The amount and variety of data have been increasing drastically for several years. These data are often represented as networks and explored with approaches arising from network theory. Recent years have witnessed the extension of network exploration approaches to capitalize on more complex and richer network frameworks. Random walks, for instance, have been extended to explore multilayer networks. However, current random walk approaches are limited in the combination and heterogeneity of networks they can handle. New analytical and numerical random walk methods are needed to cope with the increasing diversity and complexity of multilayer networks. We propose here MultiXrank, a method and associated Python package that enables Random Walk with Restart on any kind of multilayer network. We evaluate MultiXrank with leave-one-out cross-validation and link prediction, and measure the impact of the addition or removal of network data on prediction performances. Finally, we measure the sensitivity of MultiXrank to input parameters by in-depth exploration of the parameter space.

Microscopic origin of the scattering pre-peak in aqueous propylamine mixtures: X-ray and neutron experiments versus simulations.The structure of aqueous propylamine mixtures is investigated through X-ray and neutron scattering experiments, and the scattered intensities compared with computer simulation data. Both sets of data show a prominent scattering pre-peak, which first appears at propylamine mole fraction x ≥ 0.1 around scattering vector k ≈ 0.2Å −1 , and evolves towards k ≈ 0.8Å −1 for neat propylamine x = 1. The existence of a scattering pre-peak in this mixture is unexpected, specifically in view of its absence in aqueous 1-propanol or aqueous DMSO mixtures. The detailed analysis of the various atom-atom structure factors and snapshots indicates that significant micro-structure exists, which produces correlation pre-peaks in the atom-atom structure factors, positive for like species atoms correlations and negative for the cross species ones. The scattering pre-peak depends on how these two contributions cancel or not. The way the amine group bond with water, produce a pre-peak through the inbalance of the positive and negative scattering contributions, unlike 1-propanol and DMSO, where these 2 contributions compensate exactly. Hence molecular simulations demonstrate how chemical details influence the microscopic segregation in different types of molecular emulsions and can be detected or not by scattering experiments Russia S(k). This simple formula also holds more complex systems, such as micelles. The reason is that micelles look just like meso-atoms floating in a structureless solvent, as usually explained in various textbooks. In this case, the form factor F(k) refers to the micelle shape, and S(k) to micelle-micelle correlations. Since micelles are composite objects, with an underlying atomic sub-structure, the corresponding I(k) will exhibit 2 scattering peaks, a main peak k M positioned at the mean atomic size, and a pre-peak at k P < k M , related to the micelle shape and size 5,6 . Pre-peaks are equally found in neat alcohols, as I(k) reveals, in addition to a main peak at k M , their existence at k P ≈ 0.4 − 0.7Å −1 7-9 . The origin of such pre-peak has been traced back to the existence of short chain-like clustering of the hydroxyl head groups, with mean size d ≈ 10Å. These experimental results have been confirmed by computer simulation, both from snapshot and cluster analysis and study of the atom-atom correlation functions and corresponding structure factor 10-12 . These analyses clearly demonstrate that the pre-peak k P is related to both the size of the chains formed in the neat liquids and their density. If one applies the same type of analysis to spherical micellar systems, for example, which are made of surfactant molecules immersed in a solution made of solvent, ions and counterions, the existence of a scattering peak around k P ≈ 0.06Å −1 is commonly interpreted in terms of spherical mi-

Complex biological processes, such as cellular differentiation, require an intricate rewiring of intra-cellular signalling networks. Previous characterisations of these networks revealed that promiscuity in signalling, quantified by a raised network entropy, underlies a less differentiated and malignant cell state. A theoretical connection between entropy and Ricci curvature has led to applications of discrete curvatures to characterise biological signalling networks at distinct time points during differentiation and malignancy. However, understanding and predicting the dynamics of biological network rewiring remains an open problem. Here we construct a framework to apply discrete Ricci curvature and Ricci flow to the problem of biological network rewiring. By investigating the relationship between network entropy and Forman-Ricci curvature, both theoretically and empirically on single-cell RNA-sequencing data, we demonstrate that the two measures do not always positively correlate, as has been previously suggested, and provide complementary rather than interchangeable information. We next employ discrete normalised Ricci flow, to derive network rewiring trajectories from transcriptomes of stem cells to differentiated cells, which accurately predict true intermediate time points of gene expression time courses. In summary, we present a differential geometry toolkit for investigation of dynamic network rewiring during cellular differentiation and cancer.

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