Search in unknown random environments

Dynamic redundancy as a mechanism to optimize collective random searches

We explore the case of a group of random walkers looking for a target randomly located in space, such that the number of walkers is not constant but new ones can join the search, or those that are active can abandon it, with constant rates r_{b} and r_{d}, respectively. Exact analytical solutions are provided both for the fastest-first-passage time and for the collective time cost required to reach the target, for the exemplifying case of Brownian walkers with r_{d}=0. We prove that even for such a simple situation there exists an optimal rate r_{b} at which walkers should join the search to minimize the collective search costs. We discuss how these results open a new line to understand the optimal regulation in searches conducted through multiparticle random walks, e.g., in chemical or biological processes.

Stochastic representation of processes with resetting

In this paper we introduce a general stochastic representation for an important class of processes with resetting. It allows to describe any stochastic process intermittently terminated and restarted from a predefined random or nonrandom point. Our approach is based on stochastic differential equations called jump-diffusion models. It allows to analyze processes with resetting both, analytically and using Monte Carlo simulation methods. To depict the strength of our approach, we derive a number of fundamental properties of Brownian motion with Poissonian resetting, such as the Itô lemma, the moment-generating function, the characteristic function, the explicit form of the probability density function, moments of all orders, various forms of the Fokker-Planck equation, infinitesimal generator of the process, and its adjoint operator. Additionally, we extend the above results to the case of time-nonhomogeneous Poissonian resetting. This way we build a general framework for the analysis of any stochastic process with intermittent random resetting.

Optimal Resetting Brownian Bridges via Enhanced Fluctuations

We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time t_{f} is finite and the searcher returns to its starting point at t_{f}. This is simply a Brownian motion with a Poissonian resetting rate r to the origin which is constrained to start and end at the origin at time t_{f}. We unveil a surprising general mechanism that enhances fluctuations of a Brownian bridge, by introducing a small amount of resetting. This is verified for different observables, such as the mean-square displacement, the hitting probability of a fixed target and the expected maximum. This mechanism, valid for a Brownian bridge in arbitrary dimensions, leads to a finite optimal resetting rate that minimizes the time to search a fixed target. The physical reason behind an optimal resetting rate in this case is entirely different from that of resetting Brownian motions without the bridge constraint. We also derive an exact effective Langevin equation that generates numerically the trajectories of a resetting Brownian bridge in all dimensions via a completely rejection-free algorithm.

Optimal escape-and-feeding dynamics of random walkers: Rethinking the convenience of ballistic strategies

Excited random walks represent a convenient model to study food intake in a media which is progressively depleted by the walker. Trajectories in the model alternate between (i) feeding and (ii) escape (when food is missed and so it must be found again) periods, each governed by different movement rules. Here, we explore the case where the escape dynamics is adaptive, so at short times an area-restricted search is carried out, and a switch to extensive or ballistic motion occurs later if necessary. We derive for this case explicit analytical expressions of the mean escape time and the asymptotic growth of the depleted region in one dimension. These, together with numerical results in two dimensions, provide surprising evidence that ballistic searches are detrimental in such scenarios, a result which could explain why ballistic movement is barely observed in animal searches at microscopic and millimetric scales, therefore providing significant implications for biological foraging.

Target competition for resources under multiple search-and-capture events with stochastic resetting

We develop a general framework for analysing the distribution of resources in a population of targets under multiple independent search-and-capture events. Each event involves a single particle executing a stochastic search that resets to a fixed location x r at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns to x r , where it is reloaded with cargo and a new round of search and capture begins. Using renewal theory, we determine the mean number of resources in each target as a function of the splitting probabilities and unconditional mean first passage times of the corresponding search process without resetting. We then use asymptotic PDE methods to determine the effects of resetting on the distribution of resources generated by diffusive search in a bounded two-dimensional domain with N small interior targets. We show that slow resetting increases the total number of resources M tot across all targets provided that ∑ j = 1 N G ( x r , x j ) < 0 , where G is the Neumann Green's function and x j is the location of the j-th target. This implies that M tot can be optimized by varying r. We also show that the k-th target has a competitive advantage if ∑ j = 1 N G ( x r , x j ) > N G ( x r , x k ) .

Bounding energy growth in frictionless stochastic oscillators

This paper presents analytical and numerical results on the energetics of nonharmonic, undamped, single-well, stochastic oscillators driven by additive Gaussian white noises. The absence of damping and the action of noise are responsible for the lack of stationary states in such systems. We explore the properties of average kinetic, potential, and total energies along with the generalized equipartition relations. It is demonstrated that in frictionless dynamics, nonequilibrium stationary states can be produced by stochastic resetting. For an appropriate resetting protocol, the average energies become bounded. If the resetting protocol is not characterized by a finite variance of renewal intervals, stochastic resetting can only slow down the growth of the average energies but it does not bound them. Under special conditions regarding the frequency of resets, the ratios of the average energies follow the generalized equipartition relations.

