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+# -*- Org -*-
+#
+#
+# Copyright (c) 2022, Ryo Nakamura.
+# All rights reserved.
+#
+# $Id: $
+#
+
+#+LATEX_CLASS: jarticle
+#+LATEX_HEADER: \usepackage[top=2.5truecm,bottom=2.5truecm,left=2truecm,right=2truecm]{geometry}
+#+LATEX_HEADER: \pagestyle{empty}
+#+LATEX_HEADER: \usepackage{amsmath}
+#+LATEX_HEADER: \usepackage{insertfig}
+#+LATEX_HEADER: \newcommand{\neighbor}{\mathcal{N}}
+#+OPTIONS: toc:nil
+#+OPTIONS: title:nil
+
+* Random Walk on a Graph
+:PROPERTIES:  
+:UNNUMBERED: t  
+:END:
+
+It is well-known that /random walk on a graph/ is one of fundamental
+research topics in the field of discrete mathematics.  In the
+literature, a variety of studies have been devoted to understand the
+property of random walk on a graph through mathematical analysis and
+simulation experiment.
+
+The rationale behind this situation is mainly caused by favorable
+property of random walk, in particular, applicability and simplicity.
+As will be described later, random walk on a graph means that a walker
+sequentially moves according to the structure of graph, which does not
+require any complex mechanisms.  From the viewpoint of computer
+science, random walk on a graph is a promising approach to solve
+various problems for graphs.  Because the graph can be regarded as an
+abstraction of /networks/ such as communication network and social
+network, random walk on a graph is applicable to, for instance,
+content discovery on P2P (Peer-to-Peer) network and graph sampling on
+unknown social network.
+
+Before describing random walk on a graph in detail, we provide a brief
+review on graph.  Graph is one of typical data structures, and it is
+comprised of two fundamental components: /vertex/ and /edge/.  Edge is
+corresponding to a line between two distinct vertices, in other words,
+edge $e$ is expressed as $e = \{u, v\}$.  Formally, graph $G$ is
+represented as $G = (V, E)$, where $V$ and $E$ represent a set of
+vertices and a set of edges, respectively.  Also, a graph can be
+categorized into /directed/ graph and /undirected/ graph, according to
+orientation of an edge.  In the following, we focus on the undirected
+graph rather than the directed graph for its simplicity.
+
+Random walk on a graph is a series of movements that an /agent/, i.e.,
+walker, traverses a graph.  Specifically, the operation of an agent is
+described as follows; (i) an agent on vertex $u$ randomly selects
+vertex $v$ of vertices that are adjacent to vertex $v$, and the agent
+moves to the randomly-selected vertex $v$ among adjacent vertices;
+(ii) procedure (i) is repeated until a given condition is satisfied.
+Strictly speaking, at step $t$, an agent on vertex $u$ selects vertex
+$v \in \neighbor(u)$ of a set of adjacent vertices $\neighbor(u)$
+which is defined as $\{ v \, | \, \{u, v\} \in E \}$; then, at step
+$t + 1$, an agent visits adjacent vertex $v$.  Let us mathematically
+consider the behavior that the agent transits to a randomly-chosen
+adjacent vertex.  We denote the probability that an agent on vertex
+$u$ transits to vertex $v$, i.e., the transition probability, as
+$p_{uv}$.  Also, we can simply formulate the transition probability as
+\begin{align}
+  p_{uv} = \frac{1}{d_u} ,
+\end{align}
+where $d_v$ represents the /degree/ of vertex $v$ and it is defined as
+the number of adjacent vertices of vertex $v$.  Note that this
+definition is valid in the case of simple undirected graph.
+
+The goodness of random walk on a graph can be evaluated with typical
+measures --- /first hitting time/ and /cover time/.  Below, we provide
+definitions of these two measures.  First, the first hitting time
+$H_{st}$ is defined as the expected number of steps that an agent
+initially visits destination vertex $t$ from source vertex $s$.
+Hence, the average first hitting time is given by the mean of the
+first hitting time $H_{st}$ of every vertices pair $(s, t)$ on graph
+$G$.  Second, the cover time is defined as the expected number of
+steps that an agent reaches all vertices on a given graph from an
+arbitrary source vertex.