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E-raamat: Mathematical Foundations and Applications of Graph Entropy

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This latest volume in the successful Network Biology series presents current methods for determining the entropy of networks, covering the analysis of mathematical properties of methods as well as applications in areas ranging from applied mathematics to chemical graph theory.

This latest addition to the successful Network Biology series presents current methods for determining the entropy of networks, making it the first to cover the recently established Quantitative Graph Theory.
An excellent international team of editors and contributors provides an up-to-date outlook for the field, covering a broad range of graph entropy-related concepts and methods. The topics range from analyzing mathematical properties of methods right up to applying them in real-life areas.
Filling a gap in the contemporary literature this is an invaluable reference for a number of disciplines, including mathematicians, computer scientists, computational biologists, and structural chemists.
List of Contributors xi
Preface xv
1 Entropy and Renormalization in Chaotic Visibility Graphs 1(40)
Bartolo Luque
Fernando Javier Ballesteros
Alberto Robledo
Lucas Lacasa
1.1 Mapping Time Series to Networks
2(8)
1.1.1 Natural and Horizontal Visibility Algorithms
4(4)
1.1.2 A Brief Overview of Some Initial Applications
8(2)
1.1.2.1 Seismicity
8(1)
1.1.2.2 Hurricanes
8(1)
1.1.2.3 Turbulence
9(1)
1.1.2.4 Financial Applications
9(1)
1.1.2.5 Physiology
9(1)
1.2 Visibility Graphs and Entropy
10(16)
1.2.1 Definitions of Entropy in Visibility Graphs
10(2)
1.2.2 Pesin Theorem in Visibility Graphs
12(7)
1.2.3 Graph Entropy Optimization and Critical Points
19(7)
1.3 Renormalization Group Transformations of Horizontal Visibility Graphs
26(10)
1.3.1 Tangent Bifurcation
29(2)
1.3.2 Period-Doubling Accumulation Point
31(1)
1.3.3 Quasi-Periodicity
32(2)
1.3.4 Entropy Extrema and RG Transformation
34(29)
1.3.4.1 Intermittency
35(1)
1.3.4.2 Period Doubling
35(1)
1.3.4.3 Quasi-periodicity
35(1)
1.4 Summary
36(1)
1.5 Acknowledgments
37(1)
References
37(4)
2 Generalized Entropies of Complex and Random Networks 41(22)
Vladimir Gudkov
2.1 Introduction
41(1)
2.2 Generalized Entropies
42(1)
2.3 Entropy of Networks: Definition and Properties
43(2)
2.4 Application of Generalized Entropy for Network Analysis
45(8)
2.5 Open Networks
53(6)
2.6 Summary
59(1)
References
60(3)
3 Information Flow and Entropy Production on Bayesian Networks 63(38)
Sosuke Ito
Takahiro Sagawa
3.1 Introduction
63(3)
3.1.1 Background
63(1)
3.1.2 Basic Ideas of Information Thermodynamics
64(1)
3.1.3 Outline of this
Chapter
65(1)
3.2 Brief Review of Information Contents
66(4)
3.2.1 Shannon Entropy
66(1)
3.2.2 Relative Entropy
67(1)
3.2.3 Mutual Information
68(1)
3.2.4 Transfer Entropy
69(1)
3.3 Stochastic Thermodynamics for Markovian Dynamics
70(6)
3.3.1 Setup
70(2)
3.3.2 Energetics
72(1)
3.3.3 Entropy Production and Fluctuation Theorem
73(3)
3.4 Bayesian Networks
76(3)
3.5 Information Thermodynamics on Bayesian Networks
79(7)
3.5.1 Setup
79(1)
3.5.2 Information Contents on Bayesian Networks
80(3)
3.5.3 Entropy Production
83(1)
3.5.4 Generalized Second Law
84(2)
3.6 Examples
86(9)
3.6.1 Example 1: Markov Chain
86(1)
