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At the beginning of 2020, the COVID-19 pandemic was able to spread quickly in Wuhan and in the province of Hubei due to a lack of experience with this novel virus. Additionally, authories had no proven experience with applying insufficient medical, communication and crisis management tools. For a considerable period of time, the actual number of people infected was unknown. There were great uncertainties regarding the dynamics and spread of the Covid-19 virus infection. In this paper, we develop a system dynamics model for the three connected regions (Wuhan, Hubei excl. Wuhan, China excl. Hubei) to understand the infection and spread dynamics of the virus and provide a more accurate estimate of the number of infected people in Wuhan and discuss the necessity and effectivity of protective measures against this epidemic, such as the quarantines imposed throughout China. We use the statistics of confirmed cases of China excl. Hubei. Also the daily data on travel activity within China was utilized, in order to determine the actual numerical development of the infected people in Wuhan City and Hubei Province. We used a multivariate Monte Carlo optimization to parameterize the model to match the official statistics. In particular, we used the model to calculate the infections, which had already broken out, but were not diagnosed for various reasons. [ABSTRACT FROM AUTHOR]

In this paper, we define novel graph measures for directed networks. The measures are based on graph polynomials utilizing the out- and in-degrees of directed graphs. Based on these polynomial, we define another polynomial and use their positive zeros as graph measures. The measures have meaningful properties that we investigate based on analytical and numerical results. As the computational complexity to compute the measures is polynomial, our approach is efficient and can be applied to large networks. We emphasize that our approach clearly complements the literature in this field as, to the best of our knowledge, existing complexity measures for directed graphs have never been applied on a large scale. [ABSTRACT FROM AUTHOR]

Gene set analysis (GSA) has become the common methodology for analyzing transcriptomics data. However, self-contained GSA techniques are rarely, if ever, used for proteomics data analysis. Here we present a self-contained proteome level GSA of four consensus molecular subtypes (CMSs) previously established by transcriptome dissection of colon carcinoma specimens. Despite notable difference in structure of proteomics and transcriptomics data, many pathway-wide characteristic features of CMSs found at the mRNA level were reproduced at the protein level. In particular, CMS1 features show heavy involvement of immune system as well as the pathways related to mismatch repair, DNA replication and functioning of proteasome, while CMS4 tumors upregulate complement pathway and proteins participating in epithelial-to-mesenchymal transition (EMT). In addition, protein level GSA yielded a set of novel observations visible at the proteome, but not at the transcriptome level, including possible involvement of major histocompatibility complex II (MHC-II) antigens in the known immunogenicity of CMS1 and a connection between cholesterol trafficking and the regulation of Integrin-linked kinase (ILK) in CMS3. Overall, this study proves utility of self-contained GSA approaches as a critical tool for analyzing proteomics data in general and dissecting protein-level molecular portraits of human tumors in particular. [ABSTRACT FROM AUTHOR]

Gene expression profiling using transcriptional drug perturbations are useful for many biomedical discovery studies including drug repurposing and elucidation of drug mechanisms (MoA) and many other pharmacogenomic applications. However, limited data availability across cell types has severely hindered our capacity to progress in these areas. To fill this gap, recently, the LINCS program generated almost 1.3 million profiles for over 40,000 drug and genetic perturbations for over 70 different human cell types, including meta information about the experimental conditions and cell lines. Unfortunately, Big Data like the ones generated from the ongoing LINCS program do not enable easy insights from the data but possess considerable challenges toward their analysis. In this paper, we address some of these challenges. Specifically, first, we study the gene expression signature profiles from all cell lines and their perturbagents in order to obtain insights in the distributional characteristics of available conditions. Second, we investigate the differential expression of genes for all cell lines obtaining an understanding of condition dependent differential expression manifesting the biological complexity of perturbagents. As a result, our analysis helps the experimental design of follow-up studies, e.g., by selecting appropriate cell lines. [ABSTRACT FROM AUTHOR]

In this paper, we study the discrimination power of graph measures that are based on graph-theoretical matrices. The paper generalizes the work of [M. Dehmer, M. Moosbrugger. Y. Shi, Encoding structural information uniquely with polynomial-based descriptors by employing the Randić matrix, Applied Mathematics and Computation, 268(2015), 164–168]. We demonstrate that by using the new functional matrix approach, exhaustively generated graphs can be discriminated more uniquely than shown in the mentioned previous work. [ABSTRACT FROM AUTHOR]

