Random walks on complex networks with first-passage resetting

Impact of stochastic resetting on resource allocation: The case of reallocating geometric Brownian motion

We study the effects of stochastic resetting on the reallocating geometric Brownian motion (RGBM), an established model for resource redistribution relevant to systems such as population dynamics, evolutionary processes, economic activity, and even cosmology. The RGBM model is inherently nonstationary and non-ergodic, leading to complex resource redistribution dynamics. By introducing stochastic resetting, which periodically returns the system to a predetermined state, we examine how this mechanism modifies RGBM behavior. Our analysis uncovers distinct long-term regimes determined by the interplay between the resetting rate, the strength of resource redistribution, and standard geometric Brownian motion parameters: the drift and the noise amplitude. Notably, we identify a critical resetting rate beyond which the self-averaging time becomes effectively infinite. In this regime, the first two moments are stationary, indicating a stabilized distribution of an initially unstable, mean-repulsive process. We demonstrate that optimal resetting can effectively balance growth and redistribution, reducing inequality in the resource distribution. These findings help us understand better the management of resource dynamics in uncertain environments.

Drift-diffusive resetting search process with stochastic returns: Speedup beyond optimal instantaneous return

Stochastic resetting has recently emerged as a proficient strategy to reduce the completion time for a broad class of first-passage processes. In the canonical setup, one intermittently resets a given system to its initial configuration only to start afresh and continue evolving in time until the target goal is met. This is, however, an instantaneous process and thus less feasible for any practical purposes. A crucial generalization in this regard is to consider a finite-time return process which has significant ramifications to the firstpassage properties. Intriguingly, it has recently been shown that for diffusive search processes, returning in finite but stochastic time can gain significant speedup over the instantaneous resetting process. Unlike diffusion which has a diverging mean completion time, in this paper, we ask whether this phenomena can also be observed for a first-passage process with finite mean completion time. To this end, we explore the setup of a classical drift-diffusive search process in one dimension with stochastic resetting and further assume that the return phase is modulated by a potential U(x)=λ|x| with λ>0. For this process, we compute the mean first-passage time exactly and underpin its characteristics with respect to the resetting rate and potential strength. We find a unified phase space that allows us to explore and identify the system parameter regions where stochastic return supersedes over both the underlying process and the process under instantaneous resetting. Furthermore and quite interestingly, we find that for a range of parameters the mean completion time under stochastic return protocol can be reduced further than the optimally restarted instantaneous processes. We thus believe that resetting with stochastic returns can serve as a better optimization strategy owing to its dominance over classical first passage under resetting.

Random walks with stochastic resetting in complex networks: A discrete-time approach

We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light- and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time probability density functions, and in the light-tailed case, we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix, which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts-Strogatz and Barabási-Albert random graphs. We show nontrivial behavior of the dependence of the mean first passage time on the proportions of the relocation nodes, target nodes, and of the resetting rates. It turns out that, in the large-world case of the Watts-Strogatz graph, the efficiency of a random searcher particularly benefits from the presence of resets.

Random walks on scale-free flowers with stochastic resetting

This study explores the impact of stochastic resetting on the random walk dynamics within scale-free (u,v)-flowers. Utilizing the generating function technique, we develop a recursive relationship for the generating function of the first passage time and establish a connection between the mean first passage time with and without resetting. Our investigation spans multiple scenarios, with the random walker starting from various positions and aiming to reach different target nodes, allowing us to identify the optimal resetting probability that minimizes the mean first passage time for each case. We demonstrate that stochastic resetting significantly improves search efficiency, especially in larger networks. These findings underscore the effectiveness of stochastic resetting as a strategy for optimizing search algorithms in complex networks, offering valuable applications in domains such as biological transport, data networks, and search processes where rapid and efficient exploration is vital.

Random Walk on T-Fractal with Stochastic Resetting

In this study, we explore the impact of stochastic resetting on the dynamics of random walks on a T-fractal network. By employing the generating function technique, we establish a recursive relation between the generating function of the first passage time (FPT) and derive the relationship between the mean first passage time (MFPT) with resetting and the generating function of the FPT without resetting. Our analysis covers various scenarios for a random walker reaching a target site from the starting position; for each case, we determine the optimal resetting probability γ* that minimizes the MFPT. We compare the results with the MFPT without resetting and find that the inclusion of resetting significantly enhances the search efficiency, particularly as the size of the network increases. Our findings highlight the potential of stochastic resetting as an effective strategy for the optimization of search processes in complex networks, offering valuable insights for applications in various fields in which efficient search strategies are crucial.

