Entropy Estimators for Markovian Sequences: A Comparative Analysis.

Entropy estimation is a fundamental problem in information theory that has applications in various fields, including physics, biology, and computer science. Estimating the entropy of discrete sequences can be challenging due to limited data and the lack of unbiased estimators. Most existing entropy estimators are designed for sequences of independent events and their performances vary depending on the system being studied and the available data size. In this work, we compare different entropy estimators and their performance when applied to Markovian sequences. Specifically, we analyze both binary Markovian sequences and Markovian systems in the undersampled regime. We calculate the bias, standard deviation, and mean squared error for some of the most widely employed estimators. We discuss the limitations of entropy estimation as a function of the transition probabilities of the Markov processes and the sample size. Overall, this paper provides a comprehensive comparison of entropy estimators and their performance in estimating entropy for systems with memory, which can be useful for researchers and practitioners in various fields.

Partisan voter model: Stochastic description and noise-induced transitions.

We give a comprehensive mean-field analysis of the partisan voter model (PVM) and report analytical results for exit probabilities, fixation times, and the quasistationary distribution. In addition, and similarly to the noisy voter model, we introduce a noisy version of the PVM, named the noisy partisan voter model (NPVM), which accounts for the preferences of each agent for the two possible states, as well as for idiosyncratic spontaneous changes of state. We find that the finite-size noise-induced transition of the noisy voter model is modified in the NPVM leading to the emergence of intermediate phases that were absent in the standard version of the noisy voter model, as well as to both continuous and discontinuous transitions.

Ordinal analysis of lexical patterns.

Words are fundamental linguistic units that connect thoughts and things through meaning. However, words do not appear independently in a text sequence. The existence of syntactic rules induces correlations among neighboring words. Using an ordinal pattern approach, we present an analysis of lexical statistical connections for 11 major languages. We find that the diverse manners that languages utilize to express word relations give rise to unique pattern structural distributions. Furthermore, fluctuations of these pattern distributions for a given language can allow us to determine both the historical period when the text was written and its author. Taken together, our results emphasize the relevance of ordinal time series analysis in linguistic typology, historical linguistics, and stylometry.

Sampling rare trajectories using stochastic bridges.

The numerical quantification of the statistics of rare events in stochastic processes is a challenging computational problem. We present a sampling method that constructs an ensemble of stochastic trajectories that are constrained to have fixed start and end points (so-called stochastic bridges). We then show that by carefully choosing a set of such bridges and assigning an appropriate statistical weight to each bridge, one can focus more processing power on the rare events of a target stochastic process while faithfully preserving the statistics of these rare trajectories. Further, we also compare the stochastic bridges we produce to the Wentzel-Kramers-Brillouin (WKB) optimal paths of the target process, derived in the limit of low noise. We see that the generated paths, encoding the full statistics of the process, collapse onto the WKB optimal path as the level of noise is reduced. We propose that the method can also be used to judge the accuracy of the WKB approximation at finite levels of noise.

Analytical and Numerical Treatment of Continuous Ageing in the Voter Model.

The conventional voter model is modified so that an agent's switching rate depends on the 'age' of the agent-that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. The thinning algorithm of Lewis and Shedler can be modified in order to provide an efficient simulation method. Analytically, we demonstrate how the asymptotic approach to an absorbing state (consensus) can be deduced. We discuss three special cases of the age-dependent switching rate: one in which the concentration of voters can be approximated by a fractional differential equation, another for which the approach to consensus is exponential in time, and a third case in which the system reaches a frozen state instead of consensus. Finally, we include the effects of a spontaneous change of opinion, i.e., we study a noisy voter model with continuous ageing. We demonstrate that this can give rise to a continuous transition between coexistence and consensus phases. We also show how the stationary probability distribution can be approximated, despite the fact that the system cannot be described by a conventional master equation.

Pair approximation for the noisy threshold q-voter model.

In the standard q-voter model, a given agent can change its opinion only if there is a full consensus of the opposite opinion within a group of influence of size q. A more realistic extension is the threshold q voter, where a minimal agreement (at least 0

Absorbing phase transition in the coupled dynamics of node and link states in random networks.

We present a stochastic dynamics model of coupled evolution for the binary states of nodes and links in a complex network. In the context of opinion formation node states represent two possible opinions and link states represent positive or negative relationships. Dynamics proceeds via node and link state update towards pairwise satisfactory relations in which nodes in the same state are connected by positive links or nodes in different states are connected by negative links. By a mean-field rate equations analysis and Monte Carlo simulations in random networks we find an absorbing phase transition from a dynamically active phase to an absorbing phase. The transition occurs for a critical value of the relative time scale for node and link state updates. In the absorbing phase the order parameter, measuring global order, approaches exponentially the final frozen configuration. Finite-size effects are such that in the absorbing phase the final configuration is reached in a characteristic time that scales logarithmically with system size, while in the active phase, finite-size fluctuations take the system to a frozen configuration in a characteristic time that grows exponentially with system size. There is also a class of finite-size topological transition associated with group splitting in the network of these final frozen configurations.

