First-passage phenomena in hierarchical networks

In this paper we study Markov processes and related first passage problems on a class of weighted, modular graphs which generalize the Dyson hierarchical model. In these networks, the coupling strength between two nodes depends on their distance and is modulated by a parameter $\sigma$. We find that, in the thermodynamic limit, ergodicity is lost and the "distant" nodes can not be reached. Moreover, for finite-sized systems, there exists a threshold value for $\sigma$ such that, when $\sigma$ is relatively large, the inhomogeneity of the coupling pattern prevails and "distant" nodes are hardly reached. The same analysis is carried on also for generic hierarchical graphs, where interactions are meant to involve $p$-plets ($p>2$) of nodes, finding that ergodicity is still broken in the thermodynamic limit, but no threshold value for $\sigma$ is evidenced, ultimately due to a slow growth of the network diameter with the size

Phase transition for the Maki-Thompson rumour model on a small-world network

We consider the Maki-Thompson model for the stochastic propagation of a rumour within a population. We extend the original hypothesis of homogenously mixed population by allowing for a small-world network embedding the model. This structure is realized starting from a $k$-regular ring and by inserting, in the average, $c$ additional links in such a way that $k$ and $c$ are tuneable parameter for the population architecture. We prove that this system exhibits a transition between regimes of localization (where the final number of stiflers is at most logarithmic in the population size) and propagation (where the final number of stiflers grows algebraically with the population size) at a finite value of the network parameter $c$. A quantitative estimate for the critical value of $c$ is obtained via extensive numerical simulations.Comment: 24 pages, 4 figure

A walk in the statistical mechanical formulation of neural networks

Neural networks are nowadays both powerful operational tools (e.g., for pattern recognition, data mining, error correction codes) and complex theoretical models on the focus of scientific investigation. As for the research branch, neural networks are handled and studied by psychologists, neurobiologists, engineers, mathematicians and theoretical physicists. In particular, in theoretical physics, the key instrument for the quantitative analysis of neural networks is statistical mechanics. From this perspective, here, we first review attractor networks: starting from ferromagnets and spin-glass models, we discuss the underlying philosophy and we recover the strand paved by Hopfield, Amit-Gutfreund-Sompolinky. One step forward, we highlight the structural equivalence between Hopfield networks (modeling retrieval) and Boltzmann machines (modeling learning), hence realizing a deep bridge linking two inseparable aspects of biological and robotic spontaneous cognition. As a sideline, in this walk we derive two alternative (with respect to the original Hebb proposal) ways to recover the Hebbian paradigm, stemming from ferromagnets and from spin-glasses, respectively. Further, as these notes are thought of for an Engineering audience, we highlight also the mappings between ferromagnets and operational amplifiers and between antiferromagnets and flip-flops (as neural networks -built by op-amp and flip-flops- are particular spin-glasses and the latter are indeed combinations of ferromagnets and antiferromagnets), hoping that such a bridge plays as a concrete prescription to capture the beauty of robotics from the statistical mechanical perspective.Comment: Contribute to the proceeding of the conference: NCTA 2014. Contains 12 pages,7 figure

Topological properties of hierarchical networks

Hierarchical networks are attracting a renewal interest for modelling the organization of a number of biological systems and for tackling the complexity of statistical mechanical models beyond mean-field limitations. Here we consider the Dyson hierarchical construction for ferromagnets, neural networks and spin-glasses, recently analyzed from a statistical-mechanics perspective, and we focus on the topological properties of the underlying structures. In particular, we find that such structures are weighted graphs that exhibit high degree of clustering and of modularity, with small spectral gap; the robustness of such features with respect to link removal is also studied. These outcomes are then discussed and related to the statistical mechanics scenario in full consistency. Lastly, we look at these weighted graphs as Markov chains and we show that in the limit of infinite size, the emergence of ergodicity breakdown for the stochastic process mirrors the emergence of meta-stabilities in the corresponding statistical mechanical analysis

Meta-stable states in the hierarchical Dyson model drive parallel processing in the hierarchical Hopfield network

In this paper we introduce and investigate the statistical mechanics of hierarchical neural networks: First, we approach these systems \`a la Mattis, by thinking at the Dyson model as a single-pattern hierarchical neural network and we discuss the stability of different retrievable states as predicted by the related self-consistencies obtained from a mean-field bound and from a bound that bypasses the mean-field limitation. The latter is worked out by properly reabsorbing fluctuations of the magnetization related to higher levels of the hierarchy into effective fields for the lower levels. Remarkably, mixing Amit's ansatz technique (to select candidate retrievable states) with the interpolation procedure (to solve for the free energy of these states) we prove that (due to gauge symmetry) the Dyson model accomplishes both serial and parallel processing. One step forward, we extend this scenario toward multiple stored patterns by implementing the Hebb prescription for learning within the couplings. This results in an Hopfield-like networks constrained on a hierarchical topology, for which, restricting to the low storage regime (where the number of patterns grows at most logarithmical with the amount of neurons), we prove the existence of the thermodynamic limit for the free energy and we give an explicit expression of its mean field bound and of the related improved boun

