Connections between Classical and Parametric Network Entropies

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.

[1]  E. Trucco.  A note on the information content of graphs , 1956 .

[2]  Danail Bonchev,et al.  Topological order in molecules 1. Molecular branching revisited , 1995 .

[3]  A. Mowshowitz,et al.  Entropy and the complexity of graphs. I. An index of the relative complexity of a graph. , 1968, The Bulletin of mathematical biophysics.

[4]  Danail Bonchev,et al.  Information theoretic indices for characterization of chemical structures , 1983 .

[5]  Stephan Borgert,et al.  On Entropy-Based Molecular Descriptors: Statistical Analysis of Real and Synthetic Chemical Structures , 2009, J. Chem. Inf. Model..

[6]  O Mason,et al.  Graph theory and networks in Biology. , 2006, IET systems biology.

[7]  N. Trinajstic,et al.  Information theory, distance matrix, and molecular branching , 1977 .

[8]  D. Bonchev,et al.  Complexity in chemistry, biology, and ecology , 2005 .

[9]  Zhen Zhang,et al.  On Characterization of Entropy Function via Information Inequalities , 1998, IEEE Trans. Inf. Theory.

[10]  Mauro Fiorentino,et al.  Scale-free networks as entropy competition. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Abbe Mowshowitz,et al.  Entropy and the complexity of graphs: IV. Entropy measures and graphical structure , 1968 .

[12]  Alexandru T. Balaban,et al.  Chemical Graphs: Looking Back and Glimpsing Ahead , 1995, J. Chem. Inf. Comput. Sci..

[13]  Ilya Shmulevich,et al.  Entropy of complex relevant components of Boolean networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  L. da F. Costa,et al.  Characterization of complex networks: A survey of measurements , 2005, cond-mat/0505185.

[15]  Sonja Nikolic,et al.  Complexity of Molecules , 2003, J. Chem. Inf. Comput. Sci..

[16]  Matthias Dehmer,et al.  A NOVEL METHOD FOR MEASURING THE STRUCTURAL INFORMATION CONTENT OF NETWORKS , 2008, Cybern. Syst..

[17]  Matthias Dehmer,et al.  Information theoretic measures of UHG graphs with low computational complexity , 2007, Appl. Math. Comput..

[18]  Steven H. Bertz,et al.  The first general index of molecular complexity , 1981 .

[19]  D. Jayasri,et al.  Phase transition in liquid crystal elastomers—A Monte Carlo study employing non-Boltzmann sampling , 2006 .

[20]  Matthias Dehmer,et al.  Information processing in the transcriptional regulatory network of yeast: Functional robustness , 2009, BMC Systems Biology.

[21]  Li Xiao,et al.  An efficient distributed algorithm for generating multicast distribution trees , 2005, 2005 International Conference on Parallel Processing Workshops (ICPPW'05).

[22]  Frank Tutzauer,et al.  Entropy as a measure of centrality in networks characterized by path-transfer flow , 2007, Soc. Networks.

[23]  Ginestra Bianconi,et al.  Entropy measures for networks: toward an information theory of complex topologies. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Matthias Dehmer,et al.  Inequalities for entropy-based measures of network information content , 2010, Appl. Math. Comput..

[25]  Matthias Dehmer,et al.  Analysis of Microarray Data , 2008 .

[26]  Petra Weiß,et al.  A Network Model of Interpersonal Alignment in Dialog , 2010, Entropy.

[27]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[28]  Danail Bonchev,et al.  Complexity in chemistry : introduction and fundamentals , 2003 .

[29]  Danail Bonchev,et al.  Chemical Reaction Networks: A Graph-Theoretical Approach , 1996 .

[30]  Danail Bonchev,et al.  Phylogenetic distances are encoded in networks of interacting pathways , 2008, Bioinform..

[31]  Andrew Streitwieser,et al.  Barriers to ring rotation in 1,1',4,4'-tetra-tert-butyluranocene and 1,1',3,3'-tetra-tert-butylferrocene , 1981 .

[32]  Ernesto Estrada,et al.  Chemical Graph Theory , 2013 .

[33]  F Emmert-Streib,et al.  Networks for systems biology: conceptual connection of data and function. , 2011, IET systems biology.

[34]  Alexander Mehler.  A Quantitative Graph Model of Social Ontologies by Example of Wikipedia , 2011 .

[35]  G. Whitesides,et al.  Complexity in chemistry. , 1999, Science.

[36]  R. Solé,et al.  Information Theory of Complex Networks: On Evolution and Architectural Constraints , 2004 .

[37]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[38]  J. Hollunder,et al.  Information theoretic description of networks , 2007 .

[39]  Russell Lyons,et al.  Identities and Inequalities for Tree Entropy , 2007, Combinatorics, Probability and Computing.

[40]  George Oster,et al.  Chemical reaction networks , 1974, Applications of Polynomial Systems.

[41]  N. Rashevsky.  Life, information theory, and topology , 1955 .

[42]  Robert E. Ulanowicz,et al.  Information Theory in Ecology , 2001, Comput. Chem..

[43]  M. Dehmer,et al.  Analysis of Microarray Data: A Network-Based Approach , 2008 .

[44]  Matthias Dehmer,et al.  A history of graph entropy measures , 2011, Inf. Sci..

[45]  Thomas Wilhelm,et al.  What is a complex graph , 2008 .

[46]  Gábor Simonyi,et al.  Graph entropy: A survey , 1993, Combinatorial Optimization.

[47]  Calvin H. Bartholomew,et al.  Introduction and Fundamentals , 2010 .