In this paper, we investigate treatment cycles inferred from diabetes data by means of graph theory. We define the term treatment cycles graph-theoretically and perform a descriptive as well as quantitative analysis thereof. Also, we interpret our findings in terms of nursing and clinical management. [ABSTRACT FROM AUTHOR]

In this paper, we derive interrelations of graph distance measures by means of inequalities. For this investigation we are using graph distance measures based on topological indices that have not been studied in this context. Specifically, we are using the well-known Wiener index, Randić index, eigenvalue-based quantities and graph entropies. In addition to this analysis, we present results from numerical studies exploring various properties of the measures and aspects of their quality. Our results could find application in chemoinformatics and computational biology where the structural investigation of chemical components and gene networks is currently of great interest. [ABSTRACT FROM AUTHOR]

Molecular descriptors have been explored extensively. From these studies, it is known that a large number of descriptors are strongly correlated and capture similar characteristics of molecules. In this paper, we evaluate 919 Dragon-descriptors of 6 different categories by means of clustering. Also, we analyze these different categories of descriptors also find a subset of descriptors which are least correlated among each other and, hence, characterize molecular graphs distinctively. [ABSTRACT FROM AUTHOR]

Identifying diagnostic biomarkers based on genomic features for an accurate disease classification is a problem of great importance for both, basic medical research and clinical practice. In this paper, we introduce quantitative network measures as structural biomarkers and investigate their ability for classifying disease states inferred from gene expression data from prostate cancer. We demonstrate the utility of our approach by using eigenvalue and entropy-based graph invariants and compare the results with a conventional biomarker analysis of the underlying gene expression data. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a biologically inspired model to generate complex networks. In contrast to many other construction procedures for growing networks introduced so far, our method generates networks from one-dimensional symbol sequences that are related to the so called Collatz problem from number theory. The major purpose of the present paper is, first, to derive a symbol sequence from the Collatz problem, we call the step sequence, and investigate its structural properties. Second, we introduce a construction procedure for growing networks that is based on these step sequences. Third, we investigate the structural properties of this new network class including their finite scaling and asymptotic behavior of their complexity, average shortest path lengths and clustering coefficients. Interestingly, in contrast to many other network models including the small-world network from Watts & Strogatz, we find that CS graphs become ‘smaller’ with an increasing size. [ABSTRACT FROM AUTHOR]

Sensitive and specific urinary biomarkers can improve patient outcomes in many diseases through informing early diagnosis. Unfortunately, to date, the accuracy and translation of diagnostic urinary biomarkers into clinical practice has been disappointing. We believe this may be due to inappropriate standardization of diagnostic urinary biomarkers. Our objective was therefore to characterize the effects of standardizing urinary levels of IL-6, IL-8, and VEGF using the commonly applied standards namely urinary creatinine, osmolarity and protein. First, we report results based on the biomarker levels measured in 120 hematuric patients, 80 with pathologically confirmed bladder cancer, 27 with confounding pathologies and 13 in whom no underlying cause for their hematuria was identified, designated "no diagnosis". Protein levels were related to final diagnostic categories (p = 0.022, ANOVA). Osmolarity (mean = 529 mOsm; median = 528 mOsm) was normally distributed, while creatinine (mean = 10163 μmol/l, median = 9350 μmol/l) and protein (0.3297, 0.1155 mg/ml) distributions were not. When we compared AUROCs for IL-6, IL-8 and VEGF levels, we found that protein standardized levels consistently resulted in the lowest AUROCs. The latter suggests that protein standardization attenuates the "true" differences in biomarker levels across controls and bladder cancer samples. Second, in 72 hematuric patients; 48 bladder cancer and 24 controls, in whom urine samples had been collected on recruitment and at follow-up (median = 11 (1 to 20 months)), we demonstrate that protein levels were approximately 24% lower at follow-up (Bland Altman plots). There was an association between differences in individual biomarkers and differences in protein levels over time, particularly in control patients. Collectively, our findings identify caveats intrinsic to the common practice of protein standardization in biomarker discovery studies conducted on urine, particularly in patients with hematuria. [ABSTRACT FROM AUTHOR]

