Entropy and the Complexity of Graphs Revisited
Abstract
:1. Introduction
2. Taxonomy
- The deterministic category encompasses the encoding, substructure count and generative approaches. Dominant in the encoding approach is Kolmogorov complexity. The second includes measures which count the number of substructures of a specified kind [20]. Generative approaches consist of measures based on operations required to generate a graph [21].
- The probabilistic category includes measures that apply an entropy function to a probability distribution associated with a graph. This category is subdivided into intrinsic and extrinsic subcategories. Intrinsic measures use structural features of a graph to partition the graph (usually the set of vertices or edges) and thereby determine a probability distribution over the components of the partition. Extrinsic measures impose an arbitrary probability distribution on graph elements [22]. Both of these categories employ the probability distribution to compute an entropy value. Shannon’s entropy function is most commonly used, but several different families of entropy functions have been considered [23]. In the next section, we provide a brief overview of the main subcategories of the deterministic class of complexity measures. The probabilistic category is our main concern and will be examined in more detail in subsequent sections.
Deterministic Measures | Probabilistic Measures |
---|---|
Encoding | Intrinsic (probability distribution derived from structural features) |
Substructure Count | Extrinsic (probability distribution externally imposed) |
Generative |
3. Deterministic Complexity Measures
4. Probabilistic Measures of Graph Complexity
4.1. Classical Graph Entropies
4.2. Körner Entropy
4.3. Parametric Graph Entropies
4.4. Non-Parametric Graph Entropies
5. Conclusions
Acknowledgements
References and Notes
- Rashevsky, N. Life, information theory, and topology. Bull. Math. Biophys. 1955, 17, 229–235. [Google Scholar] [CrossRef]
- Albert, R.; Jeong, H.; Barabási, A.L. Diameter of the world wide web. Nature 1999, 401, 130–131. [Google Scholar]
- Albert, R.; Barabási, A.L. Statistical mechanics of complex networks. Rev. Mod. Phys. 2002, 74, 47. [Google Scholar] [CrossRef]
- Bonchev, D.; Trinajstić, N. Information theory, distance matrix and molecular branching. J. Chem. Phys. 1977, 67, 4517–4533. [Google Scholar] [CrossRef]
- Bonchev, D. Information Theoretic Indices for Characterization of Chemical Structures; Research Studies Press: Chichester, West Sussex, UK, 1983. [Google Scholar]
- Butts, C.T. The complexity of social networks: Theoretical and empirical findings. Soc. Network. 2001, 23, 31–71. [Google Scholar] [CrossRef]
- Mehler, A.; Weiß, P.; Lücking, A. A network model of interpersonal alignment. Entropy 2010, 12, 1440–1483. [Google Scholar] [CrossRef]
- Solé, R.V.; Montoya, J.M. Complexity and fragility in ecological networks. Proc. Roy. Soc. Lond. B Biol. Sci. 2001, 268, 2039–2045. [Google Scholar] [CrossRef] [PubMed]
- Ulanowicz, R.E. Quantitative methods for ecological network analysis. Comput. Biol. Chem. 2004, 28, 321–339. [Google Scholar] [CrossRef]
- Dehmer, M.; Barbarini, N.; Varmuza, K.; Graber, A. Novel topological descriptors for analyzing biological networks. BMC Struct. Biol. 2010, 10. [Google Scholar] [CrossRef]
- Basak, S.C.; Balaban, A.T.; Grunwald, G.D.; Gute, B.D. Topological indices: Their nature and mutual relatedness. J. Chem. Inf. Comput. Sci. 2000, 40, 891–898. [Google Scholar] [CrossRef] [PubMed]
- Bonchev, D.; Rouvray, D.H. (Eds.) Complexity in Chemistry, Biology, and Ecology; Mathematical and Computational Chemistry; Springer: New York, NY, USA, 2005.
