One of the simplest models for the slow relaxation and aging of glasses is the trap model by Bouchaud and others, which represents a system as a point in configurationspace hopping between local energy minima. The time evolution depends on the transition rates and the network of allowed jumps between the minima. We consider the case of sparse configuration-space connectivity given by a random graph, and study the spectral properties of the resulting master operator. We develop a general approach using the cavity method that gives access to the density of states in large systems, as well as localisation properties of the eigenvectors, which are important for the dynamics. We illustrate how, for a system with sparse connectivity and finite temperature, the density of states and the average inverse participation ratio have attributes that arise from a non-trivial combination of the corresponding mean field (fully connected) and random walk (infinite temperature) limits. In particular, we find a range of eigenvalues for which the density of states is of mean-field form but localisation properties are not, and speculate that the corresponding eigenvectors may be concentrated on extensively many clusters of network sites.

The slow relaxation and aging of glassy systems can be modelled as a Markov process on a simplified rough energy landscape: energy minima where the system tends to get trapped are taken as nodes of a random network, and the dynamics are governed by the transition rates among these. In this work we consider the case of purely activated dynamics, where the transition rates only depend on the depth of the departing trap. The random connectivity and the disorder in the trap depths make it impossible to solve the model analytically, so we base our analysis on the spectrum of eigenvalues λ of the master operator. We compute the local density of states ρ(λ|τ ) for traps with a fixed lifetime τ by means of the cavity method. This exhibits a power law behaviour ρ(λ|τ ) ∼ τ |λ| T in the regime of small relaxation rates |λ|, which we rationalize using a simple analytical approximation. In the time domain, we find that the probabilities of return to a starting node have a power law-tail that is determined by the distribution of excursion times F (t) ∼ t −(T +1) . We show that these results arise only by the combination of finite configuration space connectivity and glassy disorder, and interpret them in a simple physical picture dominated by jumps to deep neighbouring traps. arXiv:1906.07434v1 [cond-mat.dis-nn]

Clinical databases typically include, for each patient, many heterogeneous features, for example blood exams, the clinical history before the onset of the disease, the evolution of the symptoms, the results of imaging exams, and many others. We here propose to exploit a recently developed statistical approach, the Information Imbalance, to compare different subsets of patient features and automatically select the set of features which is maximally informative for a given clinical purpose, especially in minority classes. We adapt the Information Imbalance approach to work in a clinical framework, where patient features are often categorical, and are generally available only for a fraction of the patients. We apply this algorithm to a data set of ~ 1,300 patients treated for COVID-19 in Udine hospital before October 2021. Using this approach, we find combinations of features which, if used in combination, are maximally informative of the clinical fate and of the severity of the disease. The optimal number of features, which is determined automatically, turns out to be between 10 and 15. These features can be measured at admission. The approach can be used also if the features are available only for a fraction of the patients, does not require imputation and, importantly, is able to automatically select features with small inter-feature correlation. Clinical insights deriving from this study are also discussed.

Clinical data bases typically include, for each patient, many heterogeneous features, for example blood exams, the clinical history before the onset of the disease, the evolution of the symptoms, the results of imaging exams, and many others. Using subsets of these features, one can measure the similarity between two patients in several different manners. We here propose to exploit a recently developed statistical approach, the information imbalance, to compare these different similarity measures, and quantify their relative information content. We apply this approach to a data set of ~ 1,300 COVID-19 patients in Udine hospital before October 2021. Using this approach we find (asymmetric) relationships between single features and systematically compare subsets of up to 20 different features as COVID-19 severity predictors. The identified features can be measured at the moment of the admission of the patient and, if used in combination, are maximally informative of the clinical fate and of the severity of the disease. The approach can be used also if the features are available only for a fraction of the patients and, importantly, is able to select automatically features with small inter-feature correlation.