Statistical analysis of random walks on network

A Kalikova

doi:10.37943/aitu.2021.99.34.007

Abstract

This paper describes an investigation of analytical formulas for parameters in random walks. Random walks are used to model situations in which an object moves in a sequence of steps in randomly chosen directions. Given a graph and a starting point, we select a neighbor of it at random, and move to this neighbor; then we select a neighbor of this point at random, and move to it etc. It is a fundamental dynamic process that arise in many models in mathematics, physics, informatics and can be used to model random processes inherent to many important applications. Different aspects of the theory of random walks on graphs are surveyed. In particular, estimates on the important parameters of hitting time, commute time, cover time are discussed in various works. In some papers, authors have derived an analytical expression for the distribution of the cover time for a random walk over an arbitrary graph that was tested for small values of n. However, this work will show the simplified analytical expressions for distribution of hitting time, commute time, cover time for bigger values of n. Moreover, this work will present the probability mass function and the cumulative distribution function for hitting time, commute time.

Highlights

Random walks arise in many models in mathematics and physics
The aim of this paper is to show a result of an investigation of simple, exact formulas for three important quantities in random walks: hitting time, commute time and cover time for big values of n and their and simulation result
The measures in the quantitative theory of random walks that are main parameters of this paper are: the hitting time is a number of steps before node j is first time visited, starting from node i; the commute time: this is a number of steps in a random walk starting at i, before node j is visited and node i is reached again; the cover time is a number of steps to reach every node [1,2]

Summary

Introduction

Random walks arise in many models in mathematics and physics This is one of those notions that tend to pop up everywhere once we begin to look for them. There can be found different works devoted to random walks From those papers, more relevant will be analyzing [4,5,6] random walks on complex networks with derived an expression for hitting time. ISSN (E): 2707-904X between two nodes and [7,8] the properties of random walks on complex trees that both will help to understand concept and develop them Another theoretical literature that has been reviewed is [1] that gives main direction for developing analytical expressions of random walk’s parameters with some given examples of special cases for different graphs. There will be presented introduction of main properties and parameters of random walk through connections with the eigenvalues of graphs by analyzing existing, relevant work. Communicating: If state j is accessible from state i and state i is accessible from state j states i and j are said to communicate [5,10]

Part 1. Hitting time

Conclusion