# Reversible Markov Chains And Random Walks On Graphs Pdf

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- A random walk model for infection on graphs: spread of epidemics & rumours with mobile agents
- UC San Diego
- Reversible Markov Chains and Random Walks on Graphs

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## A random walk model for infection on graphs: spread of epidemics & rumours with mobile agents

We apply spectral theory to study random processes involving directed graphs. In the first half of this thesis, we examine random walks on directed graphs, which is rooted in the study of non-reversible Markov chains. We prove bounds on key spectral invariants which play a role in bounding the rate of convergence of the walk and capture isoperimetric properties of the directed graph. We first focus on the principal ratio, which is the ratio of maximum to minimum values of vertices in the stationary distribution. We characterize all graphs achieving the upper bound and give explicit constructions for these extremal graphs.

## UC San Diego

We address the question of understanding the effect of the underlying network topology on the spread of a virus and the dissemination of information when users are mobile performing independent random walks on a graph. To this end, we propose a simple model of infection that enables to study the coincidence time of two random walkers on an arbitrary graph. By studying the coincidence time of a susceptible and an infected individual both moving in the graph we obtain estimates of the infection probability. The main result of this paper is to pinpoint the impact of the network topology on the infection probability. We then study the model on power-law graphs, that exhibit heterogeneous connectivity patterns, and show the existence of a phase transition for the coincidence time depending on the parameter of the power-law of the degree distribution. We finally undertake a preliminary analysis for the case with k random walkers and provide upper bounds on the convergence time for both the complete graph and regular graphs. In recent years, there has been a surge of hand-held wireless computing devices such as PDAs together with the proliferation of new services.

Metrics details. In the framework of network sampling, random walk RW based estimation techniques provide many pragmatic solutions while uncovering the unknown network as little as possible. Despite several theoretical advances in this area, RW based sampling techniques usually make a strong assumption that the samples are in stationary regime, and hence are impelled to leave out the samples collected during the burn-in period. This work proposes two sampling schemes without burn-in time constraint to estimate the average of an arbitrary function defined on the network nodes, for example, the average age of users in a social network. The central idea of the algorithms lies in exploiting regeneration of RWs at revisits to an aggregated super-node or to a set of nodes, and in strategies to enhance the frequency of such regenerations either by contracting the graph or by making the hitting set larger. Our first algorithm, which is based on reinforcement learning RL , uses stochastic approximation to derive an estimator. This method can be seen as intermediate between purely stochastic Markov chain Monte Carlo iterations and deterministic relative value iterations.

Request PDF | On Jan 1, , D. J. Aldous and others published Reversible Markov Chains and Random Walks on Graphs | Find, read and cite all the research.

## Reversible Markov Chains and Random Walks on Graphs

These are general reversible Markov chains on at most countable state space in discrete and sometimes continuous time. One of our topics of interest will concern estimates on so-called heat kernels, that is, transition densities of such walks. Note: Classes for 8th, 13th and 15th Feb will be taken by Ayan Bhattacharya. Presentation topics will be given after mid semester exams. Sohom Gaussian free field def and properties : Prop 2.

In mathematics , a random walk is a mathematical object , known as a stochastic or random process , that describes a path that consists of a succession of random steps on some mathematical space such as the integers. Other examples include the path traced by a molecule as it travels in a liquid or a gas see Brownian motion , the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler : all can be approximated by random walk models, even though they may not be truly random in reality. As illustrated by those examples, random walks have applications to engineering and many scientific fields including ecology , psychology , computer science , physics , chemistry , biology , economics , and sociology.

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