  We consider a generalization of the Hopfield model, where the entries of
patterns are Gaussian and diluted. We focus on the high-storage regime and we
investigate analytically the topological properties of the emergent network, as
well as the thermodynamic properties of the model. We find that, by properly
tuning the dilution in the pattern entries, the network can recover different
topological regimes characterized by peculiar scalings of the average
coordination number with respect to the system size. The structure is also
shown to exhibit a large degree of cliquishness, even when very sparse.
Moreover, we obtain explicitly the replica symmetric free energy and the
self-consistency equations for the overlaps (order parameters of the theory),
which turn out to be classical weighted sums of 'sub-overlaps' defined on all
possible sub-graphs. Finally, a study of criticality is performed through a
small-overlap expansion of the self-consistencies and through a whole
fluctuation theory developed for their rescaled correlations: Both approaches
show that the net effect of dilution in pattern entries is to rescale the
critical noise level at which ergodicity breaks down.
