The effect of target orientation on the mean first passage time of a Brownian particle to a small elliptical absorber

In this study, we propose a novel random walk strategy, termed “Random Walks with both Resetting and Nonnearest-Neighbor Jumps (RW-RNNJ)”. In order to determine RW-RNNJ’s capability to efficiently search for immobile targets on networks, we focused on the first passage problem and derived exact solutions for mean first passage time (MFPT) and global mean first passage time (GMFPT) on the (2,2) flowers network by the approach of probability generating function. The obtained results show that introduced resetting mechanism significantly reduces GMFPT even for large-scale networks, while nonnearest-neighbor jumps can alter the optimal choice of the resetting probability and have a linear effect on GMFPT. Therefore, the impact of resetting on search efficiency is global, while the effect of nonnearest-neighbor jumps is local. In summary, the findings obtained here may pave the way for optimizing diverse stochastic exploration processes such as animal foraging and the trapping of diffusing molecules in large networked systems.

Based on the Koch network constructed using Koch fractals, we proposed a class of expanded Koch networks in this paper. The original triangle is replaced by r-polygon, and each node generates m sub r-polygons by every step, which makes the Koch network more general. We studied the structure and properties of the networks. The exact analytical result of the degree distribution, clustering coefficient and average path length were obtained. When parameters m and r satisfy some certain conditions, the networks follow a power-law distribution and have a small average path length. Finally, we introduced the random walk on the network. Our discussions focused on the trapping problem, particularly the calculation and derivation of mean first passage time (MFPT) and global mean first passage time (GMFPT). In addition, we also gave the relationship between the above results and the network size.

We provide an explicit formula for the global mean first-passage time (GMFPT) for random walks in a general graph with a perfect trap fixed at an arbitrary node, where GMFPT is the average of mean first-passage time to the trap over all starting nodes in the whole graph. The formula is expressed in terms of eigenvalues and eigenvectors of Laplacian matrix for the graph. We then use the formula to deduce a tight lower bound for the GMFPT in terms of only the numbers of nodes and edges, as well as the degree of the trap, which can be achieved in both complete graphs and star graphs. We show that for a large sparse graph, the leading scaling for this lower bound is proportional to the system size and the reciprocal of the degree for the trap node. Particularly, we demonstrate that for a scale-free graph of size N with a degree distribution P(d) ∼ d(-γ) characterized by γ, when the trap is placed on a most connected node, the dominating scaling of the lower bound becomes N(1-1∕γ), which can be reached in some scale-free graphs. Finally, we prove that the leading behavior of upper bounds for GMFPT on any graph is at most N(3) that can be reached in the bar-bell graphs. This work provides a comprehensive understanding of previous results about trapping in various special graphs with a trap located at a specific location.

We present a general framework, applicable to a broad class of random walks on complex networks, which provides a rigorous lower bound for the mean first-passage time of a random walker to a target site averaged over its starting position, the so-called global mean first-passage time (GMFPT). This bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the GMFPT with the network size. We show that this minimal scaling, which can be arbitrarily slow, is realized under the simple condition that the random walk is transient at the target site and independently of the small-world, scale-free, or fractal properties of the network. Last, we put forward that the GMFPT to a specific target is not a representative property of the network since the target averaged GMFPT satisfies much more restrictive bounds.

In this paper, we obtain exact scalings of mean first-passage time (MFPT) of random walks on a family of small-world treelike networks formed by two parameters, which includes three kinds. First, we determine the MFPT for a trapping problem with an immobile trap located at the initial node, which is defined as the average of the first-passage times (FPTs) to the trap node over all possible starting nodes, and it scales linearly with network size N in large networks. We then analytically obtain the partial MFPT (PMFPT) which is the mean of FPTs from the trap node to all other nodes and show that it increases with N as N ln N. Finally we establish the global MFPT (GMFPT), which is the average of FPTs over all pairs of nodes. It also grows with N as N ln N in the large limit of N. For these three kinds of random walks, we all obtain the analytical expressions of the MFPT and they all increase with network parameters. In addition, our method for calculating the MFPT is based on the self-similar structure of the considered networks and avoids the calculations of the Laplacian spectra.

