Abstract

We investigate epidemic spreading in a deterministic susceptible-infected-susceptible model on uncorrelated heterogeneous networks with higher-order interactions. We provide a recipe for the construction of one-dimensional reduced model (resilience function) of the N-dimensional susceptible-infected-susceptible dynamics in the presence of higher-order interactions. Utilizing this reduction process, we are able to capture the microscopic and macroscopic behavior of infectious networks. We find that the microscopic state of nodes (fraction of stable healthy individual of each node) inversely scales with their degree, and it becomes diminished due to the presence of higher-order interactions. In this case, we analytically obtain that the macroscopic state of the system (fraction of infectious or healthy population) undergoes abrupt transition. Additionally, we quantify the network’s resilience, i.e., how the topological changes affect the stable infected population. Finally, we provide an alternative framework of dimension reduction based on the spectral analysis of the network, which can identify the critical onset of the disease in the presence or absence of higher-order interactions. Both reduction methods can be extended for a large class of dynamical models.

Abstract

Signal propagation in complex networks drives epidemics, is responsible for information going viral, promotes trust and facilitates moral behavior in social groups, enables the development of misinformation detection algorithms, and it is the main pillar supporting the fascinating cognitive abilities of the brain, to name just some examples. The geometry of signal propagation is determined as much by the network topology as it is by the diverse forms of nonlinear interactions that may take place between the nodes. Advances are therefore often system dependent and have limited translational potential across domains. Given over two decades worth of research on the subject, the time is thus certainly ripe, indeed the need is urgent, for a comprehensive review of signal propagation in complex networks. We here first survey different models that determine the nature of interactions between the nodes, including epidemic models, Kuramoto models, diffusion models, cascading failure models, and models describing neuronal dynamics. Secondly, we cover different types of complex networks and their topologies, including temporal networks, multilayer networks, and neural networks. Next, we cover network time series analysis techniques that make use of signal propagation, including network correlation analysis, information transfer and nonlinear correlation tools, network reconstruction, source localization and link prediction, as well as approaches based on artificial intelligence. Lastly, we review applications in epidemiology, social dynamics, neuroscience, engineering, and robotics. Taken together, we thus provide the reader with an up-to-date review of the complexities associated with the network’s role in propagating signals in the hope of better harnessing this to devise innovative applications across engineering, the social and natural sciences as well as to inspire future research.

Abstract

The stable functionality of networked systems is a hallmark of their natural ability to coordinate between their multiple interacting components. Yet, real-world networks often appear random and highly irregular, raising the question of what are the naturally emerging organizing principles of complex system stability. The answer is encoded within the system’s stability matrix—the Jacobian—but is hard to retrieve, due to the scale and diversity of the relevant systems, their broad parameter space and their nonlinear interaction dynamics. Here we introduce the dynamic Jacobian ensemble, which allows us to systematically investigate the fixed-point dynamics of a range of relevant network-based models. Within this ensemble, we find that complex systems exhibit discrete stability classes

Abstract

The sporadic large fluctuations seen in the stock market are due to different factors. These large fluctuations are termed extreme events (EE). We have identified fundamental, technical, and external factors and categorized positive or negative EE depending on the impact of these factors. During such events, the stock price time series is found to be nonstationary. Hence, the Hilbert–Huang transformation is used to identify EEs based on high instantaneous energy () concentration. The analysis shows that concentration in the stock price is very high during both positive and negative EE, surpassing a threshold of where and are the mean energy and standard deviation of energy, respectively. Further, support vector regression is used to predict the stock price during an EE, with the close price being found to be the most useful input than the open-high-low-close (OHLC) inputs. The maximum prediction accuracy for one step using close price and OHLC prices are 95.98% and 95.64%, respectively. Whereas, for the two step prediction, the accuracies are 94.09% and 93.58%, respectively. These results highlight that the accuracy of one-step predictions surpasses that of two-step predictions. Also, accuracy decreases when predicting stock prices closer to an EE. The EEs identified from predicted time series exhibit statistical characteristics similar to those obtained from the original data.

Abstract

It is generally known that in a mixture of coupled active and inactive nonlinear oscillators, the entire system may stop oscillating and become inactive if the fraction of active oscillators is reduced to a critical value. This emerging phenomenon, called the “aging transition,” can be analytically predicted from the view point of cluster synchronization. One can question whether a model-free, data-driven framework based on neural networks could be used to foretell when such a system will cease oscillation. Here, we demonstrate how a straightforward ESN with trained output weights can accurately forecast both the temporal evaluation and the onset of collapse in coupled paradigmatic limit-cycle oscillators. In particular, we have demonstrated that an ESN can identify the critical fraction of inactive oscillators in a large all-to-all, small-world, and scale-free network when it is trained only with two nodes (one active and the other inactive) selected from three different pre-collapse regimes. We further demonstrate that ESN can anticipate aging transition of the network when trained with the mean-field dynamics of active and inactive oscillators.

