title: Hierarchical neural networks perform both serial and parallel processing.
abstract: In this work we study a Hebbian neural network, where neurons are arranged according to a hierarchical architecture such that their couplings scale with their reciprocal distance. As a full statistical mechanics solution is not yet available, after a streamlined introduction to the state of the art via that route, the problem is consistently approached through signal-to-noise technique and extensive numerical simulations. Focusing on the low-storage regime, where the amount of stored patterns grows at most logarithmical with the system size, we prove that these non-mean-field Hopfield-like networks display a richer phase diagram than their classical counterparts. In particular, these networks are able to perform serial processing (i.e. retrieve one pattern at a time through a complete rearrangement of the whole ensemble of neurons) as well as parallel processing (i.e. retrieve several patterns simultaneously, delegating the management of different patterns to diverse communities that build network). The tune between the two regimes is given by the rate of the coupling decay and by the level of noise affecting the system. The price to pay for those remarkable capabilities lies in a network's capacity smaller than the mean field counterpart, thus yielding a new budget principle: the wider the multitasking capabilities, the lower the network load and vice versa. This may have important implications in our understanding of biological complexity.
authors: Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy.
authors: Elena
authors: E
authors: Agliari
authors: Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy.
authors: Adriano
authors: A
authors: Barra
authors: Dipartimento di Matematica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy.
authors: Andrea
authors: A
authors: Galluzzi
authors: Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy; Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Roma, Italy.
authors: Francesco
authors: F
authors: Guerra
authors: Dipartimento di Matematica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy.
authors: Daniele
authors: D
authors: Tantari
authors: Dipartimento SBAI (Ingegneria), Sapienza Università di Roma, Via A. Scarpa 14, 00185, Roma, Italy. Electronic address: flavia.tavani@sbai.uniroma1.it.
authors: Flavia
authors: F
authors: Tavani
mesh_terms: D016571:Neural Networks, Computer
keywords: Multitasking associative networks; Parallel processing; Serial processing