arXiv subjects

Adriano Barra

Publications and source records attributed to Adriano Barra.

At least 109 records · Page 6Linked to original sources

arXiv2010

Equilibrium statistical mechanics of bipartite spin systems

Aim of this paper is to give an extensive treatment of bipartite mean field spin systems, ordered and disordered: at first, bipartite ferromagnets are investigated, achieving an explicit expression for the free energy trough a new minimax variational principle. Furthermore via the Hamilton-Jacobi technique the same free energy structure 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 a control of the high temperature regime is obtained, while via the double stochastic stability technique we get also the explicit expression of the free energy at the replica symmetric level, uniquely defined by a minimax variational principle again. A general results that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts is achieved too. 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.

cond-mat.dis-nn↗

arXiv2010

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.

cond-mat.stat-mech↗

arXiv2010

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.

cond-mat.stat-mech↗

arXiv2010

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.

cond-mat.dis-nn↗

arXiv2010

A statistical mechanics approach to autopoietic immune networks

The aim of this work is to try to bridge over theoretical immunology and disordered statistical mechanics. Our long term hope is to contribute to the development of a quantitative theoretical immunology from which practical applications may stem. In order to make theoretical immunology appealing to the statistical physicist audience we are going to work out a research article which, from one side, may hopefully act as a benchmark for future improvements and developments, from the other side, it is written in a very pedagogical way both from a theoretical physics viewpoint as well as from the theoretical immunology one. Furthermore, we have chosen to test our model describing a wide range of features of the adaptive immune response in only a paper: this has been necessary in order to emphasize the benefit available when using disordered statistical mechanics as a tool for the investigation. However, as a consequence, each section is not at all exhaustive and would deserve deep investigation: for the sake of completeness, we restricted details in the analysis of each feature with the aim of introducing a self-consistent model.

cond-mat.dis-nn↗

arXiv2009

Notes on ferromagnetic diluted P-spin model

In this paper we develop the interpolating cavity field technique for the mean field ferromagnetic p-spin. The model we introduce is a natural extension of the diluted Curie-Weiss model to p>2 spin interactions. Several properties of the free energy are analyzed and, in particular, we show that it recovers the expressions already known for p=2 models and for p>2 fully connected models. Further, as the model lacks criticality, we present extensive numerical simulations to evidence the presence of a first order phase transition and deepen the behavior at the transition line. Overall, a good agreement is obtained among analytical results, numerics and previous works.

cond-mat.stat-mech↗

arXiv2009

Fluctuations induce transitions in frustrated sparse networks

We analyze, by means of statistical mechanics, a sparse network with random competitive interactions among dichotomic variables pasted on the nodes, namely a Viana-Bray model. The model is described by an infinite series of order parameters (the multi-overlaps) and has two tunable degrees of freedom: the noise level and the connectivity (the averaged number of links). We show that there are no multiple transition lines, one for every order parameter, as a naive approach would suggest, but just one corresponding to ergodicity breaking. We explain this scenario within a novel and simple mathematical technique via a driving mechanism such that, as the first order parameter (the two replica overlap) becomes different from zero due to a real second order phase transition (with properly associated diverging rescaled fluctuations), it enforces all the other multi-overlaps toward positive values thanks to the strong correlations which develop among themselves and the two replica overlap at the critical line.

cond-mat.dis-nn↗

arXiv2009

About the ergodic regime in the analogical Hopfield neural networks. Moments of the partition function

In this paper we introduce and exploit the real replica approach for a minimal generalization of the Hopfield model, by assuming the learned patterns to be distributed accordingly to a standard unit Gaussian. We consider the high storage case, when the number of patterns is linearly diverging with the number of neurons. We study the infinite volume behavior of the normalized momenta of the partition function. We find a region in the parameter space where the free energy density in the infinite volume limit is self-averaging around its annealed approximation, as well as the entropy and the internal energy density. Moreover, we evaluate the corrections to their extensive counterparts with respect to their annealed expressions. The fluctuations of properly introduced overlaps, which act as order parameters, are also discussed.

