Abstract

Our investigation centres on assessing the importance of a biased parameter (𝛼) in a multiplex Markov chain (MMC) model that is characterized by biased random walks in multiplex networks. We explore how varying complex network topologies affect the total multiplex imbalance as a function of biased parameter. Our primary finding is that the system demonstrates a gradual increase in total imbalance within both positive and negative regions of the biased parameter, with a consistent minimum value occurring at 𝛼 = −1. In contrast to the negative region, the total imbalance is consistently high when 𝛼 is significantly positive. We perform a detailed examination of four different network structures and establish three sets of multiplex networks. In each of these networks, the second layer consists of a Regular Random network, while the first layer is either a Barabási–Albert, Erdős-Rényi, or Watts Strogatz network, depending on the set. Our results demonstrate that the combination of Barabási–Albert and Random Regular Network exhibits the highest level of right saturation imbalance. Additionally, for left saturation imbalance, the Erdős–Rényi and Random Regular combination achieve a slightly higher value. We also observe that the total amount of imbalance at 𝛼 = −1 follows a decreasing trend as the size of the network of each layer increases. Furthermore, we are also able to illustrate that the second most significant eigenvalue of the supra-transition matrix exhibits a similar pattern in response to changes in the bias parameter, aligning with the overall system’s imbalance.

Abstract

We delve into the interplay between network’s symmetry and functioning for a generic class of dynamical systems. Primarily, we focus on a class of systems that characterize the spreading process, such as the spread of epidemics in complex networks, where the coupling configuration is nonlinear rather than diffusive. Through theoretical and numerical analysis, we establish a compelling connection between the symmetry of the graph and the trajectories followed by the dynamical processes for those nodes forming symmetry orbits and displaying identical eigenvector centrality. In particular, we are able to show that when the initial transitory states are removed, the symmetric group of nodes respond synchronously; nonetheless, they maintain a constant distance from each other and hence follow splay states. We have verified this phenomenon once more using two distinct kinds of networks. In one instance, every node takes part in nontrivial clusters. In the alternative scenario, we create symmetric orbits as per our target. The cluster nodes show splay states in both situations.

Abstract

Complexity is an important metric for appropriate characterization of different classes of irregular signals, observed in the laboratory or in nature. The literature is already rich in the description of such measures using a variety of entropy and disequilibrium measures, separately or in combination. Chaotic signal was given prime importance in such studies while no such measure was proposed so far, how complex were the extreme events when compared to non-extreme chaos. We address here this question of complexity in extreme events and investigate if we can distinguish them from non-extreme chaotic signal. The normalized Shannon entropy in combination with disequlibrium is used for our study and it is able to distinguish between extreme chaos and non-extreme chaos and moreover, it depicts the transition points from periodic to extremes via Pomeau-Manneville intermittency and, from small amplitude to large amplitude chaos and its transition to extremes via interior crisis. We report a general trend of complexity against a system parameter that increases during a transition to extreme events, reaches a maximum, and then starts decreasing. We employ three models, a nonautonomous Liénard system, 2-dimensional Ikeda map and a 6-dimensional coupled Hindmarh-Rose system to validate our proposition

Abstract

In a prior study, a novel deterministic compartmental model known as the SEIHRK model was introduced, shedding light on the pivotal role of test kits as an intervention strategy for mitigating epidemics. Particularly in heterogeneous networks, it was empirically demonstrated that strategically distributing a limited number of test kits among nodes with higher degrees substantially diminishes the outbreak size. The network's dynamics were explored under varying values of infection rate. In this research, we expand upon these findings to investigate the influence of migration on infection dynamics within distinct communities of the network. Notably, we observe that nodes equipped with test kits and those without tend to segregate into two separate clusters when coupling strength is low, but beyond a critical threshold coupling coefficient, they coalesce into a unified cluster. Building on this clustering phenomenon, we develop a reduced equation model and rigorously validate its accuracy through comprehensive simulations. We show that this property is observed in both complete and random graphs.