Scaled Brownian motion with renewal resetting

We investigate an intermittent stochastic process in which the diffusive motion with time-dependent diffusion coefficient D(t)∼t^{α-1} with α>0 (scaled Brownian motion) is stochastically reset to its initial position, and starts anew. In the present work we discuss the situation in which the memory on the value of the diffusion coefficient at a resetting time is erased, so that the whole process is a fully renewal one. The situation when the resetting of the coordinate does not affect the diffusion coefficient's time dependence is considered in the other work of this series [A. S. Bodrova et al., Phys. Rev. E 100, 012119 (2019)10.1103/PhysRevE.100.012119]. We show that the properties of the probability densities in such processes (erasing or retaining the memory on the diffusion coefficient) are vastly different. In addition we discuss the first-passage properties of the scaled Brownian motion with renewal resetting and consider the dependence of the efficiency of search on the parameters of the process.

Intelligent Recommender System for Big Data Applications Based on the Random Neural Network

Online market places make their profit based on their advertisements or sales commission while businesses have the commercial interest to rank higher on recommendations to attract more customers. Web users cannot be guaranteed that the products provided by recommender systems within Big Data are either exhaustive or relevant to their needs. This article analyses the product rank relevance provided by different commercial Big Data recommender systems (Grouplens film, Trip Advisor and Amazon); it also proposes an Intelligent Recommender System (IRS) based on the Random Neural Network; IRS acts as an interface between the customer and the different Recommender Systems that iteratively adapts to the perceived user relevance. In addition, a relevance metric that combines both relevance and rank is presented; this metric is used to validate and compare the performance of the proposed algorithm. On average, IRS outperforms the Big Data recommender systems after learning iteratively from its customer.

Neural Networks in Big Data and Web Search

As digitalization is gradually transforming reality into Big Data, Web search engines and recommender systems are fundamental user experience interfaces to make the generated Big Data within the Web as visible or invisible information to Web users. In addition to the challenge of crawling and indexing information within the enormous size and scale of the Internet, e-commerce customers and general Web users should not stay confident that the products suggested or results displayed are either complete or relevant to their search aspirations due to the commercial background of the search service. The economic priority of Web-related businesses requires a higher rank on Web snippets or product suggestions in order to receive additional customers. On the other hand, web search engine and recommender system revenue is obtained from advertisements and pay-per-click. The essential user experience is the self-assurance that the results provided are relevant and exhaustive. This survey paper presents a review of neural networks in Big Data and web search that covers web search engines, ranking algorithms, citation analysis and recommender systems. The use of artificial intelligence (AI) based on neural networks and deep learning in learning relevance and ranking is also analyzed, including its utilization in Big Data analysis and semantic applications. Finally, the random neural network is presented with its practical applications to reasoning approaches for knowledge extraction.

Diffusion with resetting inside a circle

We study the Brownian motion of a particle in a bounded circular two-dimensional domain in search for a stationary target on the boundary of the domain. The process switches between two modes: one where it performs a two-dimensional diffusion inside the circle and one where it diffuses along the one-dimensional boundary. During the process, the Brownian particle resets to its initial position with a constant rate r. The Fokker-Planck formalism allows us to calculate the mean time to absorption (MTA) as well as the optimal resetting rate for which the MTA is minimized. From the derived analytical results the parameter regions where resetting reduces the search time can be specified. We also provide a numerical method for the verification of our results.

Characterization of stationary states in random walks with stochastic resetting

It is known that introducing a stochastic resetting in a random-walk process can lead to the emergence of a stationary state. Here we study this point from a general perspective through the derivation and analysis of mesoscopic (continuous-time random walk) equations for both jump and velocity models with stochastic resetting. In the case of jump models it is shown that stationary states emerge for any shape of the waiting-time and jump length distributions. The existence of such state entails the saturation of the mean square displacement to an universal value that depends on the second moment of the jump distribution and the resetting probability. The transient dynamics towards the stationary state depends on how the waiting time probability density function decays with time. If the moments of the jump distribution are finite then the tail of the stationary distributions is universally exponential, but for Lévy flights these tails decay as a power law whose exponent coincides with that from the jump distribution. For velocity models we observe that the stationary state emerges only if the distribution of flight durations has finite moments of lower order; otherwise, as occurs for Lévy walks, the stationary state does not exist, and the mean square displacement grows ballistically or superdiffusively, depending on the specific shape of the distribution of movement durations.

