In this work, we study the dynamics of multiple random walkers on networks subject to a simultaneous resetting protocol, whereby all walkers are synchronously returned to their respective initial nodes. For this collective Markovian process, we derive exact analytical expressions for the mean first-encounter time, defined as the average time required for all walkers to meet for the first time at a given node. These results are formulated in terms of the eigenvalues and eigenvectors of the transition matrices governing the dynamics without resetting, providing a clear spectral interpretation of the impact of resetting on encounter processes. We further establish a general criterion for finite networks that determines when the introduction of a nonzero resetting probability reduces the mean first-encounter time and leads to an optimal resetting strategy. The theoretical predictions are illustrated through numerical results on regular and heterogeneous networks, for encounters involving two or more walkers, and for combinations of local and nonlocal dynamics. Our findings demonstrate that simultaneous resetting can significantly reduce encounter times for specific target nodes and initial conditions, while becoming ineffective for highly exploratory dynamics or distant targets. A comparison with independent resetting shows that simultaneous resetting is more efficient in homogeneous networks, whereas independent resetting can outperform it in heterogeneous structures, thereby revealing a trade-off between synchronization and exploration. The framework provides a unified approach to collective search and encounter problems on networks with resetting.

We solve a model of sluggish stochastic motion in which a Brownian particle diffuses with a diffusion coefficient that decays algebraically with the distance to the origin, as |x|(-alpha). Additionally, the particle resets with a constant rate r to positions previously visited in the past, so that frequently visited regions are more likely to be revisited. An exact expression is obtained at all times for the position distribution in arbitrary spatial dimensions. At late times, the typical displacement of the walker from the origin grows extremely slowly, as [ln(rt)](1/(alpha+2)), and the position distribution tends to a scaling law. For any alpha > 0, the scaling function has a bimodal shape with a minimum at x = 0 and non-Gaussian tails. Although the mean square displacement is hard to compute, some generalised moments of this process can be calculated exactly at all times in one dimension, and are shown to be closely related to the moments of the well-studied model with a constant diffusion coefficient.

We study the emergence of correlations between N components of the position of a diffusive walker in N dimensions that starts at the origin and resets to previously visited sites with certain probabilities. This is equivalent to N independent one-dimensional diffusive processes starting from the origin and being subject to simultaneous resetting to positions visited in the past. Resetting follows a memory kernel that interpolates between resetting to the origin only, and the preferential relocation model, a path-dependent process which is highly non-Markov. For weak memory, the correlation coefficient between two components of the N-dimensional process grows monotonously with time and tends at late times to a constant bounded by 1/5, the value corresponding to the non-equilibrium steady state of resetting to the origin. When memory is sufficiently long-ranged, the correlation is non-monotonous and reaches a maximum at a finite time before converging to its asymptotic value. These two regimes are separated by a critical memory parameter value. In the limiting case of the preferential relocation model, the components become uncorrelated at both short and long times, but the correlation vanishes logarithmically slowly at late times. The emergence of correlations through resetting can be described in a unified way in all cases by noticing that the processes are conditionally independent and identically distributed, even in the presence of memory. In the non-Markovian case, the conditioning parameter is the duration of a Brownian path composed of several parts of the full trajectory of a fixed duration t.

Collectives are often able to process information in a distributed fashion, surpassing each individual member’s processing capacity. In fission-fusion dynamics, where group members come together and split from others often, sharing complementary information about uniquely known foraging areas could allow a group to track a heterogenous foraging environment better than any group member on its own. We analyse the partial overlaps between individual spider monkey core ranges, which we assume represent the knowledge of an individual during a given season. Sets of individuals with complementary overlaps are identified, showing a balance between redundantly and uniquely known portions, and we use simplicial complexes to represent these higher-order interactions. The structures of the simplicial complexes show holes in various dimensions, revealing complementarity in the foraging information that is being shared. We propose that the complex spatial networks arising from fission-fusion dynamics allow for adaptive, collective processing of foraging information in dynamic environments.

