Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2012-04-02
Physics
Condensed Matter
Disordered Systems and Neural Networks
Scientific paper
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-consistence 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 accordingly the critical noise level at which ergodicity breaks down.
Agliari Elena
Asti Lorenzo
Barra Adriano
Burioni Raffaella
Uguzzoni Guido
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