  We study a one-parameter family of Eikonal Hamilton-Jacobi equations on an
embedded network, and prove that there exists a unique critical value for which
the corresponding equation admits global solutions, in a suitable viscosity
sense. Such a solution is identified, via an Hopf-Lax type formula, once an
admissible trace is assigned on an intrinsic boundary. The salient point of our
method is to associate to the network an abstract graph, encoding all of the
information on the complexity of the network, and to relate the differential
equation to a discrete functional equation on the graph. Comparison principles
and representation formulae are proven in the supercritical case as well.