Abstract

The diverse excitabilities of cells often produce various spiking‑bursting oscillations that are found in the neural system. We establish the ability of a fractional‑order excitable neuron model with Caputo’s fractional derivative to analyze the effects of its dynamics on the spike train features observed in our results. The significance of this generalization relies on a theoretical framework of the model in which memory and hereditary properties are considered. Employing the fractional exponent, we first provide information about the variations in electrical activities. We deal with the 2D class I and class II excitable Morris‑Lecar (M‑L) neuron models that show the alternation of spiking and bursting features including MMOs & MMBOs of an uncoupled fractional‑order neuron. We then extend the study with the 3D slow‑fast M‑L model in the fractional domain. The considered approach establishes a way to describe various characteristics similarities between fractional‑order and classical integer‑order dynamics. Using the stability and bifurcation analysis, we discuss different parameter spaces where the quiescent state emerges in uncoupled neurons. We show the characteristics consistent with the analytical results. Next, the Erdös‑Rényi network of desynchronized mixed neurons (oscillatory and excitable) is constructed that is coupled through membrane voltage. It can generate complex firing activities where quiescent neurons start to fire. Furthermore, we have shown that increasing coupling can create cluster synchronization, and eventually it can enable the network to fire in unison. Based on cluster synchronization, we develop a reduced‑order model which can capture the activities of the entire network. Our results reveal that the effect of fractional‑order depends on the synaptic connectivity and the memory trace of the system. Additionally, the dynamics captures spike frequency adaptation and spike latency that occur over multiple timescales as the effects of fractional derivative, which has been observed in neural computation.

Abstract

Extreme events are defined as events that largely deviate from the nominal state of the system as observed in a time series. Due to the rarity and uncertainty of their occurrence, predicting extreme events has been challenging. In real life, some variables (passive variables) often encode significant information about the occurrence of extreme events manifested in another variable (active variable). For example, observables such as temperature, pressure, etc., act as passive variables in case of extreme precipitation events. These passive variables do not show any large excursion from the nominal condition yet carry the fingerprint of the extreme events. In this study, we propose a reservoir computation-based framework that can predict the preceding structure or pattern in the time evolution of the active variable that leads to an extreme event using information from the passive variable. An appropriate threshold height of events is a prerequisite for detecting extreme events and improving the skill of their prediction. We demonstrate that the magnitude of extreme events and the appearance of a coherent pattern before the arrival of the extreme event in a time series aect the prediction skill. Quantitatively, we confirm this using a metric describing the mean phase dierence between the input time signals, which decreases when the magnitude of the extreme event is relatively higher, thereby increasing the predictability skill

Abstract

In the field of complex dynamics, multistable attractors have been gaining a significant attention due to its unpredictability in occurrence and extreme sensitivity to initial conditions. Co-existing attractors are abundant in diverse systems ranging from climate to finance, ecological to social systems. In this article, we investigate a data-driven approach to infer different dynamics of a multistable system using echo state network (ESN). We start with a parameter-aware reservoir and predict diverse dynamics for different parameter values. Interestingly, machine is able to reproduce the dynamics almost perfectly even at distant parameters which lie considerably far from the parameter values related to the training dynamics. In continuation, we can predict whole bifurcation diagram significant accuracy as well. We extend this study for exploring various dynamics of multistable attractors at unknown parameter value. While, we train the machine with the dynamics of only one attarctor at parameter p, it can capture the dynamics of co-existing attractor at a new parameter value p+∆p. Continuing the simulation for multiple set of initial conditions, we can identify the basins for different attractors. We generalize the results by applying the scheme on two distinct multistable systems.

Abstract

The role of topological heterogeneity in the origin of extreme events in a network is investigated here. The dynamics of the oscillators associated with the nodes are assumed to be identical and influenced by mean-field repulsive interactions. An interplay of topological heterogeneity and the repulsive interaction between the dynamical units of the network triggers extreme events in the nodes when each node succumbs to such events for discretely different ranges of repulsive coupling. A high degree node is vulnerable to weaker repulsive interactions, while a low degree node is susceptible to stronger interactions. As a result, the formation of extreme events changes position with increasing strength of repulsive interaction from high to low degree nodes. Extreme events at any node are identified with the appearance of occasional large-amplitude events (amplitude of the temporal dynamics) that are larger than a threshold height and rare in occurrence, which we confirm by estimating the probability distribution of all events. Extreme events appear at any oscillator near the boundary of transition from rotation to libration at a critical value of the repulsive coupling strength. To explore the phenomenon, a paradigmatic second-order phase model is used to represent the dynamics of the oscillator associated with each node. We make an annealed network approximation to reduce our original model and, thereby, confirm the dual role of the repulsive interaction and the degree of a node in the origin of extreme events in any oscillator associated with a node. ACKNOWLEDGMENTS

Abstract

The persistence of biodiversity of species is a challenging proposition in ecological communities in the face of Darwinian selection. The present article investigates beyond the pairwise competitive interactions and provides a novel perspective for understanding the influence of higher-order interactions on the evolution of social phenotypes. Our simple model yields a prosperous outlook to demonstrate the impact of perturbations on intransitive competitive higherorder interactions. Using a mathematical technique, we show how alone the perturbed interaction network can quickly determine the coexistence equilibrium of competing species instead of solving a large system of ordinary differential equations. It is possible to split the system into multiple feasible cluster states depending on the number of perturbations. Our analysis also reveals the ratio between the unperturbed and perturbed species is inversely proportional to the amount of employed perturbation. Our results suggest that nonlinear dynamical systems and interaction topologies can be interplayed to comprehend species’ coexistence under adverse conditions. Particularly our findings signify that less competition between two species increases their abundance and outperforms others