Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port-Hamiltonian with general linear Kirchhoff's boundary conditions can be written as a system of $2\times 2$ hyperbolic equations on a metric graph $\mathrm{\Gamma }$. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of $\mathrm{\Gamma }$ and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of $\mathrm{\Gamma }$.