3.6.2 Example 2: Feedback Control with a Single Measurement
86(3)
3.6.3 Example 3: Repeated Feedback Control with Multiple Measurements
89(2)
3.6.4 Example 4: Markovian Information Exchanges
91(3)
3.6.5 Example 5: Complex Dynamics
94(1)
3.7 Summary and Prospects
95(1)
References
96(5)
4 Entropy, Counting, and Fractional Chromatic Number 101(32)
Seyed Saeed Changiz Rezaei
4.1 Entropy of a Random Variable
102(2)
4.2 Relative Entropy and Mutual Information
104(1)
4.3 Entropy and Counting
104(3)
4.4 Graph Entropy
107(1)
4.5 Entropy of a Convex Corner
107(1)
4.6 Entropy of a Graph
108(2)
4.7 Basic Properties of Graph Entropy
110(2)
4.8 Entropy of Some Special Graphs
112(4)
4.9 Graph Entropy and Fractional Chromatic Number
116(3)
4.10 Symmetric Graphs with respect to Graph Entropy
119(1)
4.11 Conclusion
120(1)
Appendix 4.A
121(9)
References
130(3)
5 Graph Entropy: Recent Results and Perspectives 133(50)
Xueliang Li
Meiqin Wei
5.1 Introduction
133(6)
5.2 Inequalities and Extremal Properties on (Generalized) Graph Entropies
139(32)
5.2.1 Inequalities for Classical Graph Entropies and Parametric Measures
139(2)
5.2.2 Graph Entropy Inequalities with Information Functions fv, fP and fc
141(2)
5.2.3 Information Theoretic Measures of UHG Graphs
143(3)
5.2.4 Bounds for the Entropies of Rooted Trees and Generalized Trees
146(2)
5.2.5 Information Inequalities for If(G) based on Different Information Functions
148(5)
5.2.6 Extremal Properties of Degree- and Distance-Based Graph Entropies
153(4)
5.2.7 Extremality of Iflambda(G), If2(G) If3(G) and Entropy Bounds for Dendrimers
157(6)
5.2.8 Sphere-Regular Graphs and the Extremality Entropies If2(G) and Ifsigma (G)
163(3)
5.2.9 Information Inequalities for Generalized Graph Entropies
166(5)
5.3 Relationships between Graph Structures, Graph Energies, Topological Indices, and Generalized Graph Entropies
171(8)
5.4 Summary and Conclusion
179(1)
References
180(3)
6 Statistical Methods in Graphs: Parameter Estimation, Model Selection, and Hypothesis Test 183(20)
Suzana de Siqueira Santos
Daniel Yasumasa Takahashi
Joao Ricardo Sato
Carlos Eduardo Ferreira
Andre Fujita
6.1 Introduction
183(1)
6.2 Random Graphs
184(3)
6.3 Graph Spectrum
187(2)
6.4 Graph Spectral Entropy
189(3)
6.5 Kullback-Leibler Divergence
192(1)
6.6 Jensen-Shannon Divergence
192(1)
6.7 Model Selection and Parameter Estimation
193(2)
6.8 Hypothesis Test between Graph Collections
195(3)
6.9 Final Considerations
198(2)
6.9.1 Model Selection for Protein-Protein Networks
199(1)
6.9.2 Hypothesis Test between the Spectral Densities of Functional Brain Networks
200(1)
6.9.3 Entropy of Brain Networks
200(1)
6.10 Conclusions
200(1)
6.11 Acknowledgments
201(1)
References
201(2)
7 Graph Entropies in Texture Segmentation of Images 203(30)
Martin Welk
7.1 Introduction
203(6)
7.1.1 Structure of the
Chapter
203(1)
7.1.2 Quantitative Graph Theory
204(1)
7.1.3 Graph Models in Image Analysis
205(1)
7.1.4 Texture
206(3)
7.1.4.1 Complementarity of Texture and Shape
206(1)
7.1.4.2 Texture Models
207(1)
7.1.4.3 Texture Segmentation
208(1)
7.2 Graph Entropy-Based Texture Descriptors
209(5)
7.2.1 Graph Construction
210(1)
7.2.2 Entropy-Based Graph Indices
211(3)
7.2.2.1 Shannon's Entropy
212(1)
7.2.2.2 Bonchev and Trinajsties Mean Information on Distances
212(1)
7.2.2.3 Dehmer Entropies
213(1)
7.3 Geodesic Active Contours
214(3)
7.3.1 Basic GAC Evolution for Grayscale Images
