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

Criticality in diluted ferromagnet

We perform a detailed study of the critical behavior of the mean field diluted Ising ferromagnet by analytical and numerical tools. We obtain self-averaging for the magnetization and write down an expansion for the free energy close to the critical line. The scaling of the magnetization is also rigorously obtained and compared with extensive Monte Carlo simulations. We explain the transition from an ergodic region to a non trivial phase by commutativity breaking of the infinite volume limit and a suitable vanishing field. We find full agreement among theory, simulations and previous results.Comment: 23 pages, 3 figure

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

Ordering of dipolar Ising crystals

We study Ising systems of spins with dipolar interactions. We find a simple approximate relation for the interaction energy between pairs of parallel lattice columns of spins running along the Ising spin direction. This relation provides insight into the relation between lattice geometry and the nature of the ordered state. It can be used to calculate ground state energies. We have also obtained ground state energies and ordering temperatures T_0 from Monte Carlo simulations. Simple empirical relations, that give T_0 for simple and body centered tetragonal lattices in terms of lattice parameters are also established. Finally, the nature of the ordered state and T_0 are determined for Fe_8 clusters, which crystallize on a triclinic lattice.Comment: 13 pages, 4 eps figures, to be published in PRB. For related work, see http://pipe.unizar.es/~jf

Mean-field cooperativity in chemical kinetics

We consider cooperative reactions and we study the effects of the interaction strength among the system components on the reaction rate, hence realizing a connection between microscopic and macroscopic observables. Our approach is based on statistical mechanics models and it is developed analytically via mean-field techniques. First of all, we show that, when the coupling strength is set positive, the model is able to consistently recover all the various cooperative measures previously introduced, hence obtaining a single unifying framework. Furthermore, we introduce a criterion to discriminate between weak and strong cooperativity, based on a measure of "susceptibility". We also properly extend the model in order to account for multiple attachments phenomena: this is realized by incorporating within the model $p$-body interactions, whose non-trivial cooperative capability is investigated too.Comment: 25 pages, 4 figure

A higher quantum bound for the V\'ertesi-Bene-Bell-inequality and the role of POVMs regarding its threshold detection efficiency

Recently, V\'{e}rtesi and Bene [Phys. Rev. A. {\bf 82}, 062115 (2010)] derived a two-qubit Bell inequality, $I_{CH3}$, which they show to be maximally violated only when more general positive operator valued measures (POVMs) are used instead of the usual von Neumann measurements. Here we consider a general parametrization for the three-element-POVM involved in the Bell test and obtain a higher quantum bound for the $I_{CH3}$-inequality. With a higher quantum bound for $I_{CH3}$, we investigate if there is an experimental setup that can be used for observing that POVMs give higher violations in Bell tests based on this inequality. We analyze the maximum errors supported by the inequality to identify a source of entangled photons that can be used for the test. Then, we study if POVMs are also relevant in the more realistic case that partially entangled states are used in the experiment. Finally, we investigate which are the required efficiencies of the $I_{CH3}$-inequality, and the type of measurements involved, for closing the detection loophole. We obtain that POVMs allow for the lowest threshold detection efficiency, and that it is comparable to the minimal (in the case of two-qubits) required detection efficiency of the Clauser-Horne-Bell-inequality.Comment: 11 Pages, 16 Figure

Replica symmetry breaking in mean field spin glasses trough Hamilton-Jacobi technique

During the last years, through the combined effort of the insight, coming from physical intuition and computer simulation, and the exploitation of rigorous mathematical methods, the main features of the mean field Sherrington-Kirkpatrick spin glass model have been firmly established. In particular, it has been possible to prove the existence and uniqueness of the infinite volume limit for the free energy, and its Parisi expression, in terms of a variational principle, involving a functional order parameter. Even the expected property of ultrametricity, for the infinite volume states, seems to be near to a complete proof. The main structural feature of this model, and related models, is the deep phenomenon of spontaneous replica symmetry breaking (RSB), discovered by Parisi many years ago. By expanding on our previous work, the aim of this paper is to investigate a general frame, where replica symmetry breaking is embedded in a kind of mechanical scheme of the Hamilton-Jacobi type. Here, the analog of the "time" variable is a parameter characterizing the strength of the interaction, while the "space" variables rule out quantitatively the broken replica symmetry pattern. Starting from the simple cases, where annealing is assumed, or replica symmetry, we build up a progression of dynamical systems, with an increasing number of space variables, which allow to weaken the effect of the potential in the Hamilton-Jacobi equation, as the level of symmetry braking is increased. This new machinery allows to work out mechanically the general K-step RSB solutions, in a different interpretation with respect to the replica trick, and lightens easily their properties as existence or uniqueness.Comment: 24 pages, no figure

Atrial Fibrillation Ablation and Reduction of Stroke Events: Understanding the Paradoxical Lack of Evidence

Atrial fibrillation (AF) is the most prevalent chronic arrhythmia and a major cause of stroke and mortality. It is thought to confer an overall 5-fold increased risk of a cerebrovascular event, causing ≈one-third of all ischemic strokes. Half of the 2 to 3 fold higher risk of mortality among AF patients is related to AF itself, not only via fatal progression of heart failure, the most frequent mode, but also sudden death and embolic events.1,2 Importantly, AF patients who suffer a cardioembolic stroke have a worse outcome compared with stroke patients without AF. Anticoagulation has been shown to reduce the risk of a cerebrovascular event in AF patients. However, despite adequate anticoagulation, some patients remain at risk of stroke. Whether successful catheter ablation can reduce this risk remains unclear. Although there has not been any convincing evidence thus far that AF ablation leads to a reduction in the risk of stroke, no randomized study was powered to address this question. In this review article, we discuss the AF-stroke association, as well as the apparent lack of evidence supporting the use of ablation for the specific reduction of this end point

Shear Modulus of an Elastic Solid under External Pressure as a function of Temperature: The case of Helium

The energy of a dislocation loop in a continuum elastic solid under pressure is considered within the framework of classical mechanics. For a circular loop, this is a function with a maximum at pressures that are well within reach of experimental conditions for solid helium suggesting, in this case, that dislocation loops can be generated by a pressure-assisted thermally activated process. It is also pointed out that pinned dislocations segments can alter the shear response of solid helium, by an amount consistent with current measurements, without any unpinning.Comment: 5 pages, 3 figure

Measuring Dislocation Density in Aluminum with Resonant Ultrasound Spectroscopy

Dislocations in a material will, when present in enough numbers, change the speed of propagation of elastic waves. Consequently, two material samples, differing only in dislocation density, will have different elastic constants, a quantity that can be measured using Resonant Ultrasound Spectroscopy. Measurements of this effect on aluminum samples are reported. They compare well with the predictions of the theory.Comment: 4 pages, 2 figure