Channel-facilitated transport under resetting dynamics

The transport of particles through channels holds immense significance in physics, chemistry, and biological sciences. For instance, the motion of solutes through biological membranes is facilitated by specialized proteins that create water-filled channels. Valuable insights can be obtained by studying the transition paths of particles through a channel and gathering information on their lifetimes inside the channel as well as their exit probabilities. In a similar vein, we consider a one-dimensional model of channel-facilitated transport where a diffusive particle is subject to attractive interactions with the walls of the channel. We study the statistics of conditional and unconditional escape times in the presence of resetting-an intermittent dynamics that brings the particle back to its initial coordinate stochastically. We determine analytically the physical conditions under which such a resetting mechanism becomes beneficial for the faster escape of the particles from the channel, thus enhancing transport. Our theory has been verified with the aid of Brownian dynamics simulations for various interaction strengths and extents. The overall results presented herein highlight the scope of resetting-based strategies to be universally promising for complex transport processes of single or long molecules through biological membranes.

Search with stochastic home returns can expedite classical first passage under resetting

Classical first passage under resetting is a paradigm in the search process. Despite its multitude of applications across interdisciplinary sciences, experimental realizations of such resetting processes posit practical challenges in calibrating these zero time irreversible transitions. Here, we consider a strategy in which resetting is performed using finite-time return protocols in lieu of instantaneous returns. These controls could also be accompanied with random fluctuations or errors allowing target detection even during the return phase. To better understand the phenomena, we develop a unified renewal approach that can encapsulate arbitrary search processes centered around home in a fairly general topography containing targets, various resetting times, and return mechanisms in arbitrary dimensions. While such finite-time protocols would apparently seem to prolong the overall search time in comparison to the instantaneous resetting process, we show on the contrary that a significant speed-up can be gained by leveraging the stochasticity in home returns. The formalism is then explored to reveal a universal criterion distilling the benefits of this strategy. We demonstrate how this general principle can be utilized to improve overall performance of a one-dimensional diffusive search process reinforced with experimentally feasible parameters. We believe that such strategies designed with inherent randomness can be made optimal with precise controllability in complex search processes.

Kuramoto model subject to subsystem resetting: How resetting a part of the system may synchronize the whole of it

We introduce and investigate the effects of a new class of stochastic resetting protocol called subsystem resetting, whereby a subset of the system constituents in a many-body interacting system undergoes bare evolution interspersed with simultaneous resets at random times, while the remaining constituents evolve solely under the bare dynamics. Here, by reset is meant a reinitialization of the dynamics from a given state. We pursue our investigation within the ambit of the well-known Kuramoto model of coupled phase-only oscillators of distributed natural frequencies. Here, the reset protocol corresponds to a chosen set of oscillators being reset to a synchronized state at random times. We find that the mean ω_{0} of the natural frequencies plays a defining role in determining the long-time state of the system. For ω_{0}=0, the system reaches a synchronized stationary state at long times, characterized by a time-independent nonzero value of the synchronization order parameter that quantifies macroscopic order in the system. Moreover, we find that resetting even an infinitesimal fraction of the total number of oscillators, in the extreme limit of infinite resetting rate, has the drastic effect of synchronizing the entire system, even in parameter regimes in which the bare evolution does not support global synchrony. By contrast, for ω_{0}≠0, the dynamics allows at long times either a synchronized stationary state or an oscillatory synchronized state, with the latter characterized by an oscillatory behavior as a function of time of the order parameter, with a nonzero time-independent time average. Our results thus imply that the nonreset subsystem always gets synchronized at long times through the act of resetting of the reset subsystem. Our results, analytical using the Ott-Antonsen ansatz as well as those based on numerical simulations, are obtained for two representative oscillator frequency distributions, namely, a Lorentzian and a Gaussian. Given that it is easier to reset a fraction of the system constituents than the entire system, we discuss how subsystem resetting may be employed as an efficient mechanism to control attainment of global synchrony in the Kuramoto system.