A continuous-time persistent random walk model for flocking.

A classical random walker is characterized by a random position and velocity. This sort of random walk was originally proposed by Einstein to model Brownian motion and to demonstrate the existence of atoms and molecules. Such a walker represents an inanimate particle driven by environmental fluctuations. On the other hand, there are many examples of so-called "persistent random walkers," including self-propelled particles that are able to move with almost constant speed while randomly changing their direction of motion. Examples include living entities (ranging from flagellated unicellular organisms to complex animals such as birds and fish), as well as synthetic materials. Here we discuss such persistent non-interacting random walkers as a model for active particles. We also present a model that includes interactions among particles, leading to a transition to flocking, that is, to a net flux where the majority of the particles move in the same direction. Moreover, the model exhibits secondary transitions that lead to clustering and more complex spatially structured states of flocking. We analyze all these transitions in terms of bifurcations using a number of mean field strategies (all to all interaction and advection-reaction equations for the spatially structured states), and compare these results with direct numerical simulations of ensembles of these interacting active particles.

Zealots in the mean-field noisy voter model.

The influence of zealots on the noisy voter model is studied theoretically and numerically at the mean-field level. The noisy voter model is a modification of the voter model that includes a second mechanism for transitions between states: Apart from the original herding processes, voters may change their states because of an intrinsic noisy-in-origin source. By increasing the importance of the noise with respect to the herding, the system exhibits a finite-size phase transition from a quasiconsensus state, where most of the voters share the same opinion, to one with coexistence. Upon introducing some zealots, or voters with fixed opinion, the latter scenario may change significantly. We unveil new situations by carrying out a systematic numerical and analytical study of a fully connected network for voters, but allowing different voters to be directly influenced by different zealots. We show that this general system is equivalent to a system of voters without zealots, but with heterogeneous values of their parameters characterizing herding and noisy dynamics. We find excellent agreement between our analytical and numerical results. Noise and herding or zealotry acting together in the voter model yields a nontrivial mixture of the scenarios with the two mechanisms acting alone: It represents a situation where the global-local (noise-herding) competition is coupled to a symmetry breaking (zealots). In general, the zealotry enhances the effective noise of the system, which may destroy the original quasiconsensus state, and can introduce a bias towards the opinion of the majority of zealots, hence breaking the symmetry of the system and giving rise to new phases. In the most general case we find two different transitions: a discontinuous transition from an asymmetric bimodal phase to an extreme asymmetric phase and a second continuous transition from the extreme asymmetric phase to an asymmetric unimodal phase.

Synchronization of coupled noisy oscillators: Coarse graining from continuous to discrete phases.

The theoretical description of synchronization phenomena often relies on coupled units of continuous time noisy Markov chains with a small number of states in each unit. It is frequently assumed, either explicitly or implicitly, that coupled discrete-state noisy Markov units can be used to model mathematically more complex coupled noisy continuous phase oscillators. In this work we explore conditions that justify this assumption by coarse graining continuous phase units. In particular, we determine the minimum number of states necessary to justify this correspondence for Kuramoto-like oscillators.

The noisy voter model on complex networks.

We propose a new analytical method to study stochastic, binary-state models on complex networks. Moving beyond the usual mean-field theories, this alternative approach is based on the introduction of an annealed approximation for uncorrelated networks, allowing to deal with the network structure as parametric heterogeneity. As an illustration, we study the noisy voter model, a modification of the original voter model including random changes of state. The proposed method is able to unfold the dependence of the model not only on the mean degree (the mean-field prediction) but also on more complex averages over the degree distribution. In particular, we find that the degree heterogeneity--variance of the underlying degree distribution--has a strong influence on the location of the critical point of a noise-induced, finite-size transition occurring in the model, on the local ordering of the system, and on the functional form of its temporal correlations. Finally, we show how this latter point opens the possibility of inferring the degree heterogeneity of the underlying network by observing only the aggregate behavior of the system as a whole, an issue of interest for systems where only macroscopic, population level variables can be measured.

Competition of simple and complex adoption on interdependent networks.

We consider the competition of two mechanisms for adoption processes: a so-called complex threshold dynamics and a simple susceptible-infected-susceptible (SIS) model. Separately, these mechanisms lead, respectively, to first-order and continuous transitions between nonadoption and adoption phases. We consider two interconnected layers. While all nodes on the first layer follow the complex adoption process, all nodes on the second layer follow the simple adoption process. Coupling between the two adoption processes occurs as a result of the inclusion of some additional interconnections between layers. We find that the transition points and also the nature of the transitions are modified in the coupled dynamics. In the complex adoption layer, the critical threshold required for extension of adoption increases with interlayer connectivity whereas in the case of an isolated single network it would decrease with average connectivity. In addition, the transition can become continuous depending on the detailed interlayer and intralayer connectivities. In the SIS layer, any interlayer connectivity leads to the extension of the adopter phase. Besides, a new transition appears as a sudden drop of the fraction of adopters in the SIS layer. The main numerical findings are described by a mean-field type analytical approach appropriately developed for the threshold-SIS coupled system.

