The statistics of first-passage times of random walks to target sites has proved to play a key role in determining the kinetics of space exploration in various contexts. In parallel, the number of distinct sites visited by a random walker and related observables have been introduced to characterize the geometry of space exploration. Here, we address the question of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies the coupling between kinetics and geometry of search trajectories. Focusing on 1-dimensional systems, we present a general method and derive explicit expressions of this joint distribution for several representative examples of Markovian search processes. In addition, we obtain a general scaling form, which holds also for non Markovian processes and captures the general dependence of the joint distribution on its space and time variables. We argue that the joint distribution has important applications to various problems, such as a conditional form of the Rosenstock trapping model, and the persistence properties of self-interacting random walks.
We propose local-biased random walks on general networks where a Markovian walker can choose between different types of biases in each node to define transitions to its neighbors depending on their degrees. For this ergodic dynamics, we explore the capacity of the random walker to visit all the nodes characterized by a global mean first passage time. This quantity is calculated using eigenvalues and eigenvectors of the transition matrix that defines the dynamics. In the first part, we illustrate how our framework leads to optimal transport for small-size graphs through the analysis of all the …
The territory explored by a random walk is a key property that may be quantified by the number of distinct sites that the random walk visits up to a given time. The extent of this spatial exploration characterizes many important physical, chemical, and ecological phenomena. In spite of its fundamental interest and wide utility, the number of visited sites gives only an incomplete picture of this exploration. In this work, we introduce a more fundamental quantity, the elapsed time $τ_n$ between visits to the $n^{\rm th}$ and the $(n+1)^{\rm st}$ distinct sites, from which the full dynamics abou…
Self-interacting random walks are endowed with long range memory effects that emerge from the interaction of the random walker at time $t$ with the territory that it has visited at earlier times $t'<t$. This class of non Markovian random walks has applications in a broad range of examples, ranging from insects to living cells, where a random walker modifies locally its environment -- leaving behind footprints along its path, and in turn responds to its own footprints. Because of their inherent non Markovian nature, the exploration properties of self-interacting random walks have remained elusi…
Random walks are the simplest way to explore or search a graph, and have revealed a very useful tool to investigate and characterize the structural properties of complex networks from the real world, e.g. they have been used to identify the modules of a given network, its most central nodes and paths, or to determine the typical times to reach a target. Although various types of random walks whose motion is node biased have been proposed, which are still amenable to analytical solution, most if not all of them rely on the assumption of linearity and independence of the walkers. We introduce a …
Self-repelling two-leg (biped) spider walk is considered where the local stochastic movements are governed by two independent control parameters $ β_d$ and $ β_h $, so that the former controls the distance ($ d $) between the legs positions, and the latter controls the statistics of self-crossing of the traversed paths. The probability measure for local movements is supposed to be the one for the "true self-avoiding walk" multiplied by a factor exponentially decaying with $ d $. After a transient behavior for short times, a variety of behaviors have been observed for large times depending on t…
Random walks are a common model for exploration and discovery of complex networks. While numerous algorithms have been proposed to map out an unknown network, a complementary question arises: in a known network, which nodes and edges are most likely to be discovered by a random walker in finite time? Here we introduce exposure theory, a statistical mechanics framework that predicts the learning of nodes and edges across several types of networks, including weighted and temporal, and show that edge learning follows a universal trajectory. While the learning of individual nodes and edges is nois…
Different kinds of random walks have proven to be useful in the study of structural properties of complex networks. Among them, the restricted dynamics of self-avoiding random walks (SAW), which visit only at most once each vertex in the same walk, has been successfully used in network exploration. The detection of communities of strongly connected vertices in networks remains an open problem, despite its importance, due to the high computational complexity of the associated optimization problem and the lack of a unique formal definition of communities. In this work, we propose a SAW-based met…