Optimizing search processes with stochastic resetting on the pseudofractal scale-free web

The pseudofractal scale-free web (PSFW) is a well-known model for a scale-free network with small-world characteristics. Understanding the dynamic properties of this network can provide valuable insights into dynamic processes occurring in general scale-free and small-world networks. In this study we investigate search processes using discrete-time random walks on the PSFW to reveal the impact of the resetting position on optimizing search efficiency, as measured by the mean first-passage time (MFPT). At each step the walker has two options: with a probability of 1-γ, it moves to one of the neighboring sites, and with a probability of γ, it resets to the predefined resetting position. We explore various choices for the resetting position, present rigorous results for the MFPT to a given node of the network, determine the optimal resetting probability γ^{*} where the MFPT reaches its minimum, and evaluate the ratio of the minimum for MFPT to the MFPT without resetting for each case. Results show that, in large PSFWs, both the degree of the resetting position and the distance between the target and the resetting position significantly affect the search efficiency. A higher degree of the resetting position leads to a slower convergence of the walker to the target, while a greater distance between the target and the resetting position also results in a slower convergence. Additionally, we observe that resetting to a vertex randomly selected from the stationary distribution can significantly expedite the process of the walker reaching the target. The findings presented in this study shed light on optimizing stochastic search processes on large networks, offering valuable insights into improving search efficiency in real-world applications, where the target node's location is unknown.

Ornstein-Uhlenbeck process and generalizations: Particle dynamics under comb constraints and stochastic resetting

The Ornstein-Uhlenbeck process is interpreted as Brownian motion in a harmonic potential. This Gaussian Markov process has a bounded variance and admits a stationary probability distribution, in contrast to the standard Brownian motion. It also tends to a drift towards its mean function, and such a process is called mean reverting. Two examples of the generalized Ornstein-Uhlenbeck process are considered. In the first one, we study the Ornstein-Uhlenbeck process on a comb model, as an example of the harmonically bounded random motion in the topologically constrained geometry. The main dynamical characteristics (as the first and the second moments) and the probability density function are studied in the framework of both the Langevin stochastic equation and the Fokker-Planck equation. The second example is devoted to the study of the effects of stochastic resetting on the Ornstein-Uhlenbeck process, including stochastic resetting in the comb geometry. Here the nonequilibrium stationary state is the main question in task, where the two divergent forces, namely, the resetting and the drift towards the mean, lead to compelling results in the cases of both the Ornstein-Uhlenbeck process with resetting and its generalization on the two-dimensional comb structure.

Universal Framework for Record Ages under Restart

We propose a universal framework to compute record age statistics of a stochastic time series that undergoes random restarts. The proposed framework makes minimal assumptions on the underlying process and is furthermore suited to treat generic restart protocols going beyond the Markovian setting. After benchmarking the framework for classical random walks on the 1D lattice, we derive a universal criterion underpinning the impact of restart on the age of the nth record for generic time series with nearest-neighbor transitions. Crucially, the criterion contains a penalty of order n that puts strong constraints on restart expediting the creation of records, as compared to the simple first-passage completion. The applicability of our approach is further demonstrated on an aggregation-shattering process where we compute the typical growth rates of aggregate sizes. This unified framework paves the way to explore record statistics of time series under restart in a wide range of complex systems.

Kinetic reconstruction of free energies as a function of multiple order parameters

A vast array of phenomena, ranging from chemical reactions to phase transformations, are analyzed in terms of a free energy surface defined with respect to a single or multiple order parameters. Enhanced sampling methods are typically used, especially in the presence of large free energy barriers, to estimate free energies using biasing protocols and sampling of transition paths. Kinetic reconstructions of free energy barriers of intermediate height have been performed, with respect to a single order parameter, employing the steady state properties of unconstrained simulation trajectories when barrier crossing is achievable with reasonable computational effort. Considering such cases, we describe a method to estimate free energy surfaces with respect to multiple order parameters from a steady state ensemble of trajectories. The approach applies to cases where the transition rates between pairs of order parameter values considered is not affected by the presence of an absorbing boundary, whereas the macroscopic fluxes and sampling probabilities are. We demonstrate the applicability of our prescription on different test cases of random walkers executing Brownian motion in order parameter space with an underlying (free) energy landscape and discuss strategies to improve numerical estimates of the fluxes and sampling. We next use this approach to reconstruct the free energy surface for supercooled liquid silicon with respect to the degree of crystallinity and density, from unconstrained molecular dynamics simulations, and obtain results quantitatively consistent with earlier results from umbrella sampling.

