Ginestra Bianconi

Robin Mitra, Sarah F. McGough, Tapabrata Chakraborti et al. · oxford

Missing data are an unavoidable complication in many machine learning tasks. When data are `missing at random' there exist a range of tools and techniques to deal with the issue. However, as machine learning studies become more ambitious, and seek to learn from ever-larger volumes of heterogeneous data, an increasingly encountered problem arises in which missing values exhibit an association or structure, either explicitly or implicitly. Such `structured missingness' raises a range of challenges that have not yet been systematically addressed, and presents a fundamental hindrance to machine learning at scale. Here, we outline the current literature and propose a set of grand challenges in learning from data with structured missingness.

Lucille Calmon, Michael T. Schaub, Ginestra Bianconi

Higher-order networks can sustain topological signals which are variables associated not only to the nodes, but also to the links, to the triangles and in general to the higher dimensional simplices of simplicial complexes. These topological signals can describe a large variety of real systems including currents in the ocean, synaptic currents between neurons and biological transportation networks. In real scenarios topological signal data might be noisy and an important task is to process these signals by improving their signal to noise ratio. So far topological signals are typically processed independently of each other. For instance, node signals are processed independently of link signals, and algorithms that can enforce a consistent processing of topological signals across different dimensions are largely lacking. Here we propose Dirac signal processing, an adaptive, unsupervised signal processing algorithm that learns to jointly filter topological signals supported on nodes, links and triangles of simplicial complexes in a consistent way. The proposed Dirac signal processing algorithm is formulated in terms of the discrete Dirac operator which can be interpreted as "square root" of a higher-order Hodge Laplacian. We discuss in detail the properties of the Dirac operator including its spectrum and the chirality of its eigenvectors and we adopt this operator to formulate Dirac signal processing that can filter noisy signals defined on nodes, links and triangles of simplicial complexes. We test our algorithms on noisy synthetic data and noisy data of drifters in the ocean and find that the algorithm can learn to efficiently reconstruct the true signals outperforming algorithms based exclusively on the Hodge Laplacian.

Runyue Wang, Yu Tian, Pietro Liò et al.

Topological signals are variables or features associated with both nodes and edges of a network. Recently, in the context of Topological Machine Learning, great attention has been devoted to signal processing of such topological signals. Most of the previous topological signal processing algorithms treat node and edge signals separately and work under the hypothesis that the true signal is smooth and/or well approximated by a harmonic eigenvector of the Hodge-Laplacian, which may be violated in practice. Here we propose Dirac-equation signal processing, a framework for efficiently reconstructing true signals on nodes and edges, also if they are not smooth or harmonic, by processing them jointly. The proposed physics-inspired algorithm is based on the spectral properties of the topological Dirac operator. It leverages the mathematical structure of the topological Dirac equation to boost the performance of the signal processing algorithm. We discuss how the relativistic dispersion relation obeyed by the topological Dirac equation can be used to assess the quality of the signal reconstruction. Finally, we demonstrate the improved performance of the algorithm with respect to previous algorithms. Specifically, we show that Dirac-equation signal processing can also be used efficiently if the true signal is a non-trivial linear combination of more than one eigenstate of the Dirac equation, as it generally occurs for real signals.

Ginestra Bianconi

Recently, thanks to the development of artificial intelligence (AI) there is increasing scientific attention in establishing the connections between theoretical physics and AI. Traditionally, these connections have been focusing mostly on the relation between string theory and image processing and involve important theoretical paradigms such as holography. Recently G. Bianconi has formulated the Gravity from Entropy (GfE) approach to quantum gravity in which gravity is derived from the geometric quantum relative entropy (GQRE) between two metrics associated with the Lorentzian spacetime. Here it is demonstrated that the famous Perona-Malik algorithm for image processing is the gradient flow that maximizes the GfE action in its simple warm-up scenario. Specifically, this algorithm is the outcome of the maximization of the GfE action calculated between two Euclidean metrics: the one of the support of the image and the one induced by the image. As the Perona-Malik algorithm is known to preserve sharp contours, this implies that the GfE action, does not in general lead to uniform images upon iteration of the gradient flow dynamics as it would be intuitively expected from entropic actions maximising classical entropies. Rather, the outcome of the maximization of the GfE action is compatible with the preservation of complex structures. These results provide the geometrical and information theory foundations for the Perona-Malik algorithm and might contribute to establish deeper connections between GfE, machine learning and brain research.

Anthony Baptista, Rubén J. Sánchez-García, Anaïs Baudot et al.

Networks have provided extremely successful models of data and complex systems. Yet, as combinatorial objects, networks do not have in general intrinsic coordinates and do not typically lie in an ambient space. The process of assigning an embedding space to a network has attracted lots of interest in the past few decades, and has been efficiently applied to fundamental problems in network inference, such as link prediction, node classification, and community detection. In this review, we provide a user-friendly guide to the network embedding literature and current trends in this field which will allow the reader to navigate through the complex landscape of methods and approaches emerging from the vibrant research activity on these subjects.

Ginestra Bianconi, Sergey N. Dorogovtsev

Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the spectral dimension of the graph Laplacian of two classes of non-amenable $d$ dimensional simplicial complexes: the Apollonian networks and the pseudo-fractal networks. We analyse the scaling of the spectral dimension with the topological dimension $d$ for $d\to \infty$ and we point out that randomness such as the one present in Network Geometry with Flavor can diminish the value of the spectral dimension of these structures.

Filippo Radicchi, Dmitri Krioukov, Harrison Hartle et al.

Existing information-theoretic frameworks based on maximum entropy network ensembles are not able to explain the emergence of heterogeneity in complex networks. Here, we fill this gap of knowledge by developing a classical framework for networks based on finding an optimal trade-off between the information content of a compressed representation of the ensemble and the information content of the actual network ensemble. In this way not only we introduce a novel classical network ensemble satisfying a set of soft constraints but we are also able to calculate the optimal distribution of the constraints. We show that for the classical network ensemble in which the only constraints are the expected degrees a power-law degree distribution is optimal. Also, we study spatially embedded networks finding that the interactions between nodes naturally lead to non-uniform spread of nodes in the space, with pairs of nodes at a given distance not necessarily obeying a power-law distribution. The pertinent features of real-world air transportation networks are well described by the proposed framework.

Josephine Maria Thomas, Alessandro Muscoloni, Sara Ciucci et al.

Complex network topologies and hyperbolic geometry seem specularly connected, and one of the most fascinating and challenging problems of recent complex network theory is to map a given network to its hyperbolic space. The Popularity Similarity Optimization (PSO) model represents - at the moment - the climax of this theory. It suggests that the trade-off between node popularity and similarity is a mechanism to explain how complex network topologies emerge - as discrete samples - from the continuous world of hyperbolic geometry. The hyperbolic space seems appropriate to represent real complex networks. In fact, it preserves many of their fundamental topological properties, and can be exploited for real applications such as, among others, link prediction and community detection. Here, we observe for the first time that a topological-based machine learning class of algorithms - for nonlinear unsupervised dimensionality reduction - can directly approximate the network's node angular coordinates of the hyperbolic model into a two-dimensional space, according to a similar topological organization that we named angular coalescence. On the basis of this phenomenon, we propose a new class of algorithms that offers fast and accurate coalescent embedding of networks in the hyperbolic space even for graphs with thousands of nodes.