Recent Developments in Quantitative Graph Theory: Information Inequalities for Networks

In this article, we tackle a challenging problem in quantitative graph theory. We establish relations between graph entropy measures representing the structural information content of networks. In particular, we prove formal relations between quantitative network measures based on Shannon's entropy to study the relatedness of those measures. In order to establish such information inequalities for graphs, we focus on graph entropy measures based on information functionals. To prove such relations, we use known graph classes whose instances have been proven useful in various scientific areas. Our results extend the foregoing work on information inequalities for graphs

Information processing in the transcriptional regulatory network of yeast: Functional robustness

<p>Abstract</p> <p>Background</p> <p>Gene networks are considered to represent various aspects of molecular biological systems meaningfully because they naturally provide a systems perspective of molecular interactions. In this respect, the functional understanding of the transcriptional regulatory network is considered as key to elucidate the functional organization of an organism.</p> <p>Results</p> <p>In this paper we study the functional robustness of the transcriptional regulatory network of <it>S. cerevisiae</it>. We model the information processing in the network as a first order Markov chain and study the influence of single gene perturbations on the global, asymptotic communication among genes. Modification in the communication is measured by an information theoretic measure allowing to predict genes that are 'fragile' with respect to single gene knockouts. Our results demonstrate that the predicted set of fragile genes contains a statistically significant enrichment of so called essential genes that are experimentally found to be necessary to ensure vital yeast. Further, a structural analysis of the transcriptional regulatory network reveals that there are significant differences between fragile genes, hub genes and genes with a high betweenness centrality value.</p> <p>Conclusion</p> <p>Our study does not only demonstrate that a combination of graph theoretical, information theoretical and statistical methods leads to meaningful biological results but also that such methods allow to study information processing in gene networks instead of just their structural properties.</p

Novel topological descriptors for analyzing biological networks

<p>Abstract</p> <p>Background</p> <p>Topological descriptors, other graph measures, and in a broader sense, graph-theoretical methods, have been proven as powerful tools to perform biological network analysis. However, the majority of the developed descriptors and graph-theoretical methods does not have the ability to take vertex- and edge-labels into account, e.g., atom- and bond-types when considering molecular graphs. Indeed, this feature is important to characterize biological networks more meaningfully instead of only considering pure topological information.</p> <p>Results</p> <p>In this paper, we put the emphasis on analyzing a special type of biological networks, namely bio-chemical structures. First, we derive entropic measures to calculate the information content of vertex- and edge-labeled graphs and investigate some useful properties thereof. Second, we apply the mentioned measures combined with other well-known descriptors to supervised machine learning methods for predicting Ames mutagenicity. Moreover, we investigate the influence of our topological descriptors - measures for only unlabeled vs. measures for labeled graphs - on the prediction performance of the underlying graph classification problem.</p> <p>Conclusions</p> <p>Our study demonstrates that the application of entropic measures to molecules representing graphs is useful to characterize such structures meaningfully. For instance, we have found that if one extends the measures for determining the structural information content of unlabeled graphs to labeled graphs, the uniqueness of the resulting indices is higher. Because measures to structurally characterize labeled graphs are clearly underrepresented so far, the further development of such methods might be valuable and fruitful for solving problems within biological network analysis.</p

Structural Differentiation of Graphs Using Hosoya-Based Indices

In this paper, we introduce the Hosoya-Spectral indices and the Hosoya information content of a graph. The first measure combines structural information captured by partial Hosoya polynomials and graph spectra. The latter is a graph entropy measure which is based on blocks consisting of vertices with the same partial Hosoya polynomial. We evaluate the discrimination power of these quantities by interpreting numerical results

Constrained Covariance Matrices With a Biologically Realistic Structure: Comparison of Methods for Generating High-Dimensional Gaussian Graphical Models

High-dimensional data from molecular biology possess an intricate correlation structure that is imposed by the molecular interactions between genes and their products forming various different types of gene networks. This fact is particularly well-known for gene expression data, because there is a sufficient number of large-scale data sets available that are amenable for a sensible statistical analysis confirming this assertion. The purpose of this paper is two fold. First, we investigate three methods for generating constrained covariance matrices with a biologically realistic structure. Such covariance matrices are playing a pivotal role in designing novel statistical methods for high-dimensional biological data, because they allow to define Gaussian graphical models (GGM) for the simulation of realistic data; including their correlation structure. We study local and global characteristics of these covariance matrices, and derived concentration/partial correlation matrices. Second, we connect these results, obtained from a probabilistic perspective, to statistical results of studies aiming to estimate gene regulatory networks from biological data. This connection allows to shed light on the well-known heterogeneity of statistical estimation methods for inferring gene regulatory networks and provides an explanation for the difficulties inferring molecular interactions between highly connected genes

Entropy Bounds for Hierarchical Molecular Networks

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