Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets

Higher-order clustering patterns in simplicial financial systems

The interest in induced higher-order relational and multidimensional structures embedded in the financial complex dataset is considered within the applied algebraic topology framework. The aim is to transcend the binary correlations when the interactions of the underlying system are stored in the entries of the cross-correlation matrix. By applying different criteria, we examined aggregations of firms through higher-order clustering of the financial system. The outcome is the extraction of patterns that appear in assemblages of firms due to their multidimensional properties embedded in the cross-correlation matrices. Results are compatible with classifying firms into clusters due to the industry they belong to. Furthermore, the novel and mixed collections of firms are revealed based on the applied mathematical approach. In the broader context, results shed light on the higher-order organization of interactions embedded in the cross-correlation matrix and, as a consequence, extract patterns of collective behavior within a complex system.

Potential grouping of nodes induced by higher-order structures in complex networks

Complex networks display an organization of elements into nontrivial structures at versatile inherent scales, imposing challenges on a more complete understanding of their behavior. The interest of the research presented here is in the characterization of potential mesoscale structures as building blocks of generalized communities in complex networks, with an integrated property that goes beyond the pairwise collections of nodes. For this purpose, a simplicial complex is obtained from a mathematical graph, and indirectly from time series, producing the so-called clique complex from the complex network. As the higher-order organizational structures are naturally embedded in the hierarchical strata of a simplicial complex, the relationships between aggregation of nodes are stored in the higher-order combinatorial Laplacian. Based on the postulate that aggregation of nodes represents integrated configuration of information, the observability parameter is defined for the characterization of potential configurations, computed from the entries of the combinatorial Laplacian matrix. The framework introduced here is used to characterize nontrivial inherent organizational patterns embedded in two real-world complex networks and three complex networks obtained from heart rate time series recordings of three different subject's meditative states.

The Poincaré-Shannon Machine: Statistical Physics and Machine Learning Aspects of Information Cohomology

Previous works established that entropy is characterized uniquely as the first cohomology class in a topos and described some of its applications to the unsupervised classification of gene expression modules or cell types. These studies raised important questions regarding the statistical meaning of the resulting cohomology of information and its interpretation or consequences with respect to usual data analysis and statistical physics. This paper aims to present the computational methods of information cohomology and to propose its interpretations in terms of statistical physics and machine learning. In order to further underline the cohomological nature of information functions and chain rules, the computation of the cohomology in low degrees is detailed to show more directly that the k multivariate mutual information (Ik) are (k−1)-coboundaries. The (k−1)-cocycles condition corresponds to Ik=0, which generalizes statistical independence to arbitrary degree k. Hence, the cohomology can be interpreted as quantifying the statistical dependences and the obstruction to factorization. I develop the computationally tractable subcase of simplicial information cohomology represented by entropy Hk and information Ik landscapes and their respective paths, allowing investigation of Shannon’s information in the multivariate case without the assumptions of independence or of identically distributed variables. I give an interpretation of this cohomology in terms of phase transitions in a model of k-body interactions, holding both for statistical physics without mean field approximations and for data points. The I1 components define a self-internal energy functional Uk and (−1)kIk,k≥2 components define the contribution to a free energy functional Gk (the total correlation) of the k-body interactions. A basic mean field model is developed and computed on genetic data reproducing usual free energy landscapes with phase transition, sustaining the analogy of clustering with condensation. The set of information paths in simplicial structures is in bijection with the symmetric group and random processes, providing a trivial topological expression of the second law of thermodynamics. The local minima of free energy, related to conditional information negativity and conditional independence, characterize a minimum free energy complex. This complex formalizes the minimum free-energy principle in topology, provides a definition of a complex system and characterizes a multiplicity of local minima that quantifies the diversity observed in biology. I give an interpretation of this complex in terms of unsupervised deep learning where the neural network architecture is given by the chain complex and conclude by discussing future supervised applications.

The multidimensional brain

Brain activity takes place in three spatial-plus time dimensions. This rather obvious claim has been recently questioned by papers that, taking into account the big data outburst and novel available computational tools, are starting to unveil a more intricate state of affairs. Indeed, various brain activities and their correlated mental functions can be assessed in terms of trajectories embedded in phase spaces of dimensions higher than the canonical ones. In this review, I show how further dimensions may not just represent a convenient methodological tool that allows a better mathematical treatment of otherwise elusive cortical activities, but may also reflect genuine functional or anatomical relationships among real nervous functions. I then describe how to extract hidden multidimensional information from real or artificial neurodata series, and make clear how our mind dilutes, rather than concentrates as currently believed, inputs coming from the environment. Finally, I argue that the principle "the higher the dimension, the greater the information" may explain the occurrence of mental activities and elucidate the mechanisms of human diseases associated with dimensionality reduction.