The <Noisy voter model> combines copying with spontaneous opinion changes. On a finite connected undirected <graph> with $n\geq2$ <graph vertices>[vertices], give each <graph vertex>[vertex] a spin in $\{0,1\}$. At each discrete step choose a <graph vertex>[vertex] uniformly. With <Probability> $\varepsilon\in(0,1]$, replace its spin by an <independent> fair bit; otherwise copy a uniformly chosen neighbour. This is a <Markov chain> on <spin configuration>[spin configurations].<Asymptotic behaviour of the noisy voter model density process by Richard Pymar and Nicolás Rivera>[Sections 1.1–1.2: discrete-time update convention]{ref}

Positive noise makes the finite chain <irreducible Markov chain>[irreducible]: a sequence of refreshes can produce any <spin configuration>[configuration] with positive <Probability>. There is also a positive <holding probability of a Markov chain>[holding probability], so it is <aperiodic Markov chain>[aperiodic] and has a unique <Stationary distribution>. At zero noise, the two consensus configurations are absorbing; they cease to be absorbing under positive noise.

A weighted <coupling> explains convergence even when the <graph> is not <regular graph>[regular]. Let $R(v,u)$ be the <simple random walk>[simple-random-walk] <transition matrix> and $w_v=\deg(v)/(2|E|)$ its <Stationary distribution>. For two <spin configuration>[spin configurations] use the <Weighted Hamming distance for the noisy voter model>[weighted Hamming distance]
$$
D(\eta,\xi)=\sum_vw_v\mathbf1_{\{\eta(v)\ne\xi(v)\}}.
$$
Couple the two chains with the same updated <graph vertex>[vertex], refresh decision, fresh bit and copied neighbour. If $\delta_v$ is the current disagreement <indicator function>[indicator], the expected one-step change in $D$ is
$$
\frac1n\left[-\sum_vw_v\delta_v+(1-\varepsilon)\sum_{v,u}w_vR(v,u)\delta_u\right]
=-\frac\varepsilon n D,
$$
since $wR=w$. Thus $\mathbb E D_t=(1-\varepsilon/n)^tD_0$. With $w_{\min}=\min_vw_v>0$, a remaining disagreement costs at least $w_{\min}$, so <coupling> with a stationary chain yields worst-case <total variation distance> at most $w_{\min}^{-1}(1-\varepsilon/n)^t$. The time here counts single-vertex updates; switching to a rate-one clock at each <graph vertex>[vertex] changes time by a factor of $n$.


 Noisy voter model (source code)