Background: Emphasis is increasingly being placed on the monitoring and comparison of clinical outcomes between healthcare providers. Funnel plots have become a standard graphical methodology to identify outliers and comprise plotting an outcome statistic from each provider against a specified 'target' together with upper and lower control limits. With discrete probability distributions it is not possible to specify the exact probability that an observation from an 'in-control' provider will fall outside the control limits. However, general probability characteristics can be set and specified using interpolation methods. Guidelines recommend that providers falling outside such control limits should be investigated, potentially with significant consequences, so it is important that the properties of the limits are understood. Methods: Control limits for funnel plots for the Standardised Mortality Ratio (SMR) based on the Poisson distribution were calculated using three proposed interpolation methods and the probability calculated of an 'in-control' provider falling outside of the limits. Examples using published data were shown to demonstrate the potential differences in the identification of outliers. Results: The first interpolation method ensured that the probability of an observation of an 'in control' provider falling outside either limit was always less than a specified nominal probability (p). The second method resulted in such an observation falling outside either limit with a probability that could be either greater or less than p, depending on the expected number of events. The third method led to a probability that was always greater than, or equal to, p. Conclusion: The use of different interpolation methods can lead to differences in the identification of outliers. This is particularly important when the expected number of events is small. We recommend that users of these methods be aware of the differences, and specify which interpolation method is to be used prior to any analysis. [ABSTRACT FROM AUTHOR]

GWAS has facilitated greatly the discovery of risk SNPs associated with complex diseases. Traditional methods analyze SNP individually and are limited by low power and reproducibility since correction for multiple comparisons is necessary. Several methods have been proposed based on grouping SNPs into SNP sets using biological knowledge and/or genomic features. In this article, we compare the linear kernel machine based test (LKM) and principal components analysis based approach (PCA) using simulated datasets under the scenarios of 0 to 3 causal SNPs, as well as simple and complex linkage disequilibrium (LD) structures of the simulated regions. Our simulation study demonstrates that both LKM and PCA can control the type I error at the significance level of 0.05. If the causal SNP is in strong LD with the genotyped SNPs, both the PCA with a small number of principal components (PCs) and the LKM with kernel of linear or identical-by-state function are valid tests. However, if the LD structure is complex, such as several LD blocks in the SNP set, or when the causal SNP is not in the LD block in which most of the genotyped SNPs reside, more PCs should be included to capture the information of the causal SNP. Simulation studies also demonstrate the ability of LKM and PCA to combine information from multiple causal SNPs and to provide increased power over individual SNP analysis. We also apply LKM and PCA to analyze two SNP sets extracted from an actual GWAS dataset on non-small cell lung cancer. [ABSTRACT FROM AUTHOR]

In the present work we extend and analyze the scope of our recently proposed stochastic model for transcriptional regulation, which considers an arbitrarily complex cis-regulatory system using only elementary reactions. Previously, we determined the role of cooperativity on the intrinsic fluctuations of gene expression for activating transcriptional switches, by means of master equation formalism and computer simulation. This model allowed us to distinguish between two cooperative binding mechanisms and, even though the mean expression levels were not affected differently by the acting mechanism, we showed that the associated fluctuations were different. In the present generalized model we include other regulatory functions in addition to those associated to an activator switch. Namely, we introduce repressive regulatory functions and two theoretical mechanisms that account for the biphasic response that some cis-regulatory systems show to the transcription factor concentration. We have also extended our previous master equation formalism in order to include protein production by stochastic translation of mRNA. Furthermore, we examine the graded/binary scenarios in the context of the interaction energy between transcription factors. In this sense, this is the first report to show that the cooperative binding of transcription factors to DNA promotes the ''all-or-none'' phenomenon observed in eukaryotic systems. In addition, we confirm that gene expression fluctuation levels associated with one of two cooperative binding mechanism never exceed the fluctuation levels of the other. [ABSTRACT FROM AUTHOR]

