A Hebbian approach to complex network generation

Through a redefinition of patterns in an Hopfield-like model, we introduce and develop an approach to model discrete systems made up of many, interacting components with inner degrees of freedom. Our approach clarifies the intrinsic connection between the kind of interactions among components and the emergent topology describing the system itself; also, it allows to effectively address the statistical mechanics on the resulting networks. Indeed, a wide class of analytically treatable, weighted random graphs with a tunable level of correlation can be recovered and controlled. We especially focus on the case of imitative couplings among components endowed with similar patterns (i.e. attributes), which, as we show, naturally and without any a-priori assumption, gives rise to small-world effects. We also solve the thermodynamics (at a replica symmetric level) by extending the double stochastic stability technique: free energy, self consistency relations and fluctuation analysis for a picture of criticality are obtained

Irreducible free energy expansion and overlaps locking in mean field spin glasses

We introduce a diagrammatic formulation for a cavity field expansion around the critical temperature. This approach allows us to obtain a theory for the overlap's fluctuations and, in particular, the linear part of the Ghirlanda-Guerra relationships (GG) (often called Aizenman-Contucci polynomials (AC)) in a very simple way. We show moreover how these constraints are "superimposed" by the symmetry of the model with respect to the restriction required by thermodynamic stability. Within this framework it is possible to expand the free energy in terms of these irreducible overlaps fluctuations and in a form that simply put in evidence how the complexity of the solution is related to the complexity of the entropy.Comment: 19 page

Renormalization of Yukawa model with sterile scalar in curved space-time

We explore the classical and quantum properties of a sterile scalar field coupled to $N$ copies of Dirac fermions in an external gravitational field. We find that the self-interaction scalar potential of a model that is consistent at the quantum level, includes odd (first and third) powers of a scalar. In particular, one has to consider, besides the standard non-minimal coupling of the form $\xi \varphi^2 R$, the new type of non-minimal coupling of the form $f\varphi R$ with new non-minimal parameter $f$. We study the one-loop renormalization of such a theory including renormalization of the new non-minimal coupling. Also, we calculate the one-loop effective potential using the renormalization group and show how the renormalization group analysis should be extended compared to the standard expression which was derived in 1980-ies. This conclusion is supported by the direct calculation of effective potential using normal coordinates and covariant cut-off regularization. The important features of the classical theory with a sterile scalar are related to the presence of the qualitatively new terms in the induced action of gravity, coming from the odd terms. We show that this new feature of the theory may have phenomenologically relevant consequences, both in the low-energy gravitational physics and at the high energies, corresponding to inflation.Comment: Extended version, includes more detailed discussions and the preliminary analysis of inflation. Accepted in EPJC. Small misprints correcte

Ferromagnetic models for cooperative behavior: Revisiting Universality in complex phenomena

Ferromagnetic models are harmonic oscillators in statistical mechanics. Beyond their original scope in tackling phase transition and symmetry breaking in theoretical physics, they are nowadays experiencing a renewal applicative interest as they capture the main features of disparate complex phenomena, whose quantitative investigation in the past were forbidden due to data lacking. After a streamlined introduction to these models, suitably embedded on random graphs, aim of the present paper is to show their importance in a plethora of widespread research fields, so to highlight the unifying framework reached by using statistical mechanics as a tool for their investigation. Specifically we will deal with examples stemmed from sociology, chemistry, cybernetics (electronics) and biology (immunology).Comment: Contributing to the proceedings of the Conference "Mathematical models and methods for Planet Heart", INdAM, Rome 201

The replica symmetric behavior of the analogical neural network

In this paper we continue our investigation of the analogical neural network, paying interest to its replica symmetric behavior in the absence of external fields of any type. Bridging the neural network to a bipartite spin-glass, we introduce and apply a new interpolation scheme to its free energy that naturally extends the interpolation via cavity fields or stochastic perturbations to these models. As a result we obtain the free energy of the system as a sum rule, which, at least at the replica symmetric level, can be solved exactly. As a next step we study its related self-consistent equations for the order parameters and their rescaled fluctuations, found to diverge on the same critical line of the standard Amit-Gutfreund-Sompolinsky theory.Comment: 17 page

Equilibrium statistical mechanics on correlated random graphs

Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents - crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality. At least at equilibrium, dilution simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved.Comment: 30 pages, 4 figure

Diabolical points in the magnetic spectrum of Fe_8 molecules

The magnetic molecule Fe_8 has been predicted and observed to have a rich pattern of degeneracies in its spectrum as an external magnetic field is varied. These degeneracies have now been recognized to be diabolical points. This paper analyzes the diabolicity and all essential properties of this system using elementary perturbation theory. A variety of arguments is gievn to suggest that an earlier semiclassical result for a subset of these points may be exactly true for arbitrary spinComment: uses europhys.sty package; 3 embedded ps figure