In this paper, we introduce a general computation method to systematically determine the mean first-passage time (MFPT), the global mean first-passage time (GMFPT) and splitting probabilities for a continuous Brownian motion in a confined 2D or 3D domain with multiple absorbing targets in the bulk or on the boundary. This method is applied to spherically symmetric domains in the limit of small-sized targets and asymptotic expansions of the MPFT, GMFPT and splitting probabilities are obtained in four distinct cases: 3D domains with targets in the bulk, 3D domains with targets on the boundary, 2D domains with targets on the bulk and 2D domains with targets on the boundary. This approach gives a unified description of existing exact results which were obtained using specific technics, and also yields new results, in particular for N targets splitting probabilities.

We study an unbiased random walk on dual Sierpinski gaskets embedded in d-dimensional Euclidean spaces. We first determine the mean first-passage time (MFPT) between a particular pair of nodes based on the connection between the MFPTs and the effective resistance. Then, by using the Laplacian spectra, we evaluate analytically the global MFPT (GMFPT), i.e., MFPT between two nodes averaged over all node pairs. Concerning these two quantities, we obtain explicit solutions and show how they vary with the number of network nodes. Finally, we relate our results for the case of d = 2 to the well-known Hanoi Towers problem.

First-passage processes on fractals are of particular importance since fractals are ubiquitous in nature, and first-passage processes are fundamental dynamic processes that have wide applications. The global mean first-passage time (GMFPT), which is the expected time for a walker (or a particle) to first reach the given target site while the probability distribution for the position of target site is uniform, is a useful indicator for the transport efficiency of the whole network. The smaller the GMFPT, the faster the mass is transported on the network. In this work, we consider the first-passage process on a class of fractal scale-free trees (FSTs), aiming at speeding up the first-passage process on the FSTs. Firstly, we analyze the global mean first-passage time (GMFPT) for unbiased random walks on the FSTs. Then we introduce proper weight, dominated by a parameter w (w > 0), to each edge of the FSTs and construct a biased random walks strategy based on these weights. Next, we analytically evaluated the GMFPT for biased random walks on the FSTs. The exact results of the GMFPT for unbiased and biased random walks on the FSTs are both obtained. Finally, we view the GMFPT as a function of parameter w and find the point where the GMFPT achieves its minimum. The exact result is obtained and a way to optimize and speed up the first-passage process on the FSTs is presented.

Mean first passage time for random walk on dual structure of dendrimer

The narrow escape problem refers to the problem of calculating the mean first passage time (MFPT) needed for an average Brownian particle to leave a domain with an insulating boundary containing N small well-separated absorbing windows, or traps. This mean first passage time satisfies the Poisson partial differential equation subject to a mixed Dirichlet-Neumann boundary condition on the domain boundary, with the Dirichlet condition corresponding to absorbing traps. In the limit of small total trap size, a common asymptotic theory is presented to calculate the MFPT in two-dimensional domains and in the unit sphere. The asymptotic MFPT formulas depend on mutual trap locations, allowing for global optimization of trap locations. Although the asymptotic theory for the MFPT was developed in the limit of asymptotically small trap radii, and under the assumption that the traps are well-separated, a comprehensive study involving comparison with full numerical simulations shows that the full numerical and asymptotic results for the MFPT are within 1% accuracy even when total trap size is only moderately small, and for traps that may be rather close together. This close agreement between asymptotic and numerical results at finite, and not necessarily asymptotically small, values of the trap size clearly illustrates one of the key side benefits of a theory based on a systematic asymptotic analysis. In addition, for the unit sphere, numerical results are given for the optimal configuration of a collection of traps on the surface of a sphere that minimizes the average MFPT. The case of N identical traps and a pattern of traps with two different sizes are considered. The effect of trap fragmentation on the average MFPT is also discussed.

