Many complex systems can be represented as networks consisting of distinct types of interactions, which can be categorized as links belonging to different layers. For example, a good description of the full protein–protein interactome requires, for some organisms, up to seven distinct network layers, accounting for different genetic and physical interactions, each containing thousands of protein–protein relationships. A fundamental open question is then how many layers are indeed necessary to accurately represent the structure of a multilayered complex system. Here we introduce a method based on quantum theory to reduce the number of layers to a minimum while maximizing the distinguishability between the multilayer network and the corresponding aggregated graph. We validate our approach on synthetic benchmarks and we show that the number of informative layers in some real multilayer networks of protein–genetic interactions, social, economical and transportation systems can be reduced by up to 75%. Multilayer networks have been used to capture the structure of complex systems with different types of interactions, but often contain redundant information. Here, De Domenico et al. present a method based on quantum information, to identify the minimal configuration of layers to retain.

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Structural reducibility of multilayer networks

We examine the scaling law $B \propto M^{\alpha}$ which connects organismal metabolic rate $B$ with organismal mass $M$, where $\alpha$ is commonly held to be 3/4. Since simple dimensional analysis suggests $\alpha=2/3$, we consider this to be a null hypothesis testable by empirical studies. We re-analyze data sets for mammals and birds compiled by Heusner, Bennett and Harvey, Bartels, Hemmingsen, Brody, and Kleiber, and find little evidence for rejecting $\alpha=2/3$ in favor of $\alpha=3/4$. For mammals, we find a possible breakdown in scaling for larger masses reflected in a systematic increase in $\alpha$. We also review theoretical justifications of $\alpha=3/4$ based on dimensional analysis, nutrient-supply networks, and four-dimensional biology. We find that present theories for $\alpha=3/4$ require assumptions that render them unconvincing for rejecting the null hypothesis that $\alpha=2/3$.

https://arxiv.org/pdf/physics/0007096.pdf

Re-examination of the "3/4-law" of Metabolism

Multilayer relationships among entities and information about entities must be accompanied by the means to analyze, visualize, and obtain insights from such data. We present open-source software (muxViz) that contains a collection of algorithms for the analysis of multilayer networks, which are an important way to represent a large variety of complex systems throughout science and engineering. We demonstrate the ability of muxViz to analyze and interactively visualize multilayer data using empirical genetic, neuronal, and transportation networks. Our software is available at this https URL.

MuxViz: A Tool for Multilayer Analysis and Visualization of Networks

Multilayer relationships among entities and information about entities must be accompanied by the means to analyse, visualize and obtain insights from such data. We present open-source software (muxViz) that contains a collection of algorithms for the analysis of multilayer networks, which are an important way to represent a large variety of complex systems throughout science and engineering. We demonstrate the ability of muxViz to analyse and interactively visualize multilayer data using empirical genetic, neuronal and transportation networks. Our software is available at https://github.com/manlius/muxViz.

/pdf/muxviz-a-tool-for-multilayer-analysis-and-visualization-of-4bg4mewd7f.pdf

MuxViz: a tool for multilayer analysis and visualization of networks

This paper develops systematic approaches to obtain $f$ -divergence inequalities, dealing with pairs of probability measures defined on arbitrary alphabets. Functional domination is one such approach, where special emphasis is placed on finding the best possible constant upper bounding a ratio of $f$ -divergences. Another approach used for the derivation of bounds among $f$ -divergences relies on moment inequalities and the logarithmic-convexity property, which results in tight bounds on the relative entropy and Bhattacharyya distance in terms of $\chi ^{2}$ divergences. A rich variety of bounds are shown to hold under boundedness assumptions on the relative information. Special attention is devoted to the total variation distance and its relation to the relative information and relative entropy, including “reverse Pinsker inequalities,” as well as on the $E_\gamma $ divergence, which generalizes the total variation distance. Pinsker’s inequality is extended for this type of $f$ -divergence, a result which leads to an inequality linking the relative entropy and relative information spectrum. Integral expressions of the Renyi divergence in terms of the relative information spectrum are derived, leading to bounds on the Renyi divergence in terms of either the variational distance or relative entropy.