Abstract

Statistical analysis of high-frequency stock market order transaction data is conducted to understand order transition dynamics. We employ a first-order time-homogeneous discrete-time Markov chain model to the sequence of orders of stocks belonging to six different sectors during the US–China trade war of 2018. The Markov property of the order sequence is validated by the Chi-square test. We estimate the transition probability matrix of the sequence using maximum likelihood estimation. From the heatmap of these matrices, we found the presence of active participation by different types of traders during high volatility days. On such days, these traders place limit orders primarily with the intention of deleting the majority of them to influence the market. These findings are supported by high stationary distribution and low mean recurrence values of add and delete orders. Further, we found similar spectral gap and entropy rate values, which indicates that similar trading strategies are employed on both high and low volatility days during the trade war. Among all the sectors considered in this study, we observe that there is a recurring pattern of full execution orders in the Finance & Banking sector. This shows that the banking stocks are resilient during the trade war. Hence, this study may be useful in understanding stock market order dynamics and devise trading strategies accordingly on high and low volatility days during extreme macroeconomic events.

Abstract

In an ecosystem comprising coexisting species, mutations frequently occur. These mutations can be induced by various factors such as errors in DNA replication and exposure to chemicals. They constitute an intrinsic element of species evolution. This study investigates the impact of mutations on ecosystems, employing Gillespie simulations and the formulation of the First-passage extinction problem to assess their effects and examine extinction events. Our findings suggest first-extinction time and state distribution in a system with mutation follows intriguing behavior which promotes co-existence. There also exists a depression in the state space post which mutation extends the first-extinction time. Moreover, a system devoid of mutation exhibits a discernible inclination towards probabilities that lean in the direction of an endangered state space.

Abstract

Positive phase coupling plays an attractive role in inducing in-phase synchrony in an ensemble of phase oscillators. Positive coupling involving both amplitude and phase continues to be attractive, leading to complete synchrony in identical oscillators (limit cycle or chaotic) or phase coherence in oscillators with heterogeneity of parameters. In contrast, purely positive phase velocity coupling may originate a repulsive effect on pendulumlike oscillators (with rotational motion) to bring them into a state of diametrically opposite phases or a splay state. Negative phase velocity coupling is necessary to induce synchrony or coherence in the general sense. The contrarian roles of phase coupling and phase velocity coupling on the synchrony of networks of second-order phase oscillators have been explored here. We explain our proposition using networks of two model systems, a second-order phase oscillator representing the pendulum or the superconducting Josephson junction dynamics, and a voltage-controlled oscillations in neurons model. Numerical as well as semianalytical approaches are used to confirm our result

Abstract

Achieving perfect synchronization in a complex network, specially in the presence of higher-order interactions (HOIs) at a targeted point in the parameter space, is an interesting, yet challenging task. Here we present a theoretical framework to achieve the same under the paradigm of the Sakaguchi-Kuramoto (SK) model. We analytically derive a frequency set to achieve perfect synchrony at some desired point in a complex network of SK oscillators with higher-order interactions. Considering the SK model with HOIs on top of the scale-free, random, and small world networks, we perform extensive numerical simulations to verify the proposed theory. Numerical simulations show that the analytically derived frequency set not only provides stable perfect synchronization in the network at a desired point but also proves to be very effective in achieving a high level of synchronization around it compared to the other choices of frequency sets. The stability and the robustness of the perfect synchronization state of the system are determined using the low-dimensional reduction of the network and by introducing a Gaussian noise around the derived frequency set, respectively.

Abstract

Symmetries in a network regulate its organization into functional clustered states. Given a generic ensemble of nodes and a desirable cluster (or group of clusters), we exploit the direct connection between the elements of the eigenvector centrality and the graph symmetries to generate a network equipped with the desired cluster(s), with such a synthetical structure being furthermore perfectly reflected in the modular organization of the network's functioning. Our results solve a relevant problem of designing a desired set of clusters and are of generic application in all cases where a desired parallel functioning needs to be blueprinted.

Abstract

A balanced ecosystem with coexisting constituent species is often perturbed by different natural events that persist only for a finite duration of time. What becomes important is whether, in the aftermath, the ecosystem recovers its balance or not. Here we study the fate of an ecosystem by monitoring the dynamics of a particular species that encounters a sudden increase in death rate. For exploration of the fate of the species, we use Monte-Carlo simulation on a three-species cyclic rock-paper-scissor model. The density of the affected (by perturbation) species is found to drop exponentially immediately after the pulse is applied. In spite of showing this exponential decay as a short-time behavior, there exists a region in parameter space where this species surprisingly remains as a single survivor, wiping out the other two which had not been directly affected by the perturbation. Numerical simulations using stochastic differential equations of the species give consistency to our results.