Entropy rate of random walks on complex networks under stochastic resetting

Stochastic processes under resetting at random times have attracted a lot of attention in recent years and served as illustrations of nontrivial and interesting static and dynamic features of stochastic dynamics. In this paper, we aim to address how the entropy rate is affected by stochastic resetting in discrete-time Markovian processes, and we explore nontrivial effects of the resetting in the mixing properties of a stochastic process. In particular, we consider resetting random walks (RRWs) with a single resetting node on three different types of networks: degree-regular random networks, a finite-size Cayley tree, and a Barabási-Albert (BA) scale-free network, and we compute the entropy rate as a function of the resetting probability γ. Interestingly, for the first two types of networks, the entropy rate shows a nonmonotonic dependence on γ. There exists an optimal value of γ at which the entropy rate reaches a maximum. Such a maximum is larger than that of maximal-entropy random walks (MREWs) and standard random walks (SRWs) on the same topology, while for the BA network the entropy rate of RRWs either shows a unique maximum or decreases monotonically with γ, depending upon the choice of the resetting node. When the maximum entropy rate of RRWs exists, it can be higher or lower than that of MREWs or SRWs. Our study reveals a nontrivial effect of stochastic resetting on nonequilibrium statistical physics.

Random walks on complex networks under time-dependent stochastic resetting

We study discrete-time random walks on networks subject to a time-dependent stochastic resetting, where the walker either hops randomly between neighboring nodes with a probability 1-ϕ(a) or is reset to a given node with a complementary probability ϕ(a). The resetting probability ϕ(a) depends on the time a since the last reset event (also called a, the age of the walker). Using the renewal approach and spectral decomposition of the transition matrix, we formulate the stationary occupation probability of the walker at each node and the mean first passage time between two arbitrary nodes. Concretely, we consider two different time-dependent resetting protocols that are both exactly solvable. One is that ϕ(a) is a step-shaped function of a and the other one is that ϕ(a) is a rational function of a. We demonstrate the theoretical results on several different networks, also validated by numerical simulations, and find that the time-modulated resetting protocols can be more advantageous than the constant-probability resetting in accelerating the completion of a target search process.

First passage of a diffusing particle under stochastic resetting in bounded domains with spherical symmetry

We investigate the first passage properties of a Brownian particle diffusing freely inside a d-dimensional sphere with absorbing spherical surface subject to stochastic resetting. We derive the mean time to absorption (MTA) as functions of resetting rate γ and initial distance r of the particle to the center of the sphere. We find that when r>r_{c} there exists a nonzero optimal resetting rate γ_{opt} at which the MTA is a minimum, where r_{c}=sqrt[d/(d+4)]R and R is the radius of the sphere. As r increases, γ_{opt} exhibits a continuous transition from zero to nonzero at r=r_{c}. Furthermore, we consider that the particle lies between two two-dimensional or three-dimensional concentric spheres with absorbing boundaries, and obtain the domain in which resetting expedites the MTA, which is (R_{1},r_{c_{1}})∪(r_{c_{2}},R_{2}), with R_{1} and R_{2} being the radii of inner and outer spheres, respectively. Interestingly, when R_{1}/R_{2} is less than a critical value, γ_{opt} exhibits a discontinuous transition at r=r_{c_{1}}; otherwise, such a transition is continuous. However, at r=r_{c_{2}} the transition is always continuous.

Random walks on complex networks with multiple resetting nodes: A renewal approach

Due to wide applications in diverse fields, random walks subject to stochastic resetting have attracted considerable attention in the last decade. In this paper, we study discrete-time random walks on complex networks with multiple resetting nodes. Using a renewal approach, we derive exact expressions of the occupation probability of the walker in each node and mean first-passage time between arbitrary two nodes. All the results can be expressed in terms of the spectral properties of the transition matrix in the absence of resetting. We demonstrate our results on circular networks, stochastic block models, and Barabási-Albert scale-free networks and find the advantage of the resetting processes to multiple resetting nodes in a global search on such networks. Finally, the distribution of resetting probabilities is optimized via a simulated annealing algorithm, so as to minimize the mean first-passage time averaged over arbitrary two distinct nodes.