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We study phase ordering on networks and we establish a relation between theexponent $a_\chi$ 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 atfinite temperature. For discrete symmetry models on finite ramified structureswith $T_c = 0$, which are at the lower critical dimension $d_L$, we show that$a_\chi$ is expected to vanish. We provide numerical results for the physicallyinteresting case of the $2-d$ percolation cluster at or above the percolationthreshold, i.e. at or above $d_L$, and for other networks, showing that thevalue 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 andthat $a_\chi$ only depends on the spectral dimension of the network.Comment: LateX file, 4 eps figure

Aging Dynamics and the Topology of Inhomogenous Networks