cond-mat.dis-nn↗

arXiv2009

Constraints for the order parameters in analogical neural networks

In this paper we study, via equilibrium statistical mechanics, the properties of the internal energy of an Hopfield neural network whose patterns are stored continuously (Gaussian distributed). The model is shown to be equivalent to a bipartite spin glass in which one party is given by dichotomic neurons and the other party by Gaussian spin variables. Dealing with replicated systems, beyond the Mattis magnetization, we introduce two overlaps, one for each party, as order parameters of the theory: The first is a standard overlap among neural configurations on different replicas, the second is an overlap among the Gaussian spins of different replicas. The aim of this work is to show the existence of constraints for these order parameters close to ones found in many other complex systems as spin glasses and diluted networks: we find a class of Ghirlanda-Guerra-like identities for both the overlaps, generalizing the well known constraints to the neural networks, as well as new identities where noise is involved explicitly.

cond-mat.dis-nn↗

arXiv2009

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.

cond-mat.dis-nn↗

arXiv2009

Acquaintance role for decision making and exchanges in social networks

We model a social network by a random graph whose nodes represent agents and links between two of them stand for a reciprocal interaction; each agent is also associated to a binary variable which represents a dichotomic opinion or attribute. We consider both the case of pair-wise (p=2) and multiple (p>2) interactions among agents and we study the behavior of the resulting system by means of the energy-entropy scheme, typical of statistical mechanics methods. We show, analytically and numerically, that the connectivity of the social network plays a non-trivial role: while for pair-wise interactions (p=2) the connectivity weights linearly, when interactions involve contemporary a number of agents larger than two (p>2), its weight gets more and more important. As a result, when p is large, a full consensus within the system, can be reached at relatively small critical couplings with respect to the p=2 case usually analyzed, or, otherwise stated, relatively small coupling strengths among agents are sufficient to orient most of the system.

physics.soc-ph↗

arXiv2009

Toward a quantitative approach to migrants integration

Migration phenomena and all the related issues, like integration of different social groups, are intrinsically complex problems since they strongly depend on several competitive mechanisms as economic factors, cultural differences and many others. By identifying a few essential assumptions, and using the statistical mechanics of complex systems, we propose a novel quantitative approach that provides a minimal theory for those phenomena. We show that the competitive interactions in decision making among a population of $N$ host citizens and $P$ immigrants, a bi-partite spin-glass, give rise to a "social consciousness" inside the host community in the sense of the associative memory of neural networks. The theory leads to a natural quantitative definition of migrant's "integration" inside the community. From the technical point of view this minimal picture assumes, as control parameters, only general notions like strength of the random interactions, the ratio among the two party sizes and the cultural influence. Few steps forward, toward more refined models, which include some structure on the random interaction topology (as dilution to avoid the plain mean field approach) and correlations of experiences felt among the two parties (biasing the distribution of the coupling) are discussed at the end, where we show the robustness of our approach.

physics.soc-ph↗

arXiv2009

New perspectives in the equilibrium statistical mechanics approach to social and economic sciences

In this work we review some recent development in the mathematical modelling of quantitative sociology by means of statistical mechanics. After a short pedagogical introduction to static and dynamic properties of many body systems, we develop a theory for agents interactions on random graph. Our approach is based on describing a social network as a graph whose nodes represent agents and links between two of them stand for a reciprocal interaction. Each agent has to choose among a dichotomic option (i.e. agree or disagree) with respect to a given matter and he is driven by external influences (as media) and peer to peer interactions. These mimic the imitative behavior of the collectivity and may possibly be zero if the two nodes are disconnected. For this scenario we work out both the dynamics and the corresponding equilibria (statics). Once the 2-body theory is completely explored, we analyze, on the same random graph, a "diffusive strategic dynamics" with pairwise interactions, where detailed balance constraint is relaxed. The dynamic encodes some relevant processes which are expected to play a crucial role in the approach to equilibrium in social systems, i.e. diffusion of information and strategic choices. We observe numerically that such a dynamics reaches a well defined steady state that fulfills a "shift property": the critical interaction strength for the phase transition is lower with respect to the one expected from canonical equilibrium. Finally, we show how the stationary states obtained with this dynamics can be described by statistical mechanics equilibria of a diluted p-spin model, for a suitable non-integer real p>2. Several implications from a sociological perspective are discussed together with some general outlooks.

