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A Brief Introduction to Complex Networks and Their Analysis

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Structural Analysis of Complex Networks

Abstract

In this chapter we present a brief introduction to complex networks and their analysis. We review important network classes and properties thereof as well as general analysis methods. The focus of this chapter is on the structural analysis of networks, however, information-theoretic methods are also discussed.

MSC2000: Primary 05C90; Secondary 65C60, 46N60, 94A17

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References

  1. Adamic L, Huberman B (2000) Power-law distribution of the world wide web. Science 287:2115

    Article  Google Scholar 

  2. Albert R, Barabasi AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47

    Article  MathSciNet  Google Scholar 

  3. Bakir GH, Hofmann T, Schölkopf B, Smola AJ, Taskar B, Vishwanathan SVN (eds) (2007) Predicting structured data. MIT Press, Cambridge, MA

    Google Scholar 

  4. Barabasi AL, Albert R (1999) Emergence of scaling in random networks. Science 206: 509–512

    MathSciNet  Google Scholar 

  5. Bavelas A (1948) A mathematical model for group structure. Hum Organ 7:16–30

    Google Scholar 

  6. Bavelas A (1950) Communication patterns in task-oriented groups. J Acoust Soc Am 22: 725–730

    Article  Google Scholar 

  7. Bellman R (1957) Dynamic programming. International Series. Princeton University Press, Princeton, NJ

    MATH  Google Scholar 

  8. Bonchev D (1979) Information indices for atoms and molecules. Match 7:65–113

    Google Scholar 

  9. Bonchev D (1983) Information theoretic indices for characterization of chemical structures. Research Studies Press, Chichester

    Google Scholar 

  10. Bonchev D (1995) Kolmogorov’s information, shannon’s entropy, and topological complexity of molecules. Bulg Chem Commun 28:567–582

    Google Scholar 

  11. Bonchev D (2003) Complexity in chemistry. Introduction and fundamentals. Taylor & Francis, London (Philadelphia, PA)

    Google Scholar 

  12. Bonchev D, Rouvray DH (2005) Complexity in chemistry, biology, and ecology. Mathematical and computational chemistry. Springer, Berlin

    Book  MATH  Google Scholar 

  13. Bonchev D, Trinajstić N (1977) Information theory, distance matrix and molecular branching. J Chem Phys 67:4517–4533

    Article  Google Scholar 

  14. Bonchev D, Balaban AT, Mekenyan OG (1980) Generalization of the graph center concept, and derived topological centric indexes. J Chem Inf Comput Sci 20(2):106–113

    Google Scholar 

  15. Bonacich P (1972) Factoring and weighting approaches to status scores and clique identification. J Math Sociol 2:113–120

    Article  Google Scholar 

  16. Bornholdt S, Schuster HG (2003) Handbook of graphs and networks: from the genome to the internet. Wiley, New York, NY

    MATH  Google Scholar 

  17. Bornholdt S, Schuster HG (eds) (2003) Handbook of graphs and networks: from the genome to the internet. Wiley, New York, NY

    MATH  Google Scholar 

  18. Brandes U (2001) A faster algorithm for betweenness centrality. J Math Sociol 25(2):163–177

    Article  MATH  Google Scholar 

  19. Brandes U, Erlebach T (2005) Network analysis. Lecture notes in computer science. Springer, Berlin

    Book  MATH  Google Scholar 

  20. Brandstädt A, Le VB, Sprinrand JP (1999) Graph classes. A survey. SIAM Monographs on Discrete Mathematics and Applications

    MATH  Google Scholar 

  21. Brinkmeier M, Schank T (2005) Network statistics. In Brandes U, Erlebach T (eds) Network analysis. Lecture notes in computer science. Springer, Berlin, pp 293–317

    Chapter  Google Scholar 

  22. Broder A, Kumar R, Maghoul F, Raghavan P, Rajagopalan S, Stata R, Tomkins A, Wiener J (2000) Graph structure in the web: experiments and models. In: Proceedings of the 9th WWW conference, Amsterdam

    Google Scholar 

  23. Buckley F, Harary F (1990) Distance in graphs. Addison-Wesley, Reading, MA

    MATH  Google Scholar 

  24. Bunke H (1983) What is the distance between graphs? Bull EATCS 20:35–39

    Google Scholar 

  25. Bunke H (1997) On a relation between graph edit distance and maximum common subgraph. Pattern Recognit Lett 18(9):689–694