- Emmert-Streib, F.; Dehmer, M. Information theoretic measures of UHG graphs with low computational complexity. Appl. Math. Comput. 2007, 190, 1783–1794. [Google Scholar] [CrossRef]
- Dehmer, M.; Mowshowitz, A. A history of graph entropy measures. Inform. Sci. 2011, 1, 57–78. [Google Scholar] [CrossRef]
- Mowshowitz, A. Entropy and the complexity of the graphs: I. An index of the relative complexity of a graph. Bull. Math. Biophys. 1968, 30, 175–204. [Google Scholar] [CrossRef]
- Anand, K.; Bianconi, G. Entropy measures for networks: Toward an information theory of complex topologies. Phys. Rev. E 2009, 80, 045102(R). [Google Scholar] [CrossRef]
- Solé, R.V.; Valverde, S. Information theory of complex networks: On evolution and architectural constraints. Lect. Notes Phys. 2004, 650, 189–207. [Google Scholar]
- Thurner, S. Statistical mechanics of complex networks. In Analysis of Complex Networks: From Biology to Linguistics; Dehmer, M., Emmert-Streib, F., Eds.; Wiley-VCH: Weinheim, Germany, 2009; pp. 23–45. [Google Scholar]
- Wilhelm, T.; Hollunder, J. Information theoretic description of networks. Physica A 2007, 388, 385–396. [Google Scholar] [CrossRef]
- Constantine, G. Graph complexity and the Laplacian matrix in blocked experiments. Linear and Multilinear Algebra 1990, 28, 49–56. [Google Scholar] [CrossRef]
- Bonchev, D. Kolmogorov’s information, Shannon’s entropy, and topological complexity of molecules. Bulg. Chem. Commun. 1995, 28, 567–582. [Google Scholar]
- Dehmer, M. Information processing in complex networks: Graph entropy and information functionals. Appl. Math. Comput. 2008, 201, 82–94. [Google Scholar] [CrossRef]
- Dehmer, M.; Mowshowitz, A. Generalized graph entropies. Complexity 2011, 17, 45–50. [Google Scholar] [CrossRef]
- Kolmogorov, A.N. Three approaches to the definition of information (in Russian). Probl. Peredaci Inform. 1965, 1, 3–11. [Google Scholar]
- Li, M.; Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications; Springer: New York, NY, USA, 1997. [Google Scholar]
- Bertz, S.H.; Sommmer, T.J. Rigorous mathematical approaches to strategic bonds and synthetic analysis based on conceptually simple new complexity indices. Chem. Commun. 1997. [Google Scholar] [CrossRef]
- Bonchev, D. Complexity analysis of yeast proteome network. Chem. Biodivers. 2004, 1, 312–326. [Google Scholar] [CrossRef]
- Platt, J.R. Influence of neighbor bonds on additive bond properties in paraffins. J. Chem. Phys. 1947, 15, 419–420. [Google Scholar] [CrossRef]
- Bonchev, D. Overall connectivities and topological complexities: A new powerful tool for QSPR/QSAR. J. Chem. Inf. Comput. Sci. 2000, 40, 934–941. [Google Scholar] [CrossRef] [PubMed]
- Bonchev, D. The overall Wiener index—A new tool for characterization of molecular topology. J. Chem. Inf. Comput. Sci. 2001, 41, 582–592. [Google Scholar] [CrossRef] [PubMed]
- Bonchev, D. The overall topological complexity indices. In Advances in Computational Methods in Science and Engineering; Simos, T., Maroulis, G., Eds.; VSP Publications: Melville, NY, USA, 2005; Volume 4B, pp. 1554–1557. [Google Scholar]
- Pudlák, P.; Rödl, V.; Savickiý, P. Graph complexity. Acta Informatica 1988, 25, 515–535. [Google Scholar]
- Shannon, C.E.; Weaver, W. The Mathematical Theory of Communication; University of Illinois Press: Urbana, IL, USA, 1949. [Google Scholar]
