Exact encounter times for many random walkers on regular and complex networks

Contact statistics in populations of noninteracting random walkers in two dimensions

The interaction between individuals in biological populations, dilute components of chemical systems, or particles transported by turbulent flows depends critically on their contact statistics. This work clarifies those statistics under the simplifying assumptions that the underlying motions approximate a Brownian random walk and that the particles can be treated as noninteracting. We measure the contact-interval (also called the waiting-time or interarrival-time), contact-count, and contact-duration distributions in populations of individuals undergoing noninteracting continuous-space-time random walks on a periodic two-dimensional plane (a torus) as functions of the population number density, walker radius, and random-walk step size. The contact-interval is exponentially distributed for times longer than the ballistic mean-free-collision time but not for times shorter than that, and the contact duration distribution is strongly peaked at the ballistic-crossing time for head-on collisions when the ballistic-crossing time is short compared to the mean step duration. While successive contacts between individuals are independent, the probability of repeat contact decreases with time after a previous contact. This leads to a negative duration dependence of the waiting-time interval and overdispersion of the contact-count probability density function for all time intervals. The paper demonstrates that for populations of small particles (with a walker radius that is small compared to the mean-separation or random-walk step size), the ballistic mean-free-collision interval, the ballistic-crossing time, and the random-walk-step duration can be used to construct temporal scalings which allow for common waiting-time, contact-count, and contact-duration distributions across different populations. Semi-analytic approximations for both the waiting-time and contact-duration distributions are also presented.

Mean encounter times for multiple random walkers on networks

We introduce a general approach for the study of the collective dynamics of noninteracting random walkers on connected networks. We analyze the movement of R independent (Markovian) walkers, each defined by its own transition matrix. By using the eigenvalues and eigenvectors of the R independent transition matrices, we deduce analytical expressions for the collective stationary distribution and the average number of steps needed by the random walkers to start in a particular configuration and reach specific nodes the first time (mean first-passage times), as well as global times that characterize the global activity. We apply these results to the study of mean first-encounter times for local and nonlocal random walk strategies on different types of networks, with both synchronous and asynchronous motion.

Stigmergy, collective actions, and animal social spacing

Collective animal behavior studies have led the way in developing models that account for a large number of individuals, but mostly have considered situations in which alignment and attraction play a key role, such as in schooling and flocking. By quantifying how animals react to one another's presence, when interaction is via conspecific avoidance rather than alignment or attraction, we present a mechanistic insight that enables us to link individual behavior and space use patterns. As animals respond to both current and past positions of their neighbors, the assumption that the relative location of individuals is statistically and history independent is not tenable, underscoring the limitations of traditional space use studies. We move beyond that assumption by constructing a framework to analyze spatial segregation of mobile animals when neighbor proximity may elicit a retreat, and by linking conspecific encounter rate to history-dependent avoidance behavior. Our approach rests on the knowledge that animals communicate by modifying the environment in which they live, providing a method to analyze social cohesion as stigmergy, a form of mediated animal-animal interaction. By considering a population of animals that mark the terrain as they move, we predict how the spatiotemporal patterns that emerge depend on the degree of stigmergy of the interaction processes. We find in particular that nonlocal decision rules may generate a nonmonotonic dependence of the animal encounter rate as a function of the tendency to retreat from locations recently visited by other conspecifics, which has fundamental implications for epidemic disease spread and animal sociality.

Random walk with priorities in communicationlike networks

We study a model for a random walk of two classes of particles (A and B). Where both species are present in the same site, the motion of A's takes precedence over that of B's. The model was originally proposed and analyzed in Maragakis et al. [Phys. Rev. E 77, 020103(R) (2008)]; here we provide additional results. We solve analytically the diffusion coefficients of the two species in lattices for a number of protocols. In networks, we find that the probability of a B particle to be free decreases exponentially with the node degree. In scale-free networks, this leads to localization of the B's at the hubs and arrest of their motion. To remedy this, we investigate several strategies to avoid trapping of the B's, including moving an A instead of the hindered B, allowing a trapped B to hop with a small probability, biased walk toward non-hub nodes, and limiting the capacity of nodes. We obtain analytic results for lattices and networks, and we discuss the advantages and shortcomings of the possible strategies.

Encounter times in overlapping domains: application to epidemic spread in a population of territorial animals

We develop an analytical method to calculate encounter times of two random walkers in one dimension when each individual is segregated in its own spatial domain and shares with its neighbor only a fraction of the available space, finding very good agreement with numerically exact calculations. We model a population of susceptible and infected territorial individuals with this spatial arrangement, and which may transmit an epidemic when they meet. We apply the results on encounter times to determine analytically the macroscopic propagation speed of the epidemic as a function of the microscopic characteristics: the confining geometry, the animal diffusion constant, and the infection transmission probability.