In that situation, the reduction graph $R(\mathfrak{X})$ is a metrized graph defined as follows. It has for vertices the irreducible components of the special fiber, with as many edges of length $\log\left|{\pi}\right|^{-1}$ between two vertices as the number of intersection points of the corresponding components. In an neighbourghood of a double point, $\mathfrak{X}$ looks like (i.e., has an étale map to) the scheme with equation $xy=\pi$ in the affine plane $\mathbf{A}^{2}_{K^{\circ}}$.

If one replaces the field $K$ by a finite extension $K^{\prime}$, the base change $\mathfrak{X}\otimes_{K^{\circ}}(K^{\prime})^{\circ}$ may no more be regular. Indeed, $\mathfrak{X}\otimes_{K^{\circ}}(K^{\prime})^{\circ}$ is étale locally isomorphic to $xy=(\pi^{\prime})^{e}$, where $\pi^{\prime}$ is a uniformizing element of $K^{\prime}$, and $e$ is the ramification index. When $e>1$, the origin is a singular point of that scheme and one needs to blow it up repeatedly in order to obtain a regular scheme, which is a semi-stable model of $\mathrm{X}_{K^{\prime}}$ over $(K^{\prime})^{\circ}$. The two initial components are replaced by a chain of $e+1$ components, the $e-1$ intermediate ones being projective lines. In other words, $e-1$ vertices have been added, regularly spaced along each edge. One concludes that the reduction graph has not changed, as a topological space. Its metric has not changed neither, since the $e$ edges that partition an original edge (of length $\log\left|{\pi}\right|^{-1}$) have length $\log\left|{\pi^{\prime}}\right|^{-1}=\frac{1}{e}\log\left|{\pi}\right|^{-1}$.

We say that a function on $R(\mathfrak{X})$ is piecewise linear if, up to passing to a finite extension (which replaces each edge by $e$ edges of length equal to $1/e$th of the initial one), it is linear on each edge.