Matthias Dehmer

A history of graph entropy measures

This survey seeks to describe methods for measuring the entropy of graphs and to demonstrate the wide applicability of entropy measures. Setting the scene with a review of classical measures for determining the structural information content of graphs, we discuss graph entropy measures which play an important role in a variety of problem areas, including biology, chemistry, and sociology. In addition, we examine relationships between selected entropy measures, illustrating differences quantitatively with concrete examples.

Information processing in complex networks: Graph entropy and information functionals

This paper introduces a general framework for defining the entropy of a graph. Our definition is based on a local information graph and on information functionals derived from the topological structure of a given graph. More precisely, an information functional quantifies structural information of a graph based on a derived probability distribution. Such a probability distribution leads directly to an entropy of a graph. Then, the structural information content of a graph will be is interpreted and defined as the derived graph entropy. Another major contribution of this paper is the investigation of relationships between graph entropies. In addition to this, we provide numerical results demonstrating not only the feasibility of our method, which has polynomial time complexity, but also its usefulness with regard to practical applications aiming to an understanding of information processing in complex networks.

Analysis of Complex Networks

The best way for understanding how things work is by understanding their structures [1]. Complex networks are not an exception [2]. In order to understand why some networks are more robust than others, or why the propagation of a disease in faster in one network than in another is necessary to understand how these networks are organized [3-5]. A complex network is a simplified representation of a complex system in which the entities of the system are represented by the nodes in the network and the interrelations between entities are represented by means of the links joining pairs of nodes [3-5]. In analyzing the architecture of a complex network we are concerned only with the topological organization of these nodes and links. That is to say, we are not taking care of any geometric characteristic of the systems we are representing by these networks but only on how the parts are organized or distributed to form the whole system.

INFORMATION-THEORETIC CONCEPTS FOR THE ANALYSIS OF COMPLEX NETWORKS

In this article, we present information-theoretic concepts for analyzing complex networks. We see that the application of information-theoretic concepts to networks leads to interesting tasks and gives a possibility for understanding information processing in networks. The main contribution of this article is a method for determining the structural information content of graphs that is based on a tree decomposition. It turns out that the computational complexity of the underlying algorithm is polynomial. Finally, we present some numerical results to study the influence of the used methods on the resulting information contents.

Brief Communication: Structural information content of networks: Graph entropy based on local vertex functionals

In this paper we define the structural information content of graphs as their corresponding graph entropy. This definition is based on local vertex functionals obtained by calculating j-spheres via the algorithm of Dijkstra. We prove that the graph entropy and, hence, the local vertex functionals can be computed with polynomial time complexity enabling the application of our measure for large graphs. In this paper we present numerical results for the graph entropy of chemical graphs and discuss resulting properties.

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.

A NOVEL METHOD FOR MEASURING THE STRUCTURAL INFORMATION CONTENT OF NETWORKS

In this paper we first present a novel approach to determine the structural information content (graph entropy) of a network represented by an undirected and connected graph. Such entropic measures can be very important and useful to analyze and compare complex systems by means of networks. The novel graph entropy definition is based on local vertex functionals obtained by calculating j-spheres via the algorithm of Dijkstra. We state some lower and upper bounds of the defined graph entropy to estimate the structural information content for graph classes or explicitly given graphs. Second, we present a detailed example for calculating the graph entropies of a special graph class.

Applied statistics for network biology : methods in systems biology

MODELING, SIMULATION AND MEANING OF GENE NETWORKS Network Analysis to Interpret Complex Phenotypes (Hong Yu, Jialiang Huang, Wei Zhang, and Jing-Dong J. Han) Stochastic Modelling of Regulatory Networks (Tianhai Tian) Modeling eQTL in Multiple Populations (Ching-Lin Hsiao and Cathy S.J. Fann) INFERENCE OF GENE NETWORKS Transcriptional Network Inference based on Information Theory (Patrick E. Meyer and Gianluca Bontempi) Elucidation of General and Condition-dependent Gene Pathways Using Mixture Models and Bayesian Networks (Sandra Rodriguez-Zas and Younhee Ko) Multi-scale Networks Reconstruction from Gene-expression Measurements: Correlations, Perturbations and a-priori Biological Knowledge (Daniel Remondini and Gastone Castellani) Gene Regulatory Network Inference: Combining a Genetic Programming and Hendless Filtering Approach (Lijun Qian, Haixin Wang, and Xiangfang Li) Computational Reconstruction of Protein Interaction Networks (Konrad Monks, Irmgard Muhlberger, Andreas Bernthaler, Raul Fechete, Paul Perco, Rudolf Freund, Arno Lukas, and Bernd Mayer) ANALYSIS OF GENE NETWORKS What if the Fit is Unfit? Criteria for Biological Systems Estimation Beyond Residual Errors (Eberhard O. Voit) Machine Learning Methods for Identifying Essential Genes and Proteins in Networks (Kitiporn Plaimas and Rainer Konig) Gene Co-expression Networks for the Analysis of DNA Microarray Data (Matthew Weirauch) Correlation Network Analysis and Knowledge Integration (Thomas N. Plasterer, Robert Stanley, and Erich Gombocz) Network Screening: A New Method to Identify Active Networks from an Ensemble of Known Networks (Shigeru Saito and Katsuhisa Horimoto) Community Detection in Biological Networks (Gautam S. Thakur) On Some Inverse Problems in Generating Probabilistic Boolean Networks (Xi Chen, Wai-Ki Ching and Nam-Kiu Tsing) Boolean analysis of gene-expression datasets (Debashis Sahoo) SYSTEMS APPROACH TO DISEASES Representing Cancer Cell Trajectories in a Phase-space Diagram: Switching Cellular States by Biological Phase Transitions (Mariano Bizzarri and Alessandro Giuliani) Protein Network Analysis for Disease Gene Identification and Prioritization (Jing Chen and Anil G. Jegga) Pathways and Networks as Functional Descriptors for Human Disease and Drug Response Endpoints (Y. Nikolsky and B. Bessarabova and E. Kirillov and Z. Dezso and W. Nikolskaya)

Analysis of Microarray Data

We give a brief overview over necessary steps in the analysis of microarray data. We cover quality control, preprocessing, statistical as well as enrichment analysis.

Inequalities for entropy-based measures of network information content

This paper presents a method for establishing relations between entropy-based measures applied to graphs. A special class of relations called implicit information inequalities or implicit entropy bounds is developed. A number of entropy-based measures of the structural information content of a graph have been developed over the past several decades, but little attention has been paid to relations among these measures. The research reported here aims to remedy this deficiency.