  We study phase ordering on networks and we establish a relation between the
exponent $a_\chi$ of the aging part of the integrated autoresponse function
$\chi_{ag}$ and the topology of the underlying structures. We show that $a_\chi
>0$ in full generality on networks which are above the lower critical dimension
$d_L$, i.e. where the corresponding statistical model has a phase transition at
finite temperature. For discrete symmetry models on finite ramified structures
with $T_c = 0$, which are at the lower critical dimension $d_L$, we show that
$a_\chi$ is expected to vanish. We provide numerical results for the physically
interesting case of the $2-d$ percolation cluster at or above the percolation
threshold, i.e. at or above $d_L$, and for other networks, showing that the
value of $a_\chi $ changes according to our hypothesis. For $O({\cal N})$
models we find that the same picture holds in the large-${\cal N}$ limit and
that $a_\chi$ only depends on the spectral dimension of the network.