    Article  MathSciNet  Google Scholar 

  26. Bunke H (1998) A graph distance metric based on the maximum common subgraph. Pattern Recognit Lett 19(3):255–259

    Article  MATH  Google Scholar 

  27. Bunke H, Allermann G (1983) A metric on graphs for structural pattern recognition. In: Schussler HW (ed) Proceedings of 2nd European signal processing conference EUSIPCO, pp 257–260

    Google Scholar 

  28. Bunke H, Neuhaus M (2007) Graph matching. Exact and error-tolerant methods and the automatic learning of edit costs. In: Cook D, Holder LB (eds) Mining graph data. Wiley, New York, NY, pp 17–32

    Google Scholar 

  29. Carrière SJ, Kazman R (1997) Webquery: searching and visualizing the web through connectivity. Computer Networks and ISDN Systems 29(8–13):1257–1267

    Article  Google Scholar 

  30. Cayley A (1857) On the theory of analytic forms called trees. Philos Mag 13:19–30

    Google Scholar 

  31. Cayley A (1875) On the analytical forms called trees, with application to the theory of chemical combinatorics. Report of the British Association for the Advancement of Science, pp 257–305

    Google Scholar 

  32. Chowdhury D, Stauffer D (2000) Principles of equilibrium statistical mechanics. Wiley-VCH, Weinheim

    Book  MATH  Google Scholar 

  33. Clauset A, Newman MEJ, Moore C (2004) Finding community structure in very large networks. Phys Rev E 70:066111

    Article  Google Scholar 

  34. Claussen JC (2007) Characterization of networks by the offdiagonal complexity. Physica A 365–373:321–354

    Google Scholar 

  35. Claussen JC (2007) Offdiagonal complexity: a computationally quick network complexity measure – application to protein networks and cell division. In: Deutsch A, Bravo de la Parra R et al (eds) Mathematical modeling of biological systems, vol II. Birkhäuser, Boston, MA, pp 303–311

    Google Scholar 

  36. Cook D, Holder LB (2007) Mining graph data. Wiley, New York, NY

    MATH  Google Scholar 

  37. Cormen T, Leiserson CE, Rivest RL, Leiserson C, Rivest R (2001) Introduction to algorithms. MIT Press, Cambridge, MA

    MATH  Google Scholar 

  38. Cristianini N, Shawe-Taylor J (2000) An introduction to support vector machines. Cambridge University Press, Cambridge

    Google Scholar 

  39. Dehmer M (2006) Strukturelle Analyse web-basierter Dokumente. Multimedia und Telekooperation. Deutscher Universitäts Verlag, Wiesbaden

    Google Scholar 

  40. Dehmer M (2008) A novel method for measuring the structural information content of networks. Cybern Syst 39:825–842

    Article  Google Scholar 

  41. Dehmer M, Emmert-Streib F (2008) Structural information content of chemical networks. Zeitschrift für Naturforschung, Part A 63a:155–159

    Google Scholar 

  42. Deo N, Gupta P (2001) World wide web: a graph-theoretic perspective. Technical report, Department of Computer Science, University of Central Florida

    Google Scholar 

  43. Dickinson PJ, Bunke H, Dadej A, Kraetzl M (2004) Matching graphs with unique node labels. Pattern Anal Appl 7:243–266

    MathSciNet  Google Scholar 

  44. Diestel R (2000) Graph theory. Springer, Berlin

    Google Scholar 

  45. Duch J, Arenas A (2005) Community detection in complex networks using extremal optimization. Phys Rev E, 72:027104

    Article  Google Scholar 

  46. Emmert-Streib F (2007) The chronic fatigue syndrome: a comparative pathway analysis. J Comput Biol 14(7):961–972

    Article  MathSciNet  Google Scholar 

  47. Emmert-Streib F, Chen L, Storey J (2007) Functional annotation of genes in Saccharomyces cerevisiae based on joint betweenness. arXiv:0709.3291

    Google Scholar 

  48. Emmert-Streib F, Dehmer M (2007) Global information processing in gene networks: fault tolerance. In: Proceedings of the bio-inspired models of network, information, and computing systems, Bionetics 2007, art. no. 4610138, pp 326–329

