Scaling Of Mean First Passage Time Research Articles

Tunable intracellular transport on converging microtubule morphologies

A common type of cytoskeletal morphology involves multiple microtubules converging with their minus ends at the microtubule organizing center (MTOC). Cargo-motor complex will experience ballistic transport when bound to microtubules, or diffusive transport when unbound. This machinery allows for sequestering and subsequent dispersal of dynein transported cargo. The general principles governing dynamics, efficiency and tunability of such transport in the MTOC vicinity is not fully understood. To address this, we develop a one-dimensional model that includes advective transport towards an attractor (such as the MTOC), and diffusive transport that allows particles to reach absorbing boundaries (such as cellular membranes). We calculated the mean first passage time (MFPT) for cargo to reach the boundaries as a measure of the effectiveness of sequestering (large MFPT) and diffusive dispersal (low MFPT). We show that the MFPT experiences a dramatic growth, transitioning from a low to high MFPT regime (dispersal to sequestering) over a window of cargo on-off rates that is close to in vivo values. Furthermore, increasing either the on (attachment) or off (detachment) rate can result in optimal dispersal when the attractor is placed asymmetrically. Finally, we also describe a regime of rare events where the MFPT scales exponentially with motor velocity and the escape location becomes exponentially sensitive to the attractor positioning. Our results suggest that structures such as the MTOC allow for the sensitive control of the spatial and temporal features of transport and corresponding function under physiological conditions.

Mean first-passage time of an active fluctuating membrane with stochastic resetting.

We study the mean first-passage time of a one-dimensional active fluctuating membrane that is stochastically returned to the same flat initial condition at a finite rate. We start with a Fokker-Planck equationto describe the evolution of the membrane coupled with an Ornstein-Uhlenbeck type of active noise. Using the method of characteristics, we solve the equationand obtain the joint distribution of the membrane height and active noise. In order to obtain the mean first-passage time (MFPT), we further obtain a relation between the MFPT and a propagator that includes stochastic resetting. The derived relation is then used to calculate it analytically. Our studies show that the MFPT increases with a larger resetting rate and decreases with a smaller rate, i.e., there is an optimal resetting rate. We compare the results in terms of MFPT of the membrane with active and thermal noises for different membrane properties. The optimal resetting rate is much smaller with active noise compared to thermal. When the resetting rate is much lower than the optimal rate, we demonstrate how the MFPT scales with resetting rates, distance to the target, and the properties of the membranes.

Random walks on Fibonacci treelike models

In this paper, we propose a class of growth models, named Fibonacci trees F(t), with respect to the nature of Fibonacci sequence {Ft}. First, we show that models F(t) have power-law degree distribution with exponent greater than 3. Then, we analytically study two significant topological indices, i.e., optimal mean first-passage time (OMFPT) and mean first-passage time (MFPT), for random walks on Fibonacci trees F(t), and obtain the analytical expressions using some combinatorial approaches. The methods used are widely applied for other network models with self-similar feature to derive analytical solution to OMFPT or MFPT, and we select a candidate model to validate this viewpoint. In addition, we observe from theoretical analysis and numerical simulation that the scaling of MFPT is linearly correlated with vertex number of models F(t), and show that Fibonacci trees F(t) possess more optimal topological structure than the classic scale-free tree networks.

APPLICATIONS OF LAPLACIAN SPECTRA ON A 3-PRISM GRAPH

In this paper, we calculate the Laplacian spectra of a 3-prism graph and apply them. This graph is both planar and polyhedral, and belongs to the generalized Petersen graph. Using the regular structures of this graph, we obtain the recurrent relationships for Laplacian matrix between this graph and its initial state — a triangle — and further derive the corresponding relationships for Laplacian eigenvalues between them. By these relationships, we obtain the analytical expressions for the product and the sum of the reciprocals of all nonzero Laplacian eigenvalues. Finally we apply these expressions to calculate the number of spanning trees and mean first-passage time (MFPT) and see that the scaling of MFPT with the network size N is N2, which is larger than those performed on some uniformly recursive trees.

Anomalous biased diffusion in networks

We study diffusion with a bias toward a target node in networks. This problem is relevant to efficient routing strategies in emerging communication networks like optical networks. Bias is represented by a probability p of the packet or particle to travel at every hop toward a site that is along the shortest path to the target node. We investigate the scaling of the mean first passage time (MFPT) with the size of the network. We find by using theoretical analysis and computer simulations that for random regular (RR) and Erdős-Rényi networks, there exists a threshold probability, p(th), such that for p<p(th) the MFPT scales anomalously as N(α), where N is the number of nodes, and α depends on p. For p>p(th), the MFPT scales logarithmically with N. The threshold value p(th) of the bias parameter for which the regime transition occurs is found to depend only on the mean degree of the nodes. An exact solution for every value of p is given for the scaling of the MFPT in RR networks. The regime transition is also observed for the second moment of the probability distribution function, the standard deviation. For the case of scale-free (SF) networks, we present analytical bounds and simulations results showing that the MFPT scales at most as lnN to a positive power for any finite bias, which means that in SF networks even a very small bias is considerably more efficient in comparison to unbiased walk.

Close or connected: Distance and connectivity effects on transport in networks

We develop an analytical approach that provides the dependence of the mean first-passage time (MFPT) for random walks on complex networks both on the target connectivity and on the source-target distance. Our approach puts forward two strongly different behaviors depending on the type-compact or non compact-of the random walk. In the case of non compact exploration, we show that the MFPT scales linearly with the inverse connectivity of the target and is largely independent of the starting point. On the contrary, in the compact case, the MFPT is controlled by the source-target distance, and we find that unexpectedly the target connectivity becomes irrelevant for remote targets.

Scaling of mean first-passage time as efficiency measure of nodes sending information on scale-free Koch networks

A lot of previous work showed that the sectional mean first-passage time (SMFPT), i.e., the average of mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) averaged over all starting points in scale-free small-world networks exhibits a sublinear or linear dependence on network order $N$ (number of nodes), which indicates that hub nodes are very efficient in receiving information if one looks upon the random walker as an information messenger. Thus far, the efficiency of a hub node sending information on scale-free small-world networks has not been addressed yet. In this paper, we study random walks on the class of Koch networks with scale-free behavior and small-world effect. We derive some basic properties for random walks on the Koch network family, based on which we calculate analytically the partial mean first-passage time (PMFPT) defined as the average of MFPTs from a hub node to all other nodes, excluding the hub itself. The obtained closed-form expression displays that in large networks the PMFPT grows with network order as $N \ln N$, which is larger than the linear scaling of SMFPT to the hub from other nodes. On the other hand, we also address the case with the information sender distributed uniformly among the Koch networks, and derive analytically the entire mean first-passage time (EMFPT), namely, the average of MFPTs between all couples of nodes, the leading scaling of which is identical to that of PMFPT. From the obtained results, we present that although hub nodes are more efficient for receiving information than other nodes, they display a qualitatively similar speed for sending information as non-hub nodes. Moreover, we show that the location of information sender has little effect on the transmission efficiency. The present findings are helpful for better understanding random walks performed on scale-free small-world networks.