We study the stochastic dynamics of Ising spin models with random bonds, interacting on finitely connected Poissonnian random graphs. We use the dynamical replica method to derive closed dynamical equations for the joint spin-field probability distribution, and solve these within the replica symmetry ansatz. Although the theory is developed in a general setting, with a view to future applications in various other fields, in this paper we apply it mainly to the dynamics of the Glauber algorithm (extended with cooling schedules) when running on the so-called vertex cover optimization problem. Our theoretical predictions are tested against both Monte Carlo simulations and known results from equilibrium studies. In contrast to previous dynamical analyses based on deriving closed equations for only a small numbers of scalar order parameters, the agreement between theory and experiment in the present study is nearly perfect. ‡ This involves formally the introduction of random durations for the individual spin updates which are N −1 on average, and which for finite N are drawn from a specific distribution [24].

The B cell repertoire is generated in the adult bone marrow by an ordered series of gene rearrangement processes that result in massive diversity of immunoglobulin (Ig) genes and consequently an equally large number of potential specificities for antigen. As the process is essentially random, the cells exhibiting excess reactivity with self-antigens are generated and need to be removed from the repertoire before the cells are fully mature. Some of the cells are deleted, and some will undergo receptor editing to see if changing the light chain can rescue an autoreactive antibody. As a consequence, the binding properties of the B cell receptor are changed as development progresses through pre-B ≫ immature ≫ transitional ≫ naïve phenotypes. Using long-read, high-throughput, sequencing we have produced a unique set of sequences from these four cell types in human bone marrow and matched peripheral blood, and our results describe the effects of tolerance selection on the B cell repertoire at the Ig gene level. Most strong effects of selection are seen within the heavy chain repertoire and can be seen both in gene usage and in CDRH3 characteristics. Age-related changes are small, and only the size of the CDRH3 shows constant and significant change in these data. The paucity of significant changes in either kappa or lambda light chain repertoires implies that either the heavy chain has more influence over autoreactivity than light chain and/or that switching between kappa and lambda light chains, as opposed to switching within the light chain loci, may effect a more successful autoreactive rescue by receptor editing. Our results show that the transitional cell population contains cells other than those that are part of the pre-B ≫ immature ≫ transitional ≫ naïve development pathway, since the population often shows a repertoire that is outside the trajectory of gene loss/gain between pre-B and naïve stages.

We study the Glauber dynamics of Ising spin models with random bonds, on finitely connected random graphs. We generalize a recent dynamical replica theory with which to predict the evolution of the joint spin-field distribution, to include random graphs with arbitrary degree distributions. The theory is applied to Ising ferromagnets on randomly diluted Bethe lattices, where we study the evolution of the magnetization and the internal energy. It predicts a prominent slowing down of the flow in the Griffiths phase, it suggests a further dynamical transition at lower temperatures within the Griffiths phase, and it is verified quantitatively by the results of Monte Carlo simulations. 2 c

Nearly all statistical inference methods were developed for the regime where the number N of data samples is much larger than the data dimension p. Inference protocols such as maximum likelihood (ML) or maximum a posteriori probability (MAP) are unreliable if p = O(N), due to overfitting. This limitation has for many disciplines with increasingly high-dimensional data become a serious bottleneck. We recently showed that in Cox regression for time-to-event data the overfitting errors are not just noise but take mostly the form of a bias, and how with the replica method from statistical physics one can model and predict this bias and the noise statistics. Here we extend our approach to arbitrary generalized linear regression models (GLM), with possibly correlated covariates. We analyse overfitting in ML/MAP inference without having to specify data types or regression models, relying only on the GLM form, and derive generic order parameter equations for the case of L2 priors. Second, we derive the probabilistic relationship between true and inferred regression coefficients in GLMs, and show that, for the relevant hyperparameter scaling and correlated covariates, the L2 regularization causes a predictable direction change of the coefficient vector. Our results, illustrated by application to linear, logistic, and Cox regression, enable one to correct ML and MAP inferences in GLMs systematically for overfitting bias, and thus extend their applicability into the hitherto forbidden regime p= O(N).

Computing circuits composed of noisy logical gates and their ability to represent arbitrary boolean functions with a given level of error are investigated within a statistical mechanics setting. Existing bounds on their performance are straightforwardly retrieved, generalized, and identified as the corresponding typical-case phase transitions. Results on error rates, function depth, and sensitivity, and their dependence on the gate-type and noise model used are also obtained.

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology.We identify cases where the commonly used annealed approximation is valid and others where it breaks down. Broader links between BN and general Boolean formulas are highlighted.

Properties of computing Boolean circuits composed of noisy logical gates are studied using the statistical physics methodology. A formula-growth model that gives rise to random Boolean functions is mapped onto a spin system, which facilitates the study of their typical behavior in the presence of noise. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding macroscopic phase transitions. The framework is employed for deriving results on error-rates at various function-depths and function sensitivity, and their dependence on the gate-type and noise model used. These are difficult to obtain via the traditional methods used in this field.

We show that model-based Bayesian clustering, the probabilistically most systematic approach to the partitioning of data, can be mapped into a statistical physics problem for a gas of particles, and as a result becomes amenable to a detailed quantitative analysis. A central role in the resulting statistical physics framework is played by an entropy function. We demonstrate that there is a relevant parameter regime where mean-field analysis of this function is exact, and that, under natural assumptions, the lowest entropy state of the hypothetical gas corresponds to the optimal clustering of data. The byproduct of our analysis is a simple but effective clustering algorithm, which infers both the most plausible number of clusters in the data and the corresponding partitions. Describing Bayesian clustering in statistical mechanical terms is found to be natural and surprisingly effective.