In a number of applications there is a need to determine the most likely pedigree for a group of persons based on genetic markers. Adequate models are needed to reach this goal. The markers used to perform the statistical calculations can be linked and there may also be linkage disequilibrium (LD) in the population. The purpose of this paper is to present a graphical Bayesian Network framework to deal with such data. Potential LD is normally ignored and it is important to verify that the resulting calculations are not biased. Even if linkage does not influence results for regular paternity cases, it may have substantial impact on likelihood ratios involving other, more extended pedigrees. Models for LD influence likelihoods for all pedigrees to some degree and an initial estimate of the impact of ignoring LD and/or linkage is desirable, going beyond mere rules of thumb based on marker distance. Furthermore, we show how one can readily include a mutation model in the Bayesian Network; extending other programs or formulas to include such models may require considerable amounts of work and will in many case not be practical. As an example, we consider the two STR markers vWa and D12S391. We estimate probabilities for population haplotypes to account for LD using a method based on data from trios, while an estimate for the degree of linkage is taken from the literature. The results show that accounting for haplotype frequencies is unnecessary in most cases for this specific pair of markers. When doing calculations on regular paternity cases, the markers can be considered statistically independent. In more complex cases of disputed relatedness, for instance cases involving siblings or so-called deficient cases, or when small differences in the LR matter, independence should not be assumed. (The networks are freely available at http://arken.umb.no/~dakl/BayesianNetworks.) [ABSTRACT FROM AUTHOR]

Background: Imperfect diagnostic testing reduces the power to detect significant predictors in classical cross-sectional studies. Assuming that the misclassification in diagnosis is random this can be dealt with by increasing the sample size of a study. However, the effects of imperfect tests in longitudinal data analyses are not as straightforward to anticipate, especially if the outcome of the test influences behaviour. The aim of this paper is to investigate the impact of imperfect test sensitivity on the determination of predictor variables in a longitudinal study. Methodology/Principal Findings: To deal with imperfect test sensitivity affecting the response variable, we transformed the observed response variable into a set of possible temporal patterns of true disease status, whose prior probability was a function of the test sensitivity. We fitted a Bayesian discrete time survival model using an MCMC algorithm that treats the true response patterns as unknown parameters in the model. We applied our approach to epidemiological data of bovine tuberculosis outbreaks in England and investigated the effect of reduced test sensitivity in the determination of risk factors for the disease. We found that reduced test sensitivity led to changes to the collection of risk factors associated with the probability of an outbreak that were chosen in the 'best' model and to an increase in the uncertainty surrounding the parameter estimates for a model with a fixed set of risk factors that were associated with the response variable. Conclusions/Significance: We propose a novel algorithm to fit discrete survival models for longitudinal data where values of the response variable are uncertain. When analysing longitudinal data, uncertainty surrounding the response variable will affect the significance of the predictors and should therefore be accounted for either at the design stage by increasing the sample size or at the post analysis stage by conducting appropriate sensitivity analyses. [ABSTRACT FROM AUTHOR]

One great challenge of genomic research is to efficiently and accurately identify complex gene regulatory networks. The development of high-throughput technologies provides numerous experimental data such as DNA sequences, protein sequence, and RNA expression profiles makes it possible to study interactions and regulations among genes or other substance in an organism. However, it is crucial to make inference of genetic regulatory networks from gene expression profiles and protein interaction data for systems biology. This study will develop a new approach to reconstruct time delay Boolean networks as a tool for exploring biological pathways. In the inference strategy, we will compare all pairs of input genes in those basic relationships by their corresponding p- scores for every output gene. Then, we will combine those consistent relationships to reveal the most probable relationship and reconstruct the genetic network. Specifically, we will prove that O(log n) state transition pairs are sufficient and necessary to reconstruct the time delay Boolean network of n nodes with high accuracy if the number of input genes to each gene is bounded. We also have implemented this method on simulated and empirical yeast gene expression data sets. The test results show that this proposed method is extensible for realistic networks. [ABSTRACT FROM AUTHOR]

Gene expression data can provide a very rich source of information for elucidating the biological function on the pathway level if the experimental design considers the needs of the statistical analysis methods. The purpose of this paper is to provide a comparative analysis of statistical methods for detecting the differentially expression of pathways (DEP). In contrast to many other studies conducted so far, we use three novel simulation types, producing a more realistic correlation structure than previous simulation methods. This includes also the generation of surrogate data from two large-scale microarray experiments from prostate cancer and ALL. As a result from our comprehensive analysis of 41,004 parameter configurations, we find that each method should only be applied if certain conditions of the data from a pathway are met. Further, we provide method-specific estimates for the optimal sample size for microarray experiments aiming to identify DEP in order to avoid an underpowered design. Our study highlights the sensitivity of the studied methods on the parameters of the system. [ABSTRACT FROM AUTHOR]