The use of the global mean first-passage time (GMFPT) in random walks on networks has been widely explored in the field of statistical physics, both in theory and practical applications. The GMFPT is the estimated interval of time needed to reach a state j in a system from a starting state i. In contrast, there exists an intrinsic measure for a stochastic process, known as Kemeny’s constant, which is independent of the initial state. In the literature, it has been used as a measure of network efficiency. This article deals with a graph-spectrum-based method for finding both the GMFPT and Kemeny’s constant of random walks on spiro-ring networks (that are organic compounds with a particular graph structure). Furthermore, we calculate the Laplacian matrix for some specific spiro-ring networks using the decomposition theorem of Laplacian polynomials. Moreover, using the coefficients and roots of the resulting matrices, we establish some formulae for both GMFPT and Kemeny’s constant in these spiro-ring networks.

We present a simple derivation of mean residence times (MRTs) and mean first passage times (MFPTs) for random walks in finite one-dimensional systems. The derivation is based on the analysis of the inverse matrix of transition rates which represents the random walk rate equations. The dependence of the MRT and of the MFPT on the initial condition, on the system size, and on the elementary rates is studied and a relationship to stationary solutions is established. Applications to models of light harvesting by supermolecules, and of random barriers, and to relaxation in the Ehrenfest model are discussed in detail. We propose a way to control the MFPT in supermolecules, such as dendrimers, via molecular architecture.

This work discusses the Mean First Passage Time (MFPT) and Mean Residence Time (MRT) of continuous-time nearest-neighbor random walks in a finite one-dimensional system with a trap at the origin and a reflecting barrier at the other end. The asymptotic results of the MFPT for random walks that start, for example, at the reflecting point, have a variety of dependencies with respect to its size N. For example, for the case of birth and death processes the MFPT~N; for the case of symmetric random walks the MFPT~N2 and for the case of biased random walks the MFPT~αN where α is a constant that depends on the system’s rates. In this work a transition matrix is derived in such a way that the MRT of the system is equal to (m+1)d where m is the site number, and d is any arbitrary number satisfying the condition d>0. Since the MFPT is the sum of MRTs then the corresponding MFPT for such a transition matrix is MFPT~N(1+d). Thus, one can determine the asymptotic result of the MFPT to be N1+d for any arbitrary d>0, and based on it, obtain the corresponding transition matrix. Several examples of fractional and high order asymptotic results of the MFPT such as N3.5, N5, N6, are presented.

The Kirchhoff index, global mean-first passage time, average path length and number of spanning trees are of great importance in the field of networking. The “Kirchhoff index” is known as a structure descriptor index. The “global mean-first passage time” is known as a measure for nodes that are quickly reachable from the whole network. The “average path length” is a measure of the efficiency of information or mass transport on a network, and the “number of spanning trees” is used to minimize the cost of power networks, wiring connections, etc. In this paper, we have selected a complex network based on a categorical product and have used the spectrum approach to find the Kirchhoff index, global mean-first passage time, average path length and number of spanning trees. We find the expressions for the product and sum of reciprocals of all nonzero eigenvalues of a categorical product network with the help of the eigenvalues of the path and cycles.

We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. We show that, for directed networks extracted for real-life relationships, this centrality focused resetting can improve the search to a greater extent than for the generated undirected networks. This resetting to the center advocated here can minimize the average travel time to all other nodes in real networks as well. We also present a relationship between the longest shortest path (the diameter), the average node degree and the GMFPT when the starting node is the center. We show that, for undirected scale-free networks, stochastic resetting is effective only for networks that are extremely sparse with tree-like structures as they have larger diameters and smaller average node degrees. For directed networks, the resetting is beneficial even for networks that have loops. The numerical results are confirmed by analytic solutions. Our study demonstrates that the proposed random walk approach with resetting based on centrality measures reduces the memoryless search time for targets in the examined network topologies.