Adriano Barra

Equilibrium statistical mechanics of bipartite spin systems

The aim of this paper is to give an extensive treatment of bipartite mean field spin systems, pure and disordered. At first, bipartite ferromagnets are investigated, and an explicit expression for the free energy is achieved through a new minimax variational principle. Then, via the Hamilton?Jacobi technique, the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one. The same is investigated for bipartite spin-glasses. By the Borel?Cantelli lemma we obtain the control of the high temperature regime, while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation, uniquely defined by a minimax variational principle again. We also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one-party model counterparts. For the sake of completeness, we show further that at zero temperature the replica symmetric entropy becomes negative and, consequently, such a symmetry must be broken. The treatment of the fully broken replica symmetry case is deferred to a forthcoming paper. As a first step in this direction, we start deriving the linear and quadratic constraints to overlap fluctuations.

The Replica Symmetric Approximation of the Analogical Neural Network

In this paper we continue our investigation of the analogical neural network, by introducing and studying its replica symmetric approximation in the absence of external fields. 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 from the usual spin glass case to these models.While our methods allow the formulation of a fully broken replica symmetry scheme, in this paper we limit ourselves to the replica symmetric case, in order to give the basic essence of our interpolation method. The order parameters in this case are given by the assumed averages of the overlaps for the original spin variables, and for the new Gaussian variables. 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, through a self-consistent mini-max variational principle.The so gained replica symmetric approximation turns out to be exactly correct in the ergodic region, where it coincides with the annealed expression for the free energy, and in the low density limit of stored patterns. Moreover, in the spin glass limit it gives the correct expression for the replica symmetric approximation in this case. We calculate also the entropy density in the low temperature region, where we find that it becomes negative, as expected for this kind of approximation. Interestingly, in contrast with the case where the stored patterns are digital, no phase transition is found in the low temperature limit, as a function of the density of stored patterns.

The Mean Field Ising Model trough Interpolating Techniques

Aim of this paper is to illustrate how some recent techniques developed within the framework of spin glasses do work on simpler model, focusing on the method and not on the analyzed system. To fulfill our will the candidate model turns out to be the paradigmatic mean field Ising model. The model is introduced and investigated with the interpolation techniques. We show the existence of the thermodynamic limit, bounds for the free energy density, the explicit expression for the free energy with its suitable expansion via the order parameter, the self-consistency relation, the phase transition, the critical behavior and the self-averaging properties. At the end a formulation of a Parisi-like theory is tried and discussed.

Irreducible Free Energy Expansion and Overlaps Locking in Mean Field Spin Glasses

Following the works of Guerra, 1995; Aizenmar and Contucci, J. State. Phys. 92 (5–6): 765–783 (1998), we introduce a diagrammatic formulation for a cavity field expansion around the critical temperature. This approach allows us to obtain a theory for the overlaps 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.

How glassy are neural networks

In this paper we continue our investigation on the high storage regime of a neural network with Gaussian patterns. Through an exact mapping between its partition function and one of a bipartite spin glass (whose parties consist of Ising and Gaussian spins respectively), we give a complete control of the whole annealed region. The strategy explored is based on an interpolation between the bipartite system and two independent spin glasses built respectively by dichotomic and Gaussian spins: critical line, behavior of the principal thermodynamic observables and their fluctuations as well as overlap fluctuations are obtained and discussed. Then, we move further, extending such an equivalence beyond the critical line, to explore the broken ergodicity phase under the assumption of replica symmetry and show that the quenched free energy of this (analogical) Hopfield model can be described as a linear combination of the two quenched spin glass free energies even in the replica symmetric framework.

A mechanical approach to mean field spin models

Inspired by the bridge pioneered by Guerra among statistical mechanics on lattice and analytical mechanics on 1+1 continuous Euclidean space-time, we built a self-consistent method to solve for the thermodynamics of mean-field models defined on lattice, whose order parameters self average. We show the whole procedure by analyzing in full details the simplest test case, namely the Curie-Weiss model. Further we report some applications also to models whose order parameters do not self-average, by using the Sherrington-Kirkpatrick spin glass as a guide.

Retrieval capabilities of hierarchical networks: From dyson to hopfield

Elena Agliari, Adriano Barra, Andrea Galluzzi, Francesco Guerra, Daniele Tantari, and Flavia Tavani Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy. Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185, Roma, Italy. Dipartimento SBAI (Ingegneria), Sapienza Università di Roma, Via A. Scarpa 14, 00185, Roma, Italy. (Dated: July 21, 2014)

Extensive parallel processing on scale-free networks.

Peter Sollich, Daniele Tantari, Alessia Annibale, 4 and Adriano Barra Department of Mathematics, King’s College London, Strand, London WC2R 2LS, U.K. Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185, Roma, Italy. Institute for Mathematical and Molecular Biomedicine, King’s College London, Hodking Building, London SE1 1UL, U.K. Mathematics Department, King’s College London, The Strand WC2R 2LS, London, UK. Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy. (Dated: April 15, 2014)

PARAMETER EVALUATION OF A SIMPLE MEAN-FIELD MODEL OF SOCIAL INTERACTION

The aim of this work is to implement a statistical mechanics theory of social interaction, generalizing econometric discrete choice models. A class of simple mean field discrete models is introduced and discussed both from the theoretical and phenomenological point of view. We propose a parameter evaluation procedure and test it by fitting our model against three families of data coming from different cases: the estimated interaction parameters are found to have similar positive values establishing a quantitative confirmation of the peer imitation behavior found in social psychology. Moreover all the values of the interaction parameters belong to the phase transition regime suggesting its possible role in the study of social systems.

Cancer-driven dynamics of immune cells in a microfluidic environment

Scope of the present work is to infer the migratory ability of leukocytes by stochastic processes in order to distinguish the spontaneous organization of immune cells against an insult (namely cancer). For this purpose, spleen cells from immunodeficient mice, selectively lacking the transcription factor IRF-8 (IRF-8 knockout; IRF-8 KO), or from immunocompetent animals (wild-type; WT), were allowed to interact, alternatively, with murine B16.F10 melanoma cells in an ad hoc microfluidic environment developed on a LabOnChip technology. In this setting, only WT spleen cells were able to establish physical interactions with melanoma cells. Conversely, IRF-8 KO immune cells exhibited poor dynamical reactivity towards the neoplastic cells. In the present study, we collected data on the motility of these two types of spleen cells and built a complete set of observables that recapitulate the biological complexity of the system in these experiments. With remarkable accuracy, we concluded that the IRF-8 KO cells performed pure uncorrelated random walks, while WT splenocytes were able to make singular drifted random walks that collapsed on a straight ballistic motion for the system as a whole, hence giving rise to a highly coordinate response. These results may provide a useful system to quantitatively analyse the real time cell-cell interactions and to foresee the behavior of immune cells with tumor cells at the tissue level.