title: Retrieval capabilities of hierarchical networks: from Dyson to Hopfield.
abstract: We consider statistical-mechanics models for spin systems built on hierarchical structures, which provide a simple example of non-mean-field framework. We show that the coupling decay with spin distance can give rise to peculiar features and phase diagrams much richer than their mean-field counterpart. In particular, we consider the Dyson model, mimicking ferromagnetism in lattices, and we prove the existence of a number of metastabilities, beyond the ordered state, which become stable in the thermodynamic limit. Such a feature is retained when the hierarchical structure is coupled with the Hebb rule for learning, hence mimicking the modular architecture of neurons, and gives rise to an associative network able to perform single pattern retrieval as well as multiple-pattern retrieval, depending crucially on the external stimuli and on the rate of interaction decay with distance; however, those emergent multitasking features reduce the network capacity with respect to the mean-field counterpart. The analysis is accomplished through statistical mechanics, Markov chain theory, signal-to-noise ratio technique, and numerical simulations in full consistency. Our results shed light on the biological complexity shown by real networks, and suggest future directions for understanding more realistic models.
authors: Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.
authors: Elena
authors: E
authors: Agliari
authors: Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.
authors: Adriano
authors: A
authors: Barra
authors: Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.
authors: Andrea
authors: A
authors: Galluzzi
authors: Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.
authors: Francesco
authors: F
authors: Guerra
authors: Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.
authors: Daniele
authors: D
authors: Tantari
authors: Dipartimento SBAI (Ingegneria), Sapienza Università di Roma, Via Antonio Scarpa 14, 00185 Roma, Italy.
authors: Flavia
authors: F
authors: Tavani
mesh_terms: D003198:Computer Simulation; D008280:Magnetics; D008962:Models, Theoretical; D016571:Neural Networks, Computer; D009474:Neurons / Q000166:cytology