Phase transitions in optimal search times: How random walkers should combine resetting and flight scales

Recent works have explored the properties of Lévy flights with resetting in one-dimensional domains and have reported the existence of phase transitions in the phase space of parameters which minimizes the mean first passage time (MFPT) through the origin [L. Kusmierz et al., Phys. Rev. Lett. 113, 220602 (2014)]. Here, we show how actually an interesting dynamics, including also phase transitions for the minimization of the MFPT, can also be obtained without invoking the use of Lévy statistics but for the simpler case of random walks with exponentially distributed flights of constant speed. We explore this dynamics both in the case of finite and infinite domains, and for different implementations of the resetting mechanism to show that different ways to introduce resetting consistently lead to a quite similar dynamics. The use of exponential flights has the strong advantage that exact solutions can be obtained easily for the MFPT through the origin, so a complete analytical characterization of the system dynamics can be provided. Furthermore, we discuss in detail how the phase transitions observed in random walks with resetting are closely related to several ideas recurrently used in the field of random search theory, in particular, to other mechanisms proposed to understand random search in space as mortal random walks or multiscale random walks. As a whole, we corroborate that one of the essential ingredients behind MFPT minimization lies in the combination of multiple movement scales (regardless of their specific origin).

Random walk with random resetting to the maximum position

We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability r, and with probability (1-r), it undergoes symmetric random walk, i.e., it hops to one of its neighboring sites, with equal probability (1-r)/2. For r=0, it reduces to a standard random walk whose typical distance grows as √n for large n. In the presence of a nonzero resetting rate 0<r≤1, we find that both the average maximum and the average position grow ballistically for large n, with a common speed v(r). Moreover, the fluctuations around their respective averages grow diffusively, again with the same diffusion coefficient D(r). We compute v(r) and D(r) explicitly. We also show that the probability distribution of the difference between the maximum and the location of the walker becomes stationary as n→∞. However, the approach to this stationary distribution is accompanied by a dynamical phase transition, characterized by a weakly singular large deviation function. We also show that r=0 is a special "critical" point, for which the growth laws are different from the r→0 case and we calculate the exact crossover functions that interpolate between the critical (r=0) and the off-critical (r→0) behavior for finite but large n.

Geometry-induced fluctuations of olfactory searches in bounded domains

In olfactory search an immobile target emits chemical molecules at constant rate. The molecules are transported by the medium, which is assumed to be turbulent. Considering a searcher able to detect such chemical signals and whose motion follows the infotaxis strategy, we study the statistics of the first-passage time to the target when the searcher moves on a finite two-dimensional lattice of different geometries. Far from the target, where the concentration of chemicals is low, the direction of the searcher's first movement is determined by the geometry of the domain and the topology of the lattice, inducing strong fluctuations on the average search time with respect to the initial position of the searcher. The domain is partitioned in well-defined regions characterized by the direction of the first movement. If the search starts over the interface between two different regions, large fluctuations in the search time are observed.

Olfactory searches with limited space perception

Various insects and small animals can navigate in turbulent streams to find their mates (or food) from sparse pheromone (odor) detections. Their access to internal space perception and use of cognitive maps still are heavily debated, but for some of them, limited space perception seems to be the rule. However, this poor space perception does not prevent them from impressive search capacities. Here, as an attempt to model these behaviors, we propose a scheme that can perform, even without a detailed internal space map, searches in turbulent streams. The algorithm is based on a standardized projection of the probability of the source position to remove space perception and on the evaluation of a free energy, whose minimization along the path gives direction to the searcher. An internal "temperature" allows active control of the exploration/exploitation balance during the search. We demonstrate the efficiency of the scheme numerically, with a computational model of odor plume propagation, and experimentally, with robotic searches of thermal sources in turbulent streams. In addition to being a model to describe animals' searches, this scheme may be applied to robotic searches in complex varying media without odometry error corrections and in problems in which active control of the exploration/exploitation balance is profitable.

First passages in bounded domains: when is the mean first passage time meaningful?

We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(ω) distribution of the random variable ω=τ(1)/(τ(1)+τ(2)), which is a measure for how similar the first passage times τ(1) and τ(2) are of two independent realizations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(ω) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(ω), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.

Stochastic gene expression modeling with Hill function for switch-like gene responses

Gene expression models play a key role to understand the mechanisms of gene regulation whose aspects are grade and switch-like responses. Though many stochastic approaches attempt to explain the gene expression mechanisms, the Gillespie algorithm which is commonly used to simulate the stochastic models requires additional gene cascade to explain the switch-like behaviors of gene responses. In this study, we propose a stochastic gene expression model describing the switch-like behaviors of a gene by employing Hill functions to the conventional Gillespie algorithm. We assume eight processes of gene expression and their biologically appropriate reaction rates are estimated based on published literatures. We observed that the state of the system of the toggled switch model is rarely changed since the Hill function prevents the activation of involved proteins when their concentrations stay below a criterion. In ScbA-ScbR system, which can control the antibiotic metabolite production of microorganisms, our modified Gillespie algorithm successfully describes the switch-like behaviors of gene responses and oscillatory expressions which are consistent with the published experimental study.