Order-Parameter Flow in the SK Spin-Glass II: Inclusion of Microscopic Memory Effects

We develop further a recent dynamical replica theory to describe the dynamics of the Sherrington-Kirkpatrick spin-glass in terms of closed evolution equations for macroscopic order parameters. We show how microscopic memory effects can be included in the formalism through the introduction of a dynamic order parameter function: the joint spin-field distribution. The resulting formalism describes very accurately the relaxation phenomena observed in numerical simulations, including the typical overall slowing down of the flow that was missed by the previous simple two-parameter theory. The advanced dynamical replica theory is either exact or a very good approximation.Comment: same as original, but this one is TeXabl

Tailored graph ensembles as proxies or null models for real networks II: results on directed graphs

We generate new mathematical tools with which to quantify the macroscopic topological structure of large directed networks. This is achieved via a statistical mechanical analysis of constrained maximum entropy ensembles of directed random graphs with prescribed joint distributions for in- and outdegrees and prescribed degree-degree correlation functions. We calculate exact and explicit formulae for the leading orders in the system size of the Shannon entropies and complexities of these ensembles, and for information-theoretic distances. The results are applied to data on gene regulation networks.Comment: 21 pages, 1 figure, submitted to J. Phys.

Dynamical Replica Theory for Disordered Spin Systems

We present a new method to solve the dynamics of disordered spin systems on finite time-scales. It involves a closed driven diffusion equation for the joint spin-field distribution, with time-dependent coefficients described by a dynamical replica theory which, in the case of detailed balance, incorporates equilibrium replica theory as a stationary state. The theory is exact in various limits. We apply our theory to both the symmetric- and the non-symmetric Sherrington-Kirkpatrick spin-glass, and show that it describes the (numerical) experiments very well.Comment: 7 pages RevTex, 4 figures, for PR

Smoothed Bootstrap Methods for Hypothesis Testing

This paper demonstrates the application of smoothed bootstrap methods and Efron’s methods for hypothesis testing on real-valued data, right-censored data and bivariate data. The tests include quartile hypothesis tests, two sample medians and Pearson and Kendall correlation tests. Simulation studies indicate that the smoothed bootstrap methods outperform Efron’s methods in most scenarios, particularly for small datasets. The smoothed bootstrap methods provide smaller discrepancies between the actual and nominal error rates, which makes them more reliable for testing hypotheses

Survival signature-based sensitivity analysis of systems with epistemic uncertainties

The survival signature provides a basis for efficient reliability assessment of systems with more than one component type. Often a perfect probabilistic modelling of the system is not possible due to limited information, vagueness and imprecision. Hence generalized probabilistic methods need to be used. These methods allow to explicitly model the uncertainties without the need of unjustified hypotheses and approximation. In this paper, a novel and efficient sensitivity approach is presented. The proposed approach is based on survival signature, allowing to identify and rank components in a system. A numerical example is used to illustrate the above methods

DYNAMICAL SOLUTION OF A MODEL WITHOUT ENERGY BARRIERS

In this note we study the dynamics of a model recently introduced by one of us, that displays glassy phenomena in absence of energy barriers. Using an adiabatic hypothesis we derive an equation for the evolution of the energy as a function of time that describes extremely well the glassy behaviour observed in Monte Carlo simulations.Comment: 11 pages, LaTeX, 3 uuencoded figure

Finite Size Effects in Separable Recurrent Neural Networks

We perform a systematic analytical study of finite size effects in separable recurrent neural network models with sequential dynamics, away from saturation. We find two types of finite size effects: thermal fluctuations, and disorder-induced `frozen' corrections to the mean-field laws. The finite size effects are described by equations that correspond to a time-dependent Ornstein-Uhlenbeck process. We show how the theory can be used to understand and quantify various finite size phenomena in recurrent neural networks, with and without detailed balance.Comment: 24 pages LaTex, with 4 postscript figures include