- Cover, T.M.; Thomas, J.A. Elements of Information Theory; Wiley Series in Telecommunications and Signal Processing; Wiley & Sons: Hoboken, NJ, USA, 2006. [Google Scholar]
- Butts, C.T. An axiomatic approach to network complexity. J. Math. Sociol. 2000, 24, 273–301. [Google Scholar] [CrossRef]
- Mowshowitz, A. Entropy and the complexity of graphs II: The information content of digraphs and infinite graphs. Bull. Math. Biophys. 1968, 30, 225–240. [Google Scholar] [CrossRef] [PubMed]
- Bonchev, D. Information theoretic measures of complexity. In Encyclopedia of Complexity and System Science; Meyers, R., Ed.; Springer: New York, NY, USA, 2009; Volume 5, pp. 4820–4838. [Google Scholar]
- Todeschini, R.; Consonni, V.; Mannhold, R. Handbook of Molecular Descriptors; Wiley-VCH: Weinheim, Germany, 2002. [Google Scholar]
- Bang-Jensen, J.; Gutin, G. Digraphs, Theory, Algorithms and Applications; Springer: London, UK, 2002. [Google Scholar]
- Körner, J. Coding of an information source having ambiguous alphabet and the entropy of graphs. In Proceedings of the 6th Prague Conference on Information Theory, Prague, Czech Republic, 1973; pp. 411–425.
- Simonyi, G. Graph entropy: A survey. In Combinatorial Optimization; Cook, W., Lovász, L., Seymour, P., Eds.; DIMACS Series in Discrete Mathematics and Theoretical Computer Science; American Mathematical Society: Providence, RI, USA, 1995; Volume 20, pp. 399–441. [Google Scholar]
- Cardinal, L.; Fiorini, S.; Assche, G.V. On minimum entropy graph colorings. In Proceedings of the 2004 IEEE International Symposium on Information Theory, Piscataway, NJ, USA, 2004; p. 43.
- Dehmer, M.; Varmuza, K.; Borgert, S.; Emmert-Streib, F. On entropy-based molecular descriptors: Statistical analysis of real and synthetic chemical structures. J. Chem. Inf. Model. 2009, 49, 1655–1663. [Google Scholar] [CrossRef] [PubMed]
- Dehmer, M.; Mowshowitz, A.; Emmert-Streib, F. Connections between classical and parametric network entropies. PLoS One 2011, 6, e15733. [Google Scholar] [CrossRef]
- Dehmer, M.; Sivakumar, L.; Varmuza, K. Uniquely discriminating molecular structures using novel eigenvalue-based descriptors. MATCH Commun. Math. Comput. Chem. 2012, 67, 147–172. [Google Scholar]
- Polansky, O.E.; Bonchev, D. The Wiener number of graphs. I. General theory and changes due to graph operations. MATCH MATCH Commun. Math. Comput. Chem. 1986, 21, 133–186. [Google Scholar]
- Hosoya, H. On some counting polynomials. Discrete Appl. Math. 1988, 19, 239–257. [Google Scholar] [CrossRef]
- Balaban, A.T.; Harary, F. The characteristic polynomial does not uniquely determined the topology of a molecule. J. Chem. Doc. 1971, 11, 258–259. [Google Scholar] [CrossRef]
- Randić, M.; Vracko, M.; Novic, M. Eigenvalues as molecular descriptors. In QSPR/QSAR Studies by Molecular Descriptors; Diudea, M.V., Ed.; Nova Publishing: Huntington, NJ, USA, 2001; pp. 93–120. [Google Scholar]
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Mowshowitz, A.; Dehmer, M. Entropy and the Complexity of Graphs Revisited. Entropy 2012, 14, 559-570. https://doi.org/10.3390/e14030559
Mowshowitz A, Dehmer M. Entropy and the Complexity of Graphs Revisited. Entropy. 2012; 14(3):559-570. https://doi.org/10.3390/e14030559
Chicago/Turabian StyleMowshowitz, Abbe, and Matthias Dehmer. 2012. "Entropy and the Complexity of Graphs Revisited" Entropy 14, no. 3: 559-570. https://doi.org/10.3390/e14030559