Passage Time Of Random Walk Research Articles

First passage time analysis of spatial mutation patterns reveals sub-clonal evolutionary dynamics in colorectal cancer.

The signature of early cancer dynamics on the spatial arrangement of tumour cells is poorly understood, and yet could encode information about how sub-clones grew within the expanding tumour. Novel methods of quantifying spatial tumour data at the cellular scale are required to link evolutionary dynamics to the resulting spatial architecture of the tumour. Here, we propose a framework using first passage times of random walks to quantify the complex spatial patterns of tumour cell population mixing. First, using a simple model of cell mixing we demonstrate how first passage time statistics can distinguish between different pattern structures. We then apply our method to simulated patterns of mutated and non-mutated tumour cell population mixing, generated using an agent-based model of expanding tumours, to explore how first passage times reflect mutant cell replicative advantage, time of emergence and strength of cell pushing. Finally, we explore applications to experimentally measured human colorectal cancer, and estimate parameters of early sub-clonal dynamics using our spatial computational model. We infer a wide range of sub-clonal dynamics, with mutant cell division rates varying between 1 and 4 times the rate of non-mutated cells across our sample set. Some mutated sub-clones emerged after as few as 100 non-mutant cell divisions, and others only after 50,000 divisions. The majority were consistent with boundary driven growth or short-range cell pushing. By analysing multiple sub-sampled regions in a small number of samples, we explore how the distribution of inferred dynamics could inform about the initial mutational event. Our results demonstrate the efficacy of first passage time analysis as a new methodology in spatial analysis of solid tumour tissue, and suggest that patterns of sub-clonal mixing can provide insights into early cancer dynamics.

Kemeny's constant and global mean first passage time of random walks on octagonal cell network

This paper investigates the random walks of octagonal cell network. By using the Laplacian spectrum method, we obtain the mean first passage time and Kemeny's constant between nodes. On one hand, the mean first passage time is explicitly studied in terms of the eigenvalues of a Laplacian matrix. On the other hand, Kemeny's constant is introduced to measure node strength and to determine the scaling of the random walks. We provide an explicit expression of Kemeny's constant and mean first passage time for octagonal cell network, by their Laplacian eigenvalues and the correlation among roots of characteristic polynomial. Based on the achieved results, comparative studies are also performed for and . This work also deliver an inclusive approach for exploring random walks of networks, particularly biased random walks, which likewise support to better understand and tackle some practical problems such as search and routing on networks.

Consensus and Coherence in Fractal Networks

We consider first- and second-order consensus algorithms in networks with stochastic disturbances. We quantify the deviation from consensus using the notion of network coherence, which can be expressed as an $H_{2}$ norm of the stochastic system. We use the setting of fractal networks to investigate the question of whether a purely topological measure, such as the fractal dimension, can capture the asymptotics of coherence in the large system size limit. Our analysis for first-order systems is facilitated by connections between first-order stochastic consensus and the global mean first passage time of random walks. We then show how to apply similar techniques to analyze second-order stochastic consensus systems. Our analysis reveals that two networks with the same fractal dimension can exhibit different asymptotic scalings for network coherence. Thus, this topological characterization of the network does not uniquely determine coherence behavior. The question of whether the performance of stochastic consensus algorithms in large networks can be captured by purely topological measures, such as the spatial dimension, remains open.

On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice

In this paper, we form a method to calculate the probability generating function of the total progeny of multitype branching process. As examples, we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and (2-1) random walk. Consequently, we could give the probability generating functions and the distributions of the first passage time of corresponding random walks. Especially, for recurrent random walk which could stay at its position with probability 0 < r < 1, we show that the tail probability of the first passage time decays as \(\frac{2} {{\sqrt {\pi (1 - r)} }}\frac{1} {{\sqrt n }} \) when n → ∞.

