= Weighted Hamming distance for the noisy voter model
For two configurations $\eta,\xi\in\{0,1\}^V$, the <weighted Hamming distance for the noisy voter model> is $D(\eta,\xi)=\sum_v\pi(v)\mathbf1_{\{\eta(v)\ne\xi(v)\}}$, where $\pi$ is the <stationary distribution> of the copying walk. On a connected undirected <graph>, $\pi(v)=\deg(v)/(2|E|)$. Positive weights make this a <metric>, with values between zero and one.
Couple two chains using the same uniformly selected vertex, refresh decision, fresh bit and copied neighbor. If the refresh <probability> is $\varepsilon$, stationarity $\pi P=\pi$ gives $\mathbb E[D_{t+1}\mid\eta_t,\xi_t]=(1-\varepsilon/|V|)D_t$. Copying redistributes disagreements without increasing their stationary-weighted total in expectation; refreshes remove them. The factor $|V|$ belongs to the single-vertex update convention, not a simultaneous update of every vertex.<Asymptotic behaviour of the noisy voter model density process by Richard Pymar and Nicolás Rivera>[Sections 1.1–1.2, model convention; contraction derived by the shared-update coupling]{ref}
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