreme precipitation, a limited research has been done in the Arctic due to constraints of data availability. In this backdrop, we attempt to analyze extreme precipitation events at three Arctic stations (Bjørnøya, Ny-Ålesund, and Svalbard Lufthavn) in Svalbard using extreme value theory. The analysis revealed that these high-latitudinal Arctic stations were characterized by heavy-tailed distributions for the exceedances, suggesting a higher probability of the occurrence of extreme precipitation events. Ny-Ålesund and Bjørnøya have exhibited a significant increase in return values over the last three decades. Among the three stations, Ny-Ålesund displayed the strongest return values, especially in winter post-1994 when the atmospheric temperature was characterized by an enhanced positive trend. Significant seasonal variability in return values has also been observed; the fall in Ny-Ålesund was characterized by a low-intensity regime as indicated by the shape parameter. Ny-Ålesund precipitation had shifted from heavy-tailed distribution in pre-1994 to bounded tail distribution post-1994 during spring. Bjørnøya's extremes are driven by cyclonic circulation, while southerly winds drive extremes in Ny-Ålesund and Svalbard Lufthavn. Even though, Svalbard Lufthavn, displayed regime changes, showed low variability, likely due to its position in a rain shadow region. This research highlights the nuanced responses of Arctic hydrology to warming, emphasizing the need for localized studies and active collaboration with policymakers to translate these insights into effective climate adaptation and mitigation strategies.

Abstract

We present the emergence of topological phase transition in the minimal model of two-dimensional rockpaper-
scissors cycle in the form of a doublet chain. The evolutionary dynamics of the doublet chain is obtained
by solving the anti-symmetric Lotka-Volterra equation. We show that the mass decays exponentially towards
edges and robust against small perturbation in the rate of change of mass transfer, a signature of a topological
phase. For one of the configuration of our doublet chain, the mass is transferred towards both edges and the bulk
is gaped. Further, we confirm this phase transition within the framework of topological band theory. For this,
we calculate the winding number, which change from zero to one for trivial and nontrivial topological phases,
respectively.