Aging dynamics and the topology of inhomogenous networks

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$infullgeneralityonnetworkswhichareabovethelowercriticaldimension$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 $O(\mathcal{N})$ models we find that the same picture holds in the large-$\mathcal{N}$ limit and that $a_{\chi }$ only depends on the spectral dimension of the network.