    Google Scholar 

  49. Emmert-Streib F, Dehmer M, Kilian J (2005) Classification of large graphs by a local tree decomposition. In: Arabnia HR, Scime A (eds) Proceedings of DMIN’05, international conference on data mining, Las Vegas, June 20–23, pp 200–207

    Google Scholar 

  50. Emmert-Streib F, Dehmer M (2007) Topolocial mappings between graphs, trees and generalized trees. Appl Math Comput 186(2):1326–1333

    Article  MATH  MathSciNet  Google Scholar 

  51. Emmert-Streib F, Dehmer M (eds) (2008) Analysis of microarray data: a network based approach. Wiley-VCH, Weinheim

    Google Scholar 

  52. Emmert-Streib F, Dehmer M (2005) Robustness in scale-free networks: comparing directed and undirected networks. Int J Mod Phys C 19(5):717–726

    Article  Google Scholar 

  53. Emmert-Streib F, Mushegian A (2007) A topological algorithm for identification of structural domains of proteins. BMC Bioinformatics 8:237

    Article  Google Scholar 

  54. Erdös P, Rényi A (1959) On random graphs. Publicationes Mathematicae 6:290–297

    MATH  Google Scholar 

  55. Erdös P, Rényi A (1960) On the evolution of random graphs. Publications of Mathematical Institute of the Hungarian Academy of Sciences 5:17–61

    MATH  Google Scholar 

  56. Euler L (1736) Solutio problematis ad geometriam situs pertinentis. Comentarii Academiae Scientiarum Imperialis Petropolitanae 8:128–140

    Google Scholar 

  57. Even S (1979) Algorithms. Computer Science Press, Potomac, MD

    MATH  Google Scholar 

  58. Freeman LC (1977) A set of measures of centrality based on betweenness. Sociometry 40: 35–41

    Article  Google Scholar 

  59. Freeman LC (1979) Centrality in social networks: conceptual clarification. Soc Networks 1:215–239

    Article  Google Scholar 

  60. Fujii JI, Yuki S (1997) Entropy and coding for graphs. Int J Math Stat Sci 6(1):63–77

    MATH  MathSciNet  Google Scholar 

  61. Gagneur J, Krause R, Bouwmeester T, Casari G (2004) Modular decomposition of protein–protein interaction networks. Genome Biol 5:R57

    Article  Google Scholar 

  62. Gärtner T, Flach PA, Wrobel S (2003) On graph kernels: hardness results and efficient alternatives. In: COLT, pp 129–143

    Google Scholar 

  63. Gernert D (1979) Measuring the similarity of complex structures by means of graph grammars. Bull EATCS 7:3–9

    MathSciNet  Google Scholar 

  64. Gernert D (1981) Graph grammars which generate graphs with specified properties. Bull EATCS 13:13–20

    Google Scholar 

  65. Gleiser PM, Danon L (2003) Community structure in jazz. Advances in complex systems 6(4):565–574

    Article  Google Scholar 

  66. Hage P, Harary F (1995) Eccentricity and centrality in networks. Soc Networks 17:57–63

    Article  Google Scholar 

  67. Halin R (1989) Graphentheorie. Akademie Verlag, Berlin

    MATH  Google Scholar 

  68. Harary F (1959) Status and contrastatus. Sociometry 22:23–43

    Article  MathSciNet  Google Scholar 

  69. Harary F (1965) Structural models. An introduction to the theory of directed graphs. Wiley, NY

    MATH  Google Scholar 

  70. Harary F (1967) Graph theory and theoretical physics. Academic, New York, NY

    MATH  Google Scholar 

  71. Harary F (1969) Graph theory. Addison-Wesley, Reading, MA

    Google Scholar 

  72. Hastie T, Tibshirani R, Friedman JH (2001) The elements of statistical learning. Springer, Berlin

    MATH  Google Scholar 

  73. Horváth T, Gärtner T, Wrobel S (2004) Cyclic pattern kernels for predictive graph mining. In: Proceedings of the 2004 ACM SIGKDD international conference on knowledge discovery and data mining, pp 158–167

    Google Scholar 

  74. Hsu H-P