physics.soc-ph↗

arXiv2009

A numerical investigation of the jamming transition in traffic flow on diluted planar networks

In order to develop a toy model for car's traffic in cities, in this paper we analyze, by means of numerical simulations, the transition among fluid regimes and a congested jammed phase of the flow of "kinetically constrained" hard spheres in planar random networks similar to urban roads. In order to explore as timescales as possible, at a microscopic level we implement an event driven dynamics as the infinite time limit of a class of already existing model (e.g. "Follow the Leader") on an Erdos-Renyi two dimensional graph, the crossroads being accounted by standard Kirchoff density conservations. We define a dynamical order parameter as the ratio among the moving spheres versus the total number and by varying two control parameters (density of the spheres and coordination number of the network) we study the phase transition. At a mesoscopic level it respects an, again suitable adapted, version of the Lighthill-Whitham model, which belongs to the fluid-dynamical approach to the problem. At a macroscopic level the model seems to display a continuous transition from a fluid phase to a jammed phase when varying the density of the spheres (the amount of cars in a city-like scenario) and a discontinuous jump when varying the connectivity of the underlying network.

cond-mat.stat-mech↗

arXiv2009

A certain class of Curie-Weiss models

By using a formal analogy between statistical mechanics of mean field spin systems and analytical mechanics of viscous liquids -at first pointed out by Francesco Guerra, then recently developed by the authors- we give the thermodynamic limit of the free energy and the critical behavior of Curie Weiss models for a certain class of generalized spin variables. Then, with the same techniques, we give a complete picture of the bipartite Curie-Weiss model, dealing with the same class of generalized spins. Ultimately we analyze further the existence of a minmax principle for the latter which mirrors the standard variational principle of canonical thermodynamics when generalized to multiple interacting parties.

math-ph↗

arXiv2009

Effective Interactions in Group Competition with Strategic Diffusive Dynamics

We analyze, on a random graph, a diffusive strategic dynamics with pairwise interactions, where nor Glauber prescription, neither detailed balance hold. We observe numerically that such a dynamics reaches a well defined steady state that fulfills a shift property: the critical temperature of the canonical ferromagnetic phase transition is higher with respect to the expected equilibrium one, known both numerically via Glauber relaxation or Monte Carlo simulations as well as analytically via cavity techniques or replica approaches. We show how the relaxed states of this kind of dynamics can be described by statistical mechanics equilibria of a diluted p-spin model, for a suitable non-integer real p. Several implications from both theoretical physics and quantitative sociology points of view are discussed.

cond-mat.dis-nn↗

arXiv2009

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.

math-ph↗

arXiv2009

Dilution Robustness for Mean Field Ferromagnets

In this work we compare two different random dilution of a mean field ferromagnet: the first model is built on a Bernoulli-diluted network while the second lives on a Poisson-diluted network. While it is known that the two models have in the thermodynamic limit the same free energy we investigate on the structural constraints that the two models must fulfill. We rigorously derive for each model the set of identities for the multi-overlaps distribution using different methods for the two dilutions: constraints in the former model are obtained by studying the consequences of the self-averaging of the internal energy density, while in the latter are obtained by a stochastic-stability technique. Finally we prove that the identities emerging in the two models are the same, showing "robustness" of the ferromagnetic properties of diluted networks with respect to the details of dilution.

math-ph↗