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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク voter model ： ウィキペディア英語版 voter model In the mathematical theory of probability, the voter model is a stochastic process that is a specific type of interacting particle system (see Probabilistic Cellular Automata too). A voter model is a sequential dynamical system and it is similar to the contact process. One can imagine that there is a "voter" at each point on a connected graph, where the connections indicate that there is some form of interaction between a pair of voters (nodes). The opinions of any given voter on some issue changes at random times under the influence of opinions of his neighbours. A voter's opinion at any given time can take one of two values, labelled 0 and 1. At random times, a random individual is selected and that voter's opinion are changed according to a stochastic rule. Specifically, for one of the chosen voter's neighbors is chosen according to a given set of probabilities and that individual's opinion is transferred to the chosen voter. An alternative interpretation is in terms of spatial conflict. Suppose two nations control the areas (sets of nodes) labelled 0 or 1. A flip from 0 to 1 at a given location indicates an invasion of that site by the other nation. Note that only one flip happens each time. Problems involving the voter model will often be recast in terms of the dual system of coalescing Markov chains. Frequently, these problems will then be reduced to others involving independent Markov chains. ==Definition== A voter model is a (continuous time) Markov process $\backslash scriptstyle\; \backslash eta\_t$ with state space $\backslash scriptstyle\; S=\backslash ^$ and transition rates function $\backslash scriptstyle\; c(x,\backslash eta)$, where $\backslash scriptstyle\; Z^d$ is a d-dimensional integer lattice, and $\backslash scriptstyle\; c($•,•$\backslash scriptstyle\; )$ is assumed to be nonnegative, uniformly bounded and continuous as a function of $\backslash scriptstyle\; \backslash eta$ in the product topology on $\backslash scriptstyle\; S$. Each component $\backslash scriptstyle\; \backslash eta\; \backslash in\; S$ is called a configuration. To make it clear that $\backslash scriptstyle\; \backslash eta(x)$ stands for the value of a site x in configuration $\backslash scriptstyle\; \backslash eta(.)$; while $\backslash scriptstyle\; \backslash eta\_t(x)$ means the value of a site x in configuration $\backslash scriptstyle\; \backslash eta(.)$ at time $\backslash scriptstyle\; t$. The dynamic of the process are specified by the collection of transition rates. For voter models, the rate at which there is a flip at $\backslash scriptstyle\; x$ from 0 to 1 or vice versa is given by a function $\backslash scriptstyle\; c(x,\backslash eta)$ of site $\backslash scriptstyle\; x$. It has the following properties: # $\backslash scriptstyle\; c(x,\backslash eta)=0$ for every $\backslash scriptstyle\; x\; \backslash in\; Z^d$ if $\backslash scriptstyle\; \backslash eta\; \backslash equiv\; 0$ or if $\backslash scriptstyle\; \backslash eta\; \backslash equiv\; 1$ # $\backslash scriptstyle\; c(x,\backslash eta)=c(x,\backslash zeta)$ for every $\backslash scriptstyle\; x\; \backslash in\; Z^d$ if $\backslash scriptstyle\; \backslash eta(y)+\backslash zeta(y)=1$ for all $\backslash scriptstyle\; y\; \backslash in\; Z^d$ # $\backslash scriptstyle\; c(x,\backslash eta)\backslash leq\; c(x,\backslash zeta)$ if $\backslash scriptstyle\; \backslash eta\backslash leq\; \backslash zeta$ and $\backslash scriptstyle\; \backslash eta(x)=\backslash zeta(x)=0$ # $\backslash scriptstyle\; c(x,\backslash eta)$ is invariant under shifts in $\backslash scriptstyle\; Z^d$ Property (1) says that $\backslash scriptstyle\; \backslash eta\backslash equiv\; 0$ and $\backslash scriptstyle\; \backslash eta\backslash equiv\; 1$ are fixed points for the evolution. (2) indicates that the evolution is unchanged by interchanging the roles of 0's and 1's. In property (3), $\backslash scriptstyle\; \backslash eta\backslash leq\; \backslash zeta$ means $\backslash scriptstyle\; \backslash forall\; x,\backslash eta(x)\backslash leq\backslash zeta(x)$, and $\backslash scriptstyle\; \backslash eta\; \backslash leq\; \backslash zeta$ implies $\backslash scriptstyle\; c(x,\backslash eta)\backslash leq\; c(x,\backslash zeta)$ if $\backslash scriptstyle\; \backslash eta(x)=\backslash zeta(x)=0$, and implies $\backslash scriptstyle\; c(x,\backslash eta)\backslash geq\; c(x,\backslash zeta)$ if $\backslash scriptstyle\; \backslash eta(x)=\backslash zeta(x)=1$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「voter model」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.