$f$ -Divergence Inequalities

Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the most important divergence measure of information theory, Kullback divergence. As opposed to Kullback divergence it determines in a very direct way a metric; indeed, it is the square of a metric. We consider a family of divergence measures (JD_alpha for alpha>0), the Jensen divergences of order alpha, which generalize JD as JD_1=JD. Using a result of Schoenberg, we prove that JD_alpha is the square of a metric for alpha lies in the interval (0,2], and that the resulting metric space of probability distributions can be isometrically embedded in a real Hilbert space. Quantum Jensen-Shannon divergence (QJD) is a symmetrized and smoothed version of quantum relative entropy and can be extended to a family of quantum Jensen divergences of order alpha (QJD_alpha). We strengthen results by Lamberti et al. by proving that for qubits and pure states, QJD_alpha^1/2 is a metric space which can be isometrically embedded in a real Hilbert space when alpha lies in the interval (0,2]. In analogy with Burbea and Rao's generalization of JD, we also define general QJD by associating a Jensen-type quantity to any weighted family of states. Appropriate interpretations of quantities introduced are discussed and bounds are derived in terms of the total variation and trace distance.

Properties of Classical and Quantum Jensen-Shannon Divergence

Renyi's thinning operation on a discrete random variable is a natural discrete analog of the scaling operation for continuous random variables. The properties of thinning are investigated in an information-theoretic context, especially in connection with information-theoretic inequalities related to Poisson approximation results. The classical Binomial-to-Poisson convergence (sometimes referred to as the “law of small numbers”) is seen to be a special case of a thinning limit theorem for convolutions of discrete distributions. A rate of convergence is provided for this limit, and nonasymptotic bounds are also established. This development parallels, in part, the development of Gaussian inequalities leading to the information-theoretic version of the central limit theorem. In particular, a “thinning Markov chain” is introduced, and it is shown to play a role analogous to that of the Ornstein-Uhlenbeck process in connection to the entropy power inequality.

/pdf/thinning-entropy-and-the-law-of-thin-numbers-1uiuqtggsn.pdf

Thinning, Entropy, and the Law of Thin Numbers

https://www.biologie.ens.fr/~legendre/biblio/DemetriusLegendre&Harremoes_Entropy&BodySize_BullMathBiol_2009.pdf

Evolutionary Entropy: A Predictor of Body Size, Metabolic Rate and Maximal Life Span

The "thinning" operation on a discrete random variable is the natural discrete analog of scaling a continuous variable, i.e., multiplying it by a constant. We examine the role and properties of thinning in the context of information-theoretic inequalities for Poisson approximation. The classical Binomial-to-Poisson convergence, often referred to as the "law of small numbers," is seen to be a special case of a thinning limit theorem for convolutions of discrete distributions. A rate of convergence is also provided for this limit. A Nash equilibrium is established for a channel game, where Poisson noise and a Poisson input are optimal strategies. Our development partly parallels the development of Gaussian inequalities leading to the information- theoretic version of the central limit theorem.

/pdf/thinning-and-the-law-of-small-numbers-466zclefyt.pdf

Thinning and the Law of Small Numbers

Topologies related to information divergence are introduced. The conditional limit theorem is taken as motivating example, and simplified proofs of the relevant theorems are given. Continuity properties of entropy and information divergence are discussed.

Peter Harremoës

Papers

Properties of Classical and Quantum Jensen-Shannon Divergence

Thinning, Entropy, and the Law of Thin Numbers

Evolutionary Entropy: A Predictor of Body Size, Metabolic Rate and Maximal Life Span

Thinning and the Law of Small Numbers

Information Topologies with Applications