214(1)
7.3.2 Force Terms
215(1)
7.3.3 Multichannel Images
216(1)
7.3.4 Remarks on Numerics
216(1)
7.4 Texture Segmentation Experiments
217(4)
7.4.1 First Synthetic Example
217(1)
7.4.2 Second Synthetic Example
218(2)
7.4.3 Real-World Example
220(1)
7.5 Analysis of Graph Entropy-Based Texture Descriptors
221(5)
7.5.1 Rewriting the Information Functionals
221(1)
7.5.2 Infinite Resolution Limits of Graphs
222(1)
7.5.3 Fractal Analysis
223(3)
7.6 Conclusion
226(1)
References
227(6)
8 Information Content Measures and Prediction of Physical Entropy of Organic Compounds 233(26)
Chandan Raychaudhury
Debnath Pal
8.1 Introduction
233(3)
8.2 Method
236(17)
8.2.1 Information Content Measures
236(4)
8.2.2 Information Content of Partition of a Positive Integer
240(3)
8.2.3 Information Content of Graph
243(8)
8.2.3.1 Information Content of Graph on Vertex Degree
245(1)
8.2.3.2 Information Content of Graph on Topological Distances
246(5)
8.2.3.3 Information Content of Vertex-Weighted Graph
251(1)
8.2.4 Information Content on the Shortest Molecular Path
251(2)
8.2.4.1 Computation of Example Indices
252(1)
8.3 Prediction of Physical Entropy
253(3)
8.3.1 Prediction of Entropy using Information Theoretical Indices
254(2)
8.4 Conclusion
256(1)
8.5 Acknowledgment
257(1)
References
257(2)
9 Application of Graph Entropy for Knowledge Discovery and Data Mining in Bibliometric Data 259(16)
Andre Calero Valdez
Matthias Dehmer
Andreas Holzinger
9.1 Introduction
259(2)
9.1.1 Challenges in Bibliometric Data Sets, or Why Should We Consider Entropy Measures?
260(1)
9.1.2 Structure of this
Chapter
261(1)
9.2 State of the Art
261(5)
9.2.1 Graphs and Text Mining
262(1)
9.2.2 Graph Entropy for Data Mining and Knowledge Discovery
263(1)
9.2.3 Graphs from Bibliometric Data
264(2)
9.3 Identifying Collaboration Styles using Graph Entropy from Bibliometric Data
266(1)
9.4 Method and Materials
266(1)
9.5 Results
267(4)
9.6 Discussion and Future Outlook
271(1)
9.6.1 Open Problems
271(1)
9.6.2 A Polite Warning
272(1)
References
272(3)
Index 275
Matthias Dehmer studied mathematics at the University of Siegen (Germany) and received his Ph.D. in computer science from the Technical University of Darmstadt (Germany). Afterwards, he was a research fellow at Vienna Bio Center (Austria), Vienna University of Technology, and University of Coimbra (Portugal). He obtained his habilitation in applied discrete mathematics from the Vienna University of Technology. Currently, he is Professor at UMIT - The Health and Life Sciences University (Austria) and also holds a position at the Universität der Bundeswehr München. His research interests are in applied mathematics, bioinformatics, systems biology, graph theory, complexity and information theory. He has written over 180 publications in his research areas.

Frank Emmert-Streib studied physics at the University of Siegen (Germany) gaining his PhD in theoretical physics from the University of Bremen (Germany). He received postdoctoral training from the Stowers Institute for Medical Research (Kansas City, USA) and the University of Washington (Seattle, USA). Currently, he is associate professor for Computational Biology at Tampere University of Technology (Finland). His main research interests are in the field of computational medicine, network biology and statistical genomics.
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