  We study local (the heat equation) and nonlocal (convolution type problems
with an integrable kernel) evolution problems on a metric connected finite
graph in which some of the edges have infinity length. We show that the
asymptotic behaviour of the solutions to both local and nonlocal problems is
given by the solution of the heat equation, but on a star shaped graph in which
there is only one node and as many infinite edges as in the original graph. In
this way we obtain that the compact component that consists in all the vertices
and all the edges of finite length can be reduced to a single point when
looking at the asymptotic behaviour of the solutions. For this star shaped
limit problem the asymptotic behaviour of the solutions is just given by the
solution to the heat equation in a half line with a Neumann boundary condition
at $x = 0$ and initial datum $(2M/N)\delta_{x=0}$ where M is the total mass of
the initial condition for our original problem and $N$ is the number of edges
of infinite length. In addition, we show that solutions to the nonlocal problem
converge, when we rescale the kernel, to solutions to the heat equation (the
local problem), that is, we find a relaxation limit.
