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 $D(\eta ,\xi )={\sum }_v\pi (v)1_{\{\eta (v)\ne \xi (v)\}}$, where $\pi$ is the stationary distribution of the copying walk. On a connected undirected graph, $\pi (v)=\mathrm{deg}(v)/(2|E|)$. 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 $\epsilon$, stationarity $\pi P=\pi$ gives $\mathbb{E}[D_{t+1}\mid {\eta }_t,{\xi }_t]=(1-\epsilon /|V|)D_t$. 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.^[ref]