The characterization and the definition of the complexity of objects is an important but very difficult problem that attracted much interest in many different fields. In this paper we introduce a new measure, called network diversity score (NDS), which allows us to quantify structural properties of networks. We demonstrate numerically that our diversity score is capable of distinguishing ordered, random and complex networks from each other and, hence, allowing us to categorize networks with respect to their structural complexity. We study 16 additional network complexity measures and find that none of these measures has similar good categorization capabilities. In contrast to many other measures suggested so far aiming for a characterization of the structural complexity of networks, our score is different for a variety of reasons. First, our score is multiplicatively composed of four individual scores, each assessing different structural properties of a network. That means our composite score reflects the structural diversity of a network. Second, our score is defined for a population of networks instead of individual networks. We will show that this removes an unwanted ambiguity, inherently present in measures that are based on single networks. In order to apply our measure practically, we provide a statistical estimator for the diversity score, which is based on a finite number of samples. [ABSTRACT FROM AUTHOR]

It has been generally acknowledged that the module structure of protein interaction networks plays a crucial role with respect to the functional understanding of these networks. In this paper, we study evolutionary aspects of the module structure of protein interaction networks, which forms a mesoscopic level of description with respect to the architectural principles of networks. The purpose of this paper is to investigate limitations of well known gene duplication models by showing that these models are lacking crucial structural features present in protein interaction networks on a mesoscopic scale. This observation reveals our incomplete understanding of the structural evolution of protein networks on the module level. [ABSTRACT FROM AUTHOR]

Modern biology and medicine aim at hunting molecular and cellular causes of biological functions and diseases. Gene regulatory networks (GRN) inferred from gene expression data are considered an important aid for this research by providing a map of molecular interactions. Hence, GRNs have the potential enabling and enhancing basic as well as applied research in the life sciences. In this paper, we introduce a new method called BC3NET for inferring causal gene regulatory networks from large-scale gene expression data. BC3NET is an ensemble method that is based on bagging the C3NET algorithm, which means it corresponds to a Bayesian approach with noninformative priors. In this study we demonstrate for a variety of simulated and biological gene expression data from S. cerevisiae that BC3NET is an important enhancement over other inference methods that is capable of capturing biochemical interactions from transcription regulation and proteinprotein interaction sensibly. An implementation of BC3NET is freely available as an R package from the CRAN repository. [ABSTRACT FROM AUTHOR]

The inference of gene regulatory networks from gene expression data is a difficult problem because the performance of the inference algorithms depends on a multitude of different factors. In this paper we study two of these. First, we investigate the influence of discrete mutual information (MI) estimators on the global and local network inference performance of the C3NET algorithm. More precisely, we study 4 different MI estimators (Empirical, Miller-Madow, Shrink and Schürmann- Grassberger) in combination with 3 discretization methods (equal frequency, equal width and global equal width discretization). We observe the best global and local inference performance of C3NET for the Miller-Madow estimator with an equal width discretization. Second, our numerical analysis can be considered as a systems approach because we simulate gene expression data from an underlying gene regulatory network, instead of making a distributional assumption to sample thereof. We demonstrate that despite the popularity of the latter approach, which is the traditional way of studying MI estimators, this is in fact not supported by simulated and biological expression data because of their heterogeneity. Hence, our study provides guidance for an efficient design of a simulation study in the context of network inference, supporting a systems approach. [ABSTRACT FROM AUTHOR]

Recently, the construction of networks from time series data has gained widespread interest. In this paper, we develop this area further by introducing a network construction procedure for pseudoperiodic time series. We call such networks episode networks, in which an episode corresponds to a temporal interval of a time series, and which defines a node in the network. Our model includes a number of features which distinguish it from current methods. First, the proposed construction procedure is a parametric model which allows it to adapt to the characteristics of the data; the length of an episode being the parameter. As a direct consequence, networks of minimal size containing the maximal information about the time series can be obtained. In this paper, we provide an algorithm to determine the optimal value of this parameter. Second, we employ estimates of mutual information values to define the connectivity structure among the nodes in the network to exploit efficiently the nonlinearities in the time series. Finally, we apply our method to data from electroencephalogram (EEG) experiments and demonstrate that the constructed episode networks capture discriminative information from the underlying time series that may be useful for diagnostic purposes. [ABSTRACT FROM AUTHOR]

The article discusses the guidelines to methods for the analysis of different pathways or gene sets. It states that the main objective of pathway-based schemes is to link a molecular level with a phenotype of an organism causally. Moreover, it notes that the preprocessing of the gene expression data gained addresses three issues including background correction, normalization, and summarization.

