Andrzej Kamisiński
Piotr Chołda
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Abstract
In this tutorial, 17 structural complexity indices are presented and compared, each representing one of the following categories: adjacency- and distance-based metrics, Shannon entropy-based metrics, product measures, subgraph-based metrics, and path- and walk-based metrics. The applicability of these indices to computer and communication networks is evaluated with the aid of different elementary, specifically designed, random, and real network topologies. On the grounds of the evaluation study, advantages and disadvantages of particular metrics are identified. In addition, their general properties and runtimes are assessed, and a general view on the structural network complexity is presented.
- R. Albert and A.-L. Barabási. 2002. Statistical mechanics of complex networks. Reviews of Modern Physics 74, 1, 47--97. DOI:http://dx.doi.org/10.1103/RevModPhys.74.47Google Scholar
Cross Ref
- D. Babić, D. J. Klein, I. Lukovits, S. Nikolić, and N. Trinajstić. 2002. Resistance-distance matrix: A computational algorithm and its application. International Journal of Quantum Chemistry 90, 1, 166--176. DOI:http://dx.doi.org/10.1002/qua.10057Google ScholarCross Ref
- A.-L. Barabási and R. Albert. 1999. Emergence of scaling in random networks. Science 286, 5439, 509--512. DOI:http://dx.doi.org/10.1126/science.286.5439.509Google Scholar
- A. Bavelas. 1950. Communication patterns in task-oriented groups. Journal of the Acoustical Society of America 22, 6, 725--730. DOI:http://dx.doi.org/10.1121/1.1906679Google ScholarCross Ref
- S. H. Bertz. 1981. The first general index of molecular complexity. Journal of the American Chemical Society 103, 12, 3599--3601. DOI:http://dx.doi.org/10.1021/ja00402a071Google ScholarCross Ref
- B. Bollobás. 1998. Modern Graph Theory. Graduate Texts in Mathematics, Vol. 184. Springer, New York, NY. DOI:http://dx.doi.org/10.1007/978-1-4612-0619-4Google Scholar
- D. Bonchev and G. A. Buck. 2005. Quantitative measures of network complexity. In Complexity in Chemistry, Biology, and Ecology, D. Bonchev and D. H. Rouvray (Eds.). Springer, New York, NY, 191--235. DOI:http://dx.doi.org/10.1007/0-387-25871-X_5Google Scholar
- J. R. Bunch and J. E. Hopcroft. 1974. Triangular factorization and inversion by fast matrix multiplication. Mathematics of Computation 28, 125, 231--236.Google Scholar
- G. Caldarelli. 2013. Scale-Free Networks: Complex Webs in Nature and Technology. Oxford University Press, Oxford, UK.Google Scholar
- J. C. Claussen. 2008. Offdiagonal complexity: A computationally quick network complexity measure--application to protein networks and cell division. In A. Deutsch, R. B. Parra, R. J. de Boer, O. Diekmann, P. Jagers, E. Kisdi, M. Kretzschmar, P. Lansky, and H. Metz (Eds.), Mathematical Modeling of Biological Systems, Volume II, Birkhäuser, Boston, MA, 279--287. DOI:http://dx.doi.org/10.1007/978-0-8176-4556-4_25Google Scholar
- D. M. Cvetković, M. Doob, and H. Sachs. 1998. Spectra of Graphs: Theory and Applications (3rd revised and enlarged ed.). Wiley-VCH, Weinheim, Germany.Google Scholar
- M. Dehmer and F. Emmert-Streib. 2013. Advances in Network Complexity. Wiley-VCH Verlag GmbH & Co. KGaA. DOI:http://dx.doi.org/10.1002/9783527670468Google Scholar
- Encyclopædia Britannica. 2014. Chemical compound. (2014). Retrieved April 21, 2015 from http://www.britannica.com/EBchecked/topic/108614/chemical-compound/278310/Functional-groups.Google Scholar
- P. Erdős and A. Rényi. 1959. On random graphs I. Publlicationes Mathematicae Debrecen 6, 290--297.Google Scholar
- G. H. Golub and C. F. Van Loan. 1996. Matrix Computations (3rd ed.). Johns Hopkins University Press, Baltimore, MD. Google ScholarDigital Library
- J. L. Gross and J. Yellen. 2003. Discrete mathematics and its applications. In Handbook of Graph Theory (1st ed.). CRC Press, Boca Raton, FL.Google Scholar
- I. Gutman, C. Rücker, and G. Rücker. 2001. On walks in molecular graphs. Journal of Chemical Information and Computer Sciences 41, 3, 739--745. DOI:http://dx.doi.org/10.1021/ci000149uGoogle ScholarCross Ref
