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

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

A Two-populations Ising model on diluted Random Graphs

We consider the Ising model for two interacting groups of spins embedded in an Erd\"{o}s-R\'{e}nyi random graph. The critical properties of the system are investigated by means of extensive Monte Carlo simulations. Our results evidence the existence of a phase transition at a value of the inter-groups interaction coupling $J_{12}^C$ which depends algebraically on the dilution of the graph and on the relative width of the two populations, as explained by means of scaling arguments. We also measure the critical exponents, which are consistent with those of the Curie-Weiss model, hence suggesting a wide robustness of the universality class.Comment: 11 pages, 4 figure

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

Analogue neural networks on correlated random graphs

We consider a generalization of the Hopfield model, where the entries of patterns are Gaussian and diluted. We focus on the high-storage regime and we investigate analytically the topological properties of the emergent network, as well as the thermodynamic properties of the model. We find that, by properly tuning the dilution in the pattern entries, the network can recover different topological regimes characterized by peculiar scalings of the average coordination number with respect to the system size. The structure is also shown to exhibit a large degree of cliquishness, even when very sparse. Moreover, we obtain explicitly the replica symmetric free energy and the self-consistency equations for the overlaps (order parameters of the theory), which turn out to be classical weighted sums of 'sub-overlaps' defined on all possible sub-graphs. Finally, a study of criticality is performed through a small-overlap expansion of the self-consistencies and through a whole fluctuation theory developed for their rescaled correlations: Both approaches show that the net effect of dilution in pattern entries is to rescale the critical noise level at which ergodicity breaks down.Comment: 34 pages, 3 figure

Ultrametric identities in glassy models of natural evolution

Spin-glasses constitute a well-grounded framework for evolutionary models. Of particular interest for (some of) these models is the lack of self-averaging of their order parameters (e.g. the Hamming distance between the genomes of two individuals), even in asymptotic limits, much as like what happens to the overlap between the configurations of two replica in mean-field spin-glasses. In the latter, this lack of self-averaging is related to a peculiar behavior of the overlap fluctuations, as described by the Ghirlanda–Guerra identities and by the Aizenman–Contucci polynomials, that cover a pivotal role in describing the ultrametric structure of the spin-glass landscape. As for evolutionary mod- els, such identities may therefore be related to a taxonomic classification of individuals, yet a full investigation on their validity is missing. In this paper, we study ultrametric identities in simple cases where solely random mutations take place, while selective pressure is absent, namely in flat landscape models. In particular, we study three paradigmatic models in this setting: the one parent model (which, by construction, is ultrametric at the level of single individu- als), the homogeneous population model (which is replica symmetric), and the species formation model (where a broken-replica scenario emerges at the level of species). We find analytical and numerical evidence that in the first and in the third model nor the Ghirlanda–Guerra neither the Aizenman–Contucci constraints hold, rather a new class of ultrametric identities is satisfied; in the second model all these constraints hold trivially. Very preliminary results on a real biological human genome derived by The 1000 Genome Project Consortium and on two artificial human genomes (generated by two different types neural networks) seem in better agreement with these new identities rather than the classic ones

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

Application of a Stochastic Modeling to Assess the Evolution of Tuberculous and Non-Tuberculous Mycobacterial Infection in Patients Treated with Tumor Necrosis Factor Inhibitors

Abstract In this manuscript we apply stochastic modeling to investigate the risk of reactivation of latent mycobacterial infections in patients undergoing treatment with tumor necrosis factor inhibitors. First, we review the perspective proposed by one of the authors in a previous work and which consists in predicting the occurrence of reactivation of latent tuberculosis infection or newly acquired tuberculosis during treatment; this is based on variational procedures on a simple set of parameters (e.g. rate of reactivation of a latent infection). Then, we develop a full analytical study of this approach through a Markov chain analysis and we find an exact solution for the temporal evolution of the number of cases of tuberculosis infection (re)activation. The analytical solution is compared with Monte Carlo simulations and with experimental data, showing overall excellent agreement. The generality of this theoretical framework allows to investigate also the case of nontuberculous mycobacteria infections; in particular, we show that reactivation in that context plays a minor role. This may suggest that, while the screening for tuberculous is necessary prior to initiating biologics, when considering nontuberculous mycobacteria only a watchful monitoring during the treatment is recommended. The framework outlined in this paper is quite general and could be extremely promising in further researches on drug-related adverse events

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

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.Comment: 22 pages, 14 figur