Optimal and suboptimal networks for efficient navigation measured by mean-first passage time of random walks

For a random walk on a network, the mean first-passage time from a node i to another node j chosen stochastically, according to the equilibrium distribution of Markov chain representing the random walk is called the Kemeny constant, which is closely related to the navigability on the network. Thus, the configuration of a network that provides optimal or suboptimal navigation efficiency is a question of interest. It has been proved that complete graphs have the exact minimum Kemeny constant over all graphs. In this paper, by using another method we first prove that complete graphs are the optimal networks with a minimum Kemeny constant, which grows linearly with the network size. Then, we study the Kemeny constant of a class of sparse networks that exhibit remarkable scale-free and fractal features as observed in many real-life networks, which cannot be described by complete graphs. To this end, we determine the closed-form solutions to all eigenvalues and their degeneracies of the networks. Employing these eigenvalues, we derive the exact solution to the Kemeny constant, which also behaves linearly with the network size for some particular cases of networks. We further use the eigenvalue spectra to determine the number of spanning trees in the networks under consideration, which is in complete agreement with previously reported results. Our work demonstrates that scale-free and fractal properties are favorable for efficient navigation, which could be considered when designing networks with high navigation efficiency.

Passage times of random walks and Lévy processes across power law boundaries

We establish an integral test involving only the distribution of the increments of a random walk S which determines whether lim supn→∞(Sn/n κ ) is almost surely zero, finite or infinite when 1/2 <κ <1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller (9) concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ ≥ 0. The results, and those of (9), are also extended to Levy processes. 1. Random walks In (9) an almost complete solution was given to the problem of finding analytic conditions, expressed directly in terms of the step distribution F of the random walk S = (Sn ,n ≥ 0), for first-passage times over one-sided and two-sided power law boundaries of the random walk to be almost surely (a.s.) finite. The exception was for the one-sided passage time T ∗

Approximations to expected stopping times with applications to sequential estimation

In this paper we consider sharp asymptotic aproximations to renewal functions associated with general stochastic sequences. We establish a variant of Smith's renewal theorem with a bounded remainder term imposing moment conditions on the underlying stochastic sequence. This result is then applied to an asymptotic analysis of stopping times arising in sequential estimation problems. We show that the renewal theorem can be used for deriving approximations to the expected values of first passage times for random walks with dependent increments

Cutpoint Decoupling and First Passage Times for Random Walks on Graphs

One approach to the computations for Markov chains, due to Meyer, is to break a problem down into corresponding computations for several related chains involving a smaller number of states. In this spirit, we focus on the mean first passage matrix associated with a random walk on a connected graph, and consider the problem of transforming the computation of that matrix into smaller tasks. We show that this is possible when there is a cutpoint in the graph and provide an explicit formula for the mean first passage matrix when this is the case.

Multifractal scaling in Sinai diffusion

We consider the mean first passage time of random walks to go from one end of a segment of a Sinai lattice to the other. We show that the distribution of the mean first passage time over Sinai disorder obeys a finite-size-multiscaling form.

Equivalence of rates of convergence in the CLT between maximum partial sums and first passage times of random walks

Abstract Let {X n} by a sequence of iid r.v.'s such that {EX}1=µµ0 and Var X 1=σ2 C}, c⩾0. Set The following results are proved: 1. If Δ n =O(n−γ), 0<γ⩽½ then which is equivalent to w(c)=O(c−γ). 2. If then which is equivalent of

Random walks on random trees

Let T denote one of the nn−2 trees with n labelled nodes that is rooted at a given node x (see [6] or [8] as a general reference on trees). If i and j are any two nodes of T, we write i ∼ j if they are joined by an edge in T. We want to consider random walks on T; we assume that when we are at a node i of degree d the probability that we proceed to node j at the next step is di–1 if i ∼ j and zero otherwise. Our object here is to determine the first two moments of the first return and first passage times for random walks on T when T is a specific tree and when T is chosen at random from the set of all labelled trees with certain properties.

Estimates of the Rates of Convergence in Limit Theorems for the First Passage Times of Random Walks

Let $T_r$ be the time of first passage to the level $r > 0$ by a random walk with independent and identically distributed steps and mean $\nu \geqq 0$. Estimates are given for the rate at which the distribution of $T_r$, suitably scaled and normalized, converges to the stable distribution with index $\frac{1}{2}$ when $\nu = 0$ and to the normal distribution when $\nu > 0$ as $r \rightarrow \infty$.

Domains of Attraction of First Passage Times

It is well known that the first passage times for Brownian motion have stable laws with exponent $\frac{1}{2}$. It is shown here that first passage times for random walks have distributions in the domain of attraction of a stable law with exponent $\frac{1}{2}$.