Anticipated synchronization in coupled complex Ginzburg-Landau systems.

We study the occurrence of anticipated synchronization in two complex Ginzburg-Landau systems coupled in a master-slave configuration. Master and slave systems are ruled by the same autonomous function, but the slave system receives the injection from the master and is subject to a negative delayed self-feedback loop. We give evidence that the magnitude of the largest anticipation time, obtained for complex-valued coupling constants, depends on the dynamical regime where the system operates (defect turbulence, phase turbulence, or bichaos) and scales with the linear autocorrelation time of the system. We also provide analytical conditions for the stability of the anticipated synchronization manifold that are in qualitative agreement with those obtained numerically. Finally, we report on the existence of anticipated synchronization in coupled two-dimensional complex Ginzburg-Landau systems.

Markets, Herding and Response to External Information.

We focus on the influence of external sources of information upon financial markets. In particular, we develop a stochastic agent-based market model characterized by a certain herding behavior as well as allowing traders to be influenced by an external dynamic signal of information. This signal can be interpreted as a time-varying advertising, public perception or rumor, in favor or against one of two possible trading behaviors, thus breaking the symmetry of the system and acting as a continuously varying exogenous shock. As an illustration, we use a well-known German Indicator of Economic Sentiment as information input and compare our results with Germany's leading stock market index, the DAX, in order to calibrate some of the model parameters. We study the conditions for the ensemble of agents to more accurately follow the information input signal. The response of the system to the external information is maximal for an intermediate range of values of a market parameter, suggesting the existence of three different market regimes: amplification, precise assimilation and undervaluation of incoming information.

Descending from infinity: convergence of tailed distributions.

We investigate the relaxation of long-tailed distributions under stochastic dynamics that do not support such tails. Linear relaxation is found to be a borderline case in which long tails are exponentially suppressed in time but not eliminated. Relaxation stronger than linear suppresses long tails immediately, but may lead to strong transient peaks in the probability distribution. We also find that a δ-function initial distribution under stronger than linear decay displays not one but two different regimes of diffusive spreading.

Simulating non-Markovian stochastic processes.

We present a simple and general framework to simulate statistically correct realizations of a system of non-Markovian discrete stochastic processes. We give the exact analytical solution and a practical and efficient algorithm like the Gillespie algorithm for Markovian processes, with the difference being that now the occurrence rates of the events depend on the time elapsed since the event last took place. We use our non-Markovian generalized Gillespie stochastic simulation methodology to investigate the effects of nonexponential interevent time distributions in the susceptible-infected-susceptible model of epidemic spreading. Strikingly, our results unveil the drastic effects that very subtle differences in the modeling of non-Markovian processes have on the global behavior of complex systems, with important implications for their understanding and prediction. We also assess our generalized Gillespie algorithm on a system of biochemical reactions with time delays. As compared to other existing methods, we find that the generalized Gillespie algorithm is the most general because it can be implemented very easily in cases (such as for delays coupled to the evolution of the system) in which other algorithms do not work or need adapted versions that are less efficient in computational terms.

Suppression of deterministic and stochastic extreme desynchronization events using anticipated synchronization.

We numerically show that extreme events induced by parameter mismatches or noise in coupled oscillatory systems can be anticipated and suppressed before they actually occur. We show this in a main system unidirectionally coupled to an auxiliary system subject to a negative delayed feedback. Each system consists of two electronic oscillators coupled in a master-slave configuration. Extreme events are observed in this coupled system as large and sporadic desynchronization events. Under certain conditions, the auxiliary system can predict the dynamics of the main system. We use this to efficiently suppress the extreme events by applying a direct corrective reset to the main system.

Weighted-ensemble Brownian dynamics simulation: sampling of rare events in nonequilibrium systems.

We provide an algorithm based on weighted-ensemble (WE) methods, to accurately sample systems at steady state. Applying our method to different one- and two-dimensional models, we succeed in calculating steady-state probabilities of order 10(-300) and reproduce the Arrhenius law for rates of order 10(-280). Special attention is payed to the simulation of nonpotential systems where no detailed balance assumption exists. For this large class of stochastic systems, the stationary probability distribution density is often unknown and cannot be used as preknowledge during the simulation. We compare the algorithm's efficiency with standard Brownian dynamics simulations and the original WE method.

On the effect of heterogeneity in stochastic interacting-particle systems.

We study stochastic particle systems made up of heterogeneous units. We introduce a general framework suitable to analytically study this kind of systems and apply it to two particular models of interest in economy and epidemiology. We show that particle heterogeneity can enhance or decrease the size of the collective fluctuations depending on the system, and that it is possible to infer the degree and the form of the heterogeneity distribution in the system by measuring only global variables and their fluctuations. Our work shows that, in some cases, heterogeneity among the units composing a system can be fully taken into account without losing analytical tractability.