- A. Kamisiński, P. Chołda, and A. Jajszczyk. 2015. Online Appendix. (Jan. 2015). Retrieved April 12, 2015 from http://kt.agh.edu.pl/∼kamisinski/pub/2015_csur_complexity/.Google Scholar
- V. Karyotis, E. Stai, and S. Papavassiliou. 2013. Evolutionary Dynamics of Complex Communications Networks. CRC Press, Boca Raton, FL. DOI:http://dx.doi.org/10.1201/b15505 Google ScholarDigital Library
- J. Kim and T. Wilhelm. 2008. What is a complex graph? Physica A: Statistical Mechanics and its Applications 387, 11, 2637--2652. DOI:http://dx.doi.org/10.1016/j.physa.2008.01.015Google Scholar
- V. Latora and M. Marchiori. 2003. Economic small-world behavior in weighted networks. The European Physical Journal B—Condensed Matter and Complex Systems 32, 2, 249--263. DOI:http://dx.doi.org/10.1140/epjb/e2003-00095-5Google Scholar
- F. Le Gall. 2014. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC’14). ACM, New York, NY, 296--303. DOI:http://dx.doi.org/10.1145/2608628.2608664 Google ScholarDigital Library
- I. Lukovits, A. Miličević, S. Nikolić, and N. Trinajstić. 2002. On walk counts and complexity of general graphs. Internet Electronic Journal of Molecular Design 1, 8. 388--400. http://www.biochempress.com/av01_0388.html.Google Scholar
- E. A. Moore, B. T. Sutcliffe, A. H. Pakiari, T. E. Simos, S. Wilson, M. Springborg, D. Pugh, R. A. Lewis, N. Trinajstić, and K. Travis. 2004. Chemical Modelling. SPR—Chemical Modelling, Vol. 3. The Royal Society of Chemistry. DOI:http://dx.doi.org/10.1039/9781847553331Google Scholar
- H. L. Morgan. 1965. The generation of a unique machine description for chemical structures—a technique developed at chemical abstracts service. Journal of Chemical Documentation 5, 2, 107--113. DOI:http://dx.doi.org/10.1021/c160017a018Google ScholarCross Ref
- A. Mowshowitz. 1968. Entropy and the complexity of graphs: I. An index of the relative complexity of a graph. Bulletin of Mathematical Biophysics 30, 1, 175--204. DOI:http://dx.doi.org/10.1007/BF02476948Google ScholarCross Ref
- A. Mowshowitz and M. Dehmer. 2010. A symmetry index for graphs. Symmetry: Culture and Science 21, 4, 321--327.Google Scholar
- S. Orlowski, R. Wessäly, M. Pióro, and A. Tomaszewski. 2010. SNDlib 1.0—survivable network design library. Networks 55, 3, 276--286. DOI:http://dx.doi.org/10.1002/net.v55:3 Google ScholarDigital Library
- C. Raychaudhury, S. K. Ray, J. J. Ghosh, A. B. Roy, and S. C. Basak. 1984. Discrimination of isomeric structures using information theoretic topological indices. Journal of Computational Chemistry 5, 6, 581--588. DOI:http://dx.doi.org/10.1002/jcc.540050612Google ScholarCross Ref
- M. Razinger. 1982. Extended connectivity in chemical graphs. Theoretica chimica acta 61, 6, 581--586. DOI:http://dx.doi.org/10.1007/BF00552670Google Scholar
- G. Rücker and C. Rücker. 1993. Counts of all walks as atomic and molecular descriptors. Journal of Chemical Information and Computer Sciences 33, 5, 683--695. DOI:http://dx.doi.org/10.1021/ci00015a005Google Scholar
- R. Seidel. 1995. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences 51, 3, 400--403. DOI:http://dx.doi.org/10.1006/jcss.1995.1078 Google ScholarDigital Library
- D. J. Watts and S. H. Strogatz. 1998. Collective dynamics of “small-world” networks. Nature 393, 6684, 440--442. DOI:http://dx.doi.org/10.1038/30918Google Scholar
- D. B. West. 2000. Introduction to Graph Theory (2nd ed.). Prentice Hall, Upper Saddle River, NJ.Google Scholar
- T. Wilhelm. 2003. An elementary dynamic model for non-binary food webs. Ecological Modelling 168, 1--2, 145--152. DOI:http://dx.doi.org/10.1016/S0304-3800(03)00207-2Google ScholarCross Ref
- T. Wilhelm and J. Hollunder. 2007. Information theoretic description of networks. Physica A: Statistical Mechanics and Its Applications 385, 1, 385--396. DOI:http://dx.doi.org/10.1016/j.physa.2007.06.029Google ScholarCross Ref
Index Terms
- Assessing the Structural Complexity of Computer and Communication Networks
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