Collective systems that self-organise to maximise the group’s ability to collect and distribute information can be successful in environments with high spatial and temporal variation. Such organisations are abundant in nature, as sharing information is a key benefit of many biological collective systems, and have been influential in the design of many artificial collectives such as swarm robotics. Understanding how these systems may be spatially distributed to optimise their collective potential is therefore of importance in both ecology and in collective systems design. Here, we develop a mathematical model which uses an optimisation framework to determine the higher-order spatial structure of a collective that optimises group-level knowledge transfer. The domain of the objective function is a set of weighted hypergraphs, which can fully represent the spatial structure from a topological perspective. By varying the parameters within the objective function and the constraints, we determine how the optimal spatial structure may vary when individuals differ in their information gathering ability and how this variation differs in the context of resource constraints. Our key findings are that the amount of resources in the environment can lead to specific subgroup sizes being optimal for the group as a whole when individuals are homogeneous in their information gathering abilities. Further, when there is variation in information gathering abilities, our model implies that the sharing of space between smaller subgroups of the population, rather than the whole population, is optimal for collective knowledge sharing. Our results have applications across diverse contexts from behavioural ecology to bio-inspired collective systems design.

Diffusion with an incorporated resetting mechanism provides a reference framework for modeling a wide range of natural phenomena. Within this framework, the optimal resetting rate is a key quantity that arises from the optimization of the mean first-passage time. While substantial work has focused on the study of the optimal resetting rate in unbounded one dimensional domains, little is still known about the optimization of the mean first-passage time in bounded systems, in particular when multiple resetting sites are available. In this work, we consider a particle diffusing along a circular circumference and under resetting, with an absorbing target site at a fixed location. Using the appropriate free propagator for this system, we compute the Laplace transform of the survival probability when resetting occurs to multiple sites drawn from an arbitrary probability density function. We also calculate the mean first-passage time at the target site, and study the dependence of the optimal resetting rate in terms of the relevant parameters of the system in a two-resetting site configuration. Depending on the arc length between one of the resetting sites and the absorbing target site, and the weight of the remaining resetting site, the optimal resetting rate can exhibit abrupt ("first order”) and continuous ("second order”) transitions. Moreover, the behavior of the mean first-passage time is rich enough to allow both critical and tri-critical points to exist in the parameter space. All the transitions have "mirror symmetry” around the selected target site and its corresponding diametrically opposite site.

Processes controlled by stochastic synthesis and degradation (SSD) are widespread in biology but their reaction kinetics are not well understood. Using methods borrowed from the theory of resetting processes, we determine the first-passage properties of a collection of independent particles that are synthesized and degraded at constant rates, and follow an arbitrary diffusive process in space. At equal synthesis and degradation rates, the mean reaction time with a target site can be minimized as in stochastic resetting, and a $CV$-criterion is derived. When the degradation rate is held fixed and the synthesis costs are taken into account, an optimal synthesis rate is obtained. In bounded domains, despite particle degradation, SSD improves the mean search time compared to a single non-degrading particle if the synthesis rate exceeds a critical value. The latter obeys a universal relation. We illustrate these findings with Brownian diffusion on the infinite line and in an interval.

We study the relaxation of a diffusive particle confined in an arbitrary external potential and subject to a non-Markovian resetting protocol. With a constant rate r, a previous time tau between the initial time and the present time t is chosen from a given probability distribution K(tau,t), and the particle is reset to the position that it occupied at time tau. Depending on the shape of K(tau,t), the particle either relaxes toward the Gibbs-Boltzmann distribution or toward a non-trivial stationary distribution that breaks ergodicity and depends on the initial position and the resetting protocol. From a general asymptotic theory, we find that if the kernel K(tau,t) is sufficiently localized near tau = 0, i.e. mostly the initial part of the trajectory is remembered and revisited, the steady state is non-Gibbs-Boltzmann. Conversely, if K(tau,t) decays slowly enough or increases with tau, i.e. recent positions are more likely to be revisited, the probability distribution of the particle tends toward the Gibbs-Boltzmann state at large times. In the latter case, however, the temporal approach to the stationary state is generally anomalously slow, following for instance an inverse power law or a stretched exponential, if K(tau,t) is not too strongly peaked at the current time t. These findings are verified by the analysis of several exactly solvable cases and by numerical simulations.

