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.

Abstract

The COVID-19 pandemic wrought havoc across India, particularly during its devastating second and third waves. This study undertakes a crucial epidemiological analysis of these waves, leveraging actual variant count data. Given limited sequencing efforts, variant information is sparse, prompting a novel approach to scaling up with actual case data. Employing a multi-strain variant-level SEIRS model tailored to each Indian state, we modeled the disease’s propagation. We report the estimated parameters of the SEIRS models for the two waves separately. Notably, the transmission coefficients ( ) were estimated to be 0.12 for Kappa, 0.35 for Delta, and 0.38 for Omicron. These coefficients signify the contagiousness of each variant, offering critical information for understanding and managing the pandemic’s dynamics. The findings hold significant implications for public health strategies, emphasizing the urgency of comprehensive variant tracking and proactive measures to mitigate the impact of evolving viral strains.

Abstract

In this paper, we propose a network whose nodes are labeled by the composite numbers and two nodes are connected by an undirected link if they are relatively prime to each other. As the size of the network increases, the network will be connected whenever the largest possible node index n ≥ 49. To investigate how the nodes are connected, we analytically describe that the link density saturates to 6/π2 , whereas the average degree increases linearly with slope 6/π2 with the size of the network. To investigate how the neighbors of the nodes are connected to each other, we find the shortest path length will be at most 3 for 49 ≤ n ≤ 288 and it is at most 2 for n ≥ 289. We also derive an analytic expression for the local clustering coefficients of the nodes, which quantifies how close the neighbors of a node to form a triangle. We also provide an expression for the number of r-length labeled cycles, which indicates the existence of a cycle of length at most O(log n). Finally, we show that this graph sequence is actually a sequence of weakly pseudo-random graphs. We numerically verify our observed analytical results. As a possible application, we have observed less synchronizability (the ratio of the largest and smallest positive eigenvalue of the Laplacian matrix is high) as compared to Erdős–Rényi random network and Barabási–Albert network. This unusual observation is consistent with the prolonged transient behaviors of ecological and predator–prey networks which can easily avoid the global synchronization.

Abstract

We investigate the phenomenon of transition to synchronization in the Sakaguchi-Kuramoto model in the presence of higher-order interactions and global order parameter adaptation. The investigation is done by performing extensive numerical simulations and low-dimensional modeling of the system. Numerical simulations of the full system show both continuous (second-order) as well as discontinuous transitions. The discontinuous transitions can either be associated with explosive (first-order) or tiered synchronization states depending on the choice of parameters. To develop an in depth understanding of the transition scenario in the parameter space we derive a reduced order model (ROM) using the Ott-Antonsen ansatz, the results of which closely match with those of the numerical simulations of the full system. The simplicity and analytical accessibility of the ROM help to conveniently unfold the transition scenario in the system having complex dependence on the parameters. Simultaneous analysis of the full system and the ROM clearly identifies the regions of the parameter space exhibiting different types of transitions. It is observed that the second-order transition is connected with a supercritical pitchfork bifurcation (PB) of the ROM. On the other hand, the discontinuous tiered transition is associated with multiple saddle-node (SN) bifurcations along with a supercritical PB and the first-order transition involves a subcritical PB alongside a SN bifurcation. Finally, the stability analysis of the different synchronization states of the system is performed analytically.

Abstract

This paper identifies the cryptocurrency market crashes and analyses its dynamics using the complex network. We identify three distinct crashes during 2017–20, and the analysis is carried out by dividing the time series into pre-crash, crash, and post-crash periods. Partial correlation based complex network analysis is carried out to study the crashes. Degree density (𝜌𝐷), average path length (̄𝑙), and average clustering coefficient (𝑐𝑐) are estimated from these networks. We find that both 𝜌𝐷 and 𝑐𝑐 are smallest during the pre-crash period, and spike during the crash suggesting the network is dense during a crash. Although 𝜌𝐷 and 𝑐𝑐 decrease in the postcrash period, they remain higher than pre-crash levels for the 2017–18 and 2018–19 crashes suggesting a market attempt to return to normalcy. We get ̄𝑙 is minimal during the crash period, suggesting a rapid flow of information. A dense network and rapid information flow suggest that during a crash uninformed synchronized panic sell-off happens. However, during the 2019– 20 crash, the values of 𝜌𝐷, 𝑐𝑐, and ̄𝑙 did not vary significantly, indicating minimal change in dynamics compared to other crashes. The findings of this study may guide investors in making decisions during market crashes.