Hierarchical neural networks perform both serial and parallel processing

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 diff erent 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 viceversa. This may have important implications in our understanding of biological complexity

From Dyson to Hopfield: Processing on hierarchical networks

We consider statistical-mechanical 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 that their mean-field counterpart. In particular, we consider the Dyson model, mimicking ferromagnetism in lattices, and we prove the existence of a number of meta-stabilities, beyond the ordered state, which get 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 both as a serial processor as well as a parallel processor, 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, graph theory, signal-to-noise 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

Theatricality in Installation Artworks: An Overview

The article is an investigation into theatricality from various standpoints (among others those of Michael Fried, Claire Bishop, Juliane Rebentisch and Samuel Weber) in order to focus on different views on theatricality considered as partially emancipated from theatre and to verify if and to what extent each of them can apply to installation artworks as environments and intermedial devices. Ultimately the article propounds the idea of a paradoxical anti-theatrical theatricality of installation art, grasped in its very connection to site-specificity, critically engaging Martin Heidegger’s insights regarding the «Gestell» and the «work-being» of the work of art, as a general theoretical basis through which a particular focus of ‘specificity’ of installation is endorsed

L’esperienza estetica come esperienza di immagini. Walter Benjamin e Theodor W. Adorno

Both Walter Benjamin and Theodor W. Adorno consider ‘aesthetical experience’ as an “image experience” assuming a power of images “to set free forces” directed to produce or support aesthetical-political (Benjamin) or aesthetical-critical (Adorno) requirements. Profane illumination, ‘thinkimages’, phantasmagory, dialectical images, decayed ‘aura’ and technicalized images in Benjamin’s theory of aesthetical modernity. Expressive feature or “mimetic” eloquence in nature and art countering reality, dismantled ‘aura’ in contemporary desacralized work of art, but also persisting ‘aura’ in its meaningful dimension in Adorno’s aesthetical theory

Adorno, Bloch e il campo d'azione dell'utopia. Un dialogo radiofonico.

Il saggio intende mettere a fuoco le rispettive proposte di Bloch e Adorno relative a un rinnovato pensiero dell’utopia, rinvenendo in entrambe una, sia pur differente, struttura ‘agonale’. Una breve citazione tratta da un’opera di Brecht, «manca qualcosa» (Mahagonny), è il punto di avvio della discussione radiofonica avvenuta nel 1964 tra Bloch e Adorno sulle contraddizioni e gli anacronismi dell’utopico. Secondo Bloch la frase rivela la presenza dell’utopia nel processo dell’essere, il che rende la speranza una strategia di lotta. Bloch fonda il suo argomento sulla base della tesi della ‘prossimità’ dell’utopia e attraverso una fenomenologia della coscienza utopica che si confronta con vari processi della realtà e con simboli e ‘topoi’ utopici incontrati lungo il cammino. Contro l’emergere di tali immagini del desiderio Adorno suggerisce che è possibile parlare di utopia solo negativamente e che il ‘campo d’azione’ dell’utopia consiste piuttosto nel mutamento di tutte le categorie – e non solo di quelle sociali ¬– attraverso il procedimento della negazione determinata. Sostengo infine che entrambi i filosofi non traspongono l’utopia in un tempo futuro per il fatto che essa assume un valore agonale e operativo, che vede l’utopico agire criticamente come una trascendenza ‘immanente’ dell’esperienza concreta.The essay aims at focusing on both Bloch’s and Adorno’s proposals for a renewed utopian thinking, claiming their differently ‘agonistic’ structure. Brecht’s short sentence «Something’s missing» (Mahagonny) gives way to a discussion between Ernst Bloch and Theodor W. Adorno broadcasted by radio in 1964 on the contradictions and anachronism of ‘the utopian’. The sentence reveals according to Bloch the very presence of utopia in the process of being, which makes hope a struggling strategy. Bloch substantiates his argument with the thesis of the ‘proximity’ of utopia and with the use of a phenomenology of utopian consciousness engaged with various processes of reality and with symbols and utopic ‘topoi’ encontered along the way. Against the emergence of such wish-images, Adorno suggests that we can talk about utopia only in a negative way and that the ‘operational theatre’ of utopia is rather a changing of all categories – not only in the social field – through determined negation. I argue that for both philosophers utopia is not transposed into the future because of its agonistc and operational value: it performs critically as ‘immanent’ transcendence of concrete experience