This paper explores relationships between classical and parametric measures of graph (or network) complexity. Classical measures are based on vertex decompositions induced by equivalence relations. Parametric measures, on the other hand, are constructed by using information functions to assign probabilities to the vertices. The inequalities established in this paper relating classical and parametric measures lay a foundation for systematic classification of entropy-based measures of graph complexity. [ABSTRACT FROM AUTHOR]

Background: In this paper we investigate the definition and formation of financial networks. Specifically, we study the influence of the time scale on their construction. Methodology/Principal Findings: For our analysis we use correlation-based networks obtained from the daily closing prices of stock market data. More precisely, we use the 30 stocks that currently comprise the Dow Jones Industrial Average (DJIA) and estimate financial networks where nodes correspond to stocks and edges correspond to none vanishing correlation coefficients. That means only if a correlation coefficient is statistically significant different from zero, we include an edge in the network. This construction procedure results in unweighted, undirected networks. By separating the time series of stock prices in non-overlapping intervals, we obtain one network per interval. The length of these intervals corresponds to the time scale of the data, whose influence on the construction of the networks will be studied in this paper. Conclusions/Significance: Numerical analysis of four different measures in dependence on the time scale for the construction of networks allows us to gain insights about the intrinsic time scale of the stock market with respect to a meaningful graph-theoretical analysis. [ABSTRACT FROM AUTHOR]

Background: The evaluation of the complexity of an observed object is an old but outstanding problem. In this paper we are tying on this problem introducing a measure called statistic complexity. Methodology/Principal Findings: This complexity measure is different to all other measures in the following senses. First, it is a bivariate measure that compares two objects, corresponding to pattern generating processes, on the basis of the normalized compression distance with each other. Second, it provides the quantification of an error that could have been encountered by comparing samples of finite size from the underlying processes. Hence, the statistic complexity provides a statistical quantification of the statement 'X is similarly complex as Y'. Conclusions: The presented approach, ultimately, transforms the classic problem of assessing the complexity of an object into the realm of statistics. This may open a wider applicability of this complexity measure to diverse application areas. [ABSTRACT FROM AUTHOR]

The fundamental difference between classic and modern biology is that technological innovations allow to generate highthroughput data to get insights into molecular interactions on a genomic scale. These high-throughput data can be used to infer gene networks, e.g., the transcriptional regulatory or signaling network, representing a blue print of the current dynamical state of the cellular system. However, gene networks do not provide direct answers to biological questions, instead, they need to be analyzed to reveal functional information of molecular working mechanisms. In this paper we propose a new approach to analyze the transcriptional regulatory network of yeast to predict cell cycle regulated genes. The novelty of our approach is that, in contrast to all other approaches aiming to predict cell cycle regulated genes, we do not use time series data but base our analysis on the prior information of causal interactions among genes. The major purpose of the present paper is to predict cell cycle regulated genes in S. cerevisiae. Our analysis is based on the transcriptional regulatory network, representing causal interactions between genes, and a list of known periodic genes. No further data are used. Our approach utilizes the causal membership of genes and the hierarchical organization of the transcriptional regulatory network leading to two groups of periodic genes with a well defined direction of information flow. We predict genes as periodic if they appear on unique shortest paths connecting two periodic genes from different hierarchy levels. Our results demonstrate that a classical problem as the prediction of cell cycle regulated genes can be seen in a new light if the concept of a causal membership of a gene is applied consequently. This also shows that there is a wealth of information buried in the transcriptional regulatory network whose unraveling may require more elaborate concepts than it might seem at first. [ABSTRACT FROM AUTHOR]

In this paper we derive entropy bounds for hierarchical networks. More precisely, starting from a recently introduced measure to determine the topological entropy of non-hierarchical networks, we provide bounds for estimating the entropy of hierarchical graphs. Apart from bounds to estimate the entropy of a single hierarchical graph, we see that the derived bounds can also be used for characterizing graph classes. Our contribution is an important extension to previous results about the entropy of non-hierarchical networks because for practical applications hierarchical networks are playing an important role in chemistry and biology. In addition to the derivation of the entropy bounds, we provide a numerical analysis for two special graph classes, rooted trees and generalized trees, and demonstrate hereby not only the computational feasibility of our method but also learn about its characteristics and interpretability with respect to data analysis. [ABSTRACT FROM AUTHOR]