We study an exactly solvable random walk model with long-range memory on arbitrary networks. The walker performs unbiased random steps to nearest-neighbor nodes and intermittently resets to previously visited nodes in a preferential way such that the most visited nodes have proportionally a higher probability to be chosen for revisit. The occupation probability can be expressed as a sum over the eigenmodes of the standard random walk matrix of the network, where the amplitudes slowly decay as power-laws at large times, instead of exponentially. The stationary state is the same as in the absence of memory, and detailed balance is fulfilled. However, the relaxation of the transient part becomes critically self-organized at late times, as it is dominated by a single power-law whose exponent depends on the second largest eigenvalue and on the resetting probability. We apply our findings to finite networks, such as rings, complete graphs, Watts–Strogatz, and Barabási–Albert networks, and to Barbell and comb-like graphs. Our study could be of interest for modeling complex transport phenomena, such as human mobility, epidemic spreading, or animal foraging.

We present several arguments for the preeminence of social interactions in determining and giving shape to societies. In our view, a society can emerge from social interaction and relationship patterns without the need for establishing an a priori limit on who actually belongs to it. Markers of group identity are one element among many that allow societies to persist.

Brownian diffusion subject to stochastic resetting to a fixed position has been widely studied for applications to random search processes. In an unbounded domain, the mean first-passage time at a target site can be minimized for a convenient choice of the resetting rate. Here we study this optimization problem in one dimension when resetting occurs to random positions, chosen from a probability density function with compact support that does not include the target. Depending on the shape of this distribution, the optimal resetting rate either varies smoothly with the mean distance to the target, as in single-site resetting, or exhibits a discontinuity caused by the presence of a second local minimum in the mean first-passage time. These two regimes are separated by a critical line containing a singular point that we characterize through a Ginzburg-Landau theory. To quantify how useful is a given resetting point for the search, we calculate the probability density function of the last resetting position before absorption. The discontinuous transition separates two markedly different optimal strategies: one with a small resetting rate where the last path before absorption starts from a rather distant but likely position, while the other strategy has a large resetting rate, favoring last paths starting from not-so-likely points but which are closer to the target.

Economic competition and deregulation have led to a polarization of societies between a small, increasingly powerful elite and a majority of socially excluded individuals, marginalized and unconnected to political representations. This is the breeding ground for protest movements, relayed by local ties and amplified by social networks. Based on the characteristics revealed by socio-economic research into the Arab revolutions of the 2010s, this article proposes a formalization inspired by the theory of complex systems. We discuss the conditions under which an initial localized event - for example, the suicide of a street vendor condemned to ruin in a small Tunisian town - can trigger an explosion of the number of opponents to the regime, typical of a revolutionary episode. We consider a network model of agents and oppressors where pair interactions are controlled by a fear parameter, or the inclination of the individuals to rebel despite of repression. The model exhibits a phase transition at a critical threshold above which the quiescent state becomes unstable. Furthermore, the ability of individuals to forge triadic relationships accelerates the process of joining a rising rebellion, making the transition more abrupt, in the form of a brutal discontinuity that can be described as revolutionary. The imposition of a counter-revolution by the hardening of repression can be explained by the hysteresis property displayed by the model: when mobilization has extended the initial network and made it partially permanent, repression by the authorities must be tightened to higher levels than before to regain control.

Random walks on lattices with preferential relocation to previously visited sites provide a simple framework for modeling the displacements of animals and humans. When the lattice contains a few impurities or resource sites where the walker spends more time on average at each visit than on the other sites, the long-range memory can suppress diffusion and induce by reinforcement a steady state localized around a resource. This phenomenon can be identified with a spatial learning process. Here we study theoretically and numerically how the decay of memory impacts learning in a model with one impurity. If memory decays as 1/τ or slower, where τ is the time backward into the past, the localized solution is the same as with perfect, nondecaying memory and it is linearly stable. If forgetting is faster than 1/τ, for instance exponential, an unusual regime of intermittent localization is observed, where well-localized periods of exponentially distributed duration are disrupted by possibly long intervals of diffusive motion. At the transition between the two regimes, for a kernel in 1/τ, the approach to the stable localized state is the fastest, opposite to the expected critical slowing down effect. Hence, forgetting can allow the walker to save a lot of memory without compromising learning and to achieve a faster learning process. These findings agree with biological evidence on the benefits of forgetting.

In many random search processes of interest in chemistry, biology, or during rescue operations, an entity must find a specific target site before the latter becomes inactive, no longer available for reaction or lost. We present exact results on a minimal model system, a one-dimensional searcher performing a discrete time random walk, or Lévy flight. In contrast with the case of a permanent target, the capture probability and the conditional mean first passage time can be optimized. The optimal Lévy index takes a nontrivial value, even in the long lifetime limit, and exhibits an abrupt transition as the initial distance to the target is varied. Depending on the target lifetime, this transition is discontinuous or continuous, separated by a nonconventional tricritical point. These results pave the way to the optimization of search processes under time constraints.

The study of diffusion with preferential returns to places visited in the past has attracted increased attention in recent years. In these highly non-Markov processes, a standard diffusive particle intermittently resets at a given rate to previously visited positions. At each reset, a position to be revisited is randomly chosen with a probability proportional to the accumulated amount of time spent by the particle at that position. These preferential revisits typically generate a very slow diffusion, logarithmic in time, but still with a Gaussian position distribution at late times. Here we consider an active version of this model, where between resets the particle is self-propelled with constant speed and switches direction in one dimension according to a telegraphic noise. Hence there are two sources of non-Markovianity in the problem. We exactly derive the position distribution in Fourier space, as well as the variance of the position at all times. The crossover from the short-time ballistic regime, dominated by activity, to the long-time anomalous logarithmic growth induced by memory is studied. We also analytically derive a large deviation principle for the position, which exhibits a logarithmic time scaling instead of the usual algebraic form. Interestingly, at large distances, the large deviations become independent of time and match the nonequilibrium steady state of a particle under resetting to its starting position only.

We study the problem of a target search by a Brownian particle subject to stochastic resetting to a pair of sites. The mean search time is minimized by an optimal resetting rate which does not vary smoothly, in contrast with the well-known single site case, but exhibits a discontinuous transition as the position of one resetting site is varied while keeping the initial position of the particle fixed, or vice versa. The discontinuity vanishes at a "liquid-gas" critical point in position space. This critical point exists provided that the relative weight m of the further site is comprised in the interval [2.9028...,8.5603...]. When the initial position is a random variable that follows the resetting point distribution, a discontinuous transition also exists for the optimal rate as the distance between the resetting points is varied, provided that m exceeds the critical value m_{c}=6.6008.... This setup can be mapped onto an intermittent search problem with switching diffusion coefficients and represents a minimal model for the study of distributed resetting.

High order interactions (HOIs) are increasingly being studied as a crucial feature of ecological communities, as they are ubiquitous in nature and their effects can be profound. Moreover, it has been suggested by both hypothetical and empirically informed models that HOIs may be important for promoting the existence of highly diverse systems. In a diversified coffee system in Chiapas, Mexico, observations and field experiments have also suggested that this kind of interactions are partially responsible for the persistence of the system and the autonomous regulation of its potential pests. Here, we modeled the interaction network of 22 of the best documented species in this coffee system. We tested whether the inclusion of HOIs is important to the overall performance of the community in the face of species loss, measured as mean species survival and abundance of some key components. To this end, we used the framework of extended food webs, parametrized according to the bioenergetic model, which is based on allometric relationships between species. We found that HOIs presence did not result in significant differences in species survival or abundance. While specific HOIs may have very important ecological effects, this study provides no evidence of a general positive or negative role of HOIs in species survival rate or abundance. These findings are in line with a previous study of the same network, which from a purely structural point of view, found that HOIs had positive or neutral effects on the robustness of the network under different perturbations. Further studies are still necessary in order to gain a nuanced understanding of the role of HOIs in agroecosystems and ecosystems in general.

Models of fractal growth commonly consider particles diffusing in a medium and that stick irreversibly to the forming aggregate when making contact for the first time. As shown by the well-known diffusion limited aggregation (DLA) model and its generalisations, the fractal dimension is sensitive to the nature of the stochastic motion of the particles. Here, we study the structures formed by finite-lived Brownian particles, i.e. particles constrained to find the aggregate within a prescribed time, and which are removed otherwise. This motion can be modelled by diffusion with stochastic resetting, a class of processes which has been widely studied in recent years. In the short lifetime limit, a very small fraction of the particles manage to reach the aggregate. Hence, growth is controlled by atypical Brownian trajectories, that move nearly in straight line according to a large deviation principle. In d dimensions, the resulting fractal dimension of the aggregate decreases from the DLA value and tends to 1, instead of increasing to d as expected from ballistic aggregation. In the zero lifetime limit one recovers the non-trivial model of 'aggregation by the tips' proposed long ago by R. Jullien [J. Phys. A: Math. Gen. 19, 2129 (1986)].

We experimentally study the statistics of the transition path time taken by a submicron bead to successfully traverse an energy barrier created by two optical tweezers in two prototypical viscoelastic fluids, namely, aqueous polymer and micellar solutions. We find a very good agreement between our experimental distributions and a theoretical expression derived from the generalized Langevin equation for the particle motion. Our results reveal that the mean transition path times measured in such viscoelastic fluids have a nontrivial dependence on the barrier curvature and they can be significantly reduced when compared with those determined in Newtonian fluids of the same zero-shear viscosity. We verify that the decrease of the mean transition path time can be described in terms of an effective viscosity that quantitatively coincides with that measured by linear microrheology at a frequency determined by the reactive mode that gives rise to the unstable motion over the barrier. Therefore our results uncover the linear response of the particle during its thermally activated escape from a metastable state even when taking place in a non-Markovian bath.

CONTEXT: Coffee is produced under different management systems and scales of production categorized as Syndromes of Production. The "Capitalist Syndrome" is characterized by large-scale and high-density planting farms that may promote the development of plant pathogens like coffee leaf rust (CLR). Harvesting dynamics are also affected by the syndrome of production and generate spatial trajectories that could contribute to the dispersal of pathogens across and within plantations. However, these spatial trajectories have not yet been described, nor their relationship with the syndrome of production, and even less its potential ecological implications for pathogen dispersal. OBJECTIVE: Describe and analyze the daily spatial movement of coffee harvesters in two large-scale capitalist plantations, an organic and a conventional plantation, and systematize the drivers that might explain the differences in the spatial trajectories. METHODS: Using State-Space Models, we recorded and analyzed the spatial movements of harvesters. We then constructed a driver tree for harvest dynamics, which incorporated qualitative variables related to the environment, coffee biology, and management aspects reported by the harvesters or in previous studies. RESULTS AND CONCLUSIONS: The model differentiated two kinds of movements: 1) when trees have berries, harvesters remain in the coffee rows or areas nearby (Collect state; 94-98 % of the steps); 2) when not, harvesters make longer steps within the harvesting location or move to another area (Search state; 2-6 % of the steps). In the organic plantation, the Search state had a longer-tailed step-length distribution than in the conventional plantation, resulting in a significantly larger visited area per worker (p < 0.05). This might be directly related to the lower interplant ripening percentage or smaller harvesting locations ("pantes") per number of harvesters. The number of harvested trees might be affected by the fruit load or the coffee variety, among others. Harvesting movements that explore a larger area, either by visiting more plants or by changing locations on the same day, could create more foci of CLR infection across the plantation. SIGNIFICANCE: These results constitute an initial analysis of harvesting trajectories and highlight practices that can reduce the potential impact of human dispersal of pathogens, like shorter harvesting trajectories by working fewer hours a day or avoiding harvesting at the end of the maturation season when few trees have berries and harvesters have to travel medium to long distances. This calls for organic coffee management that could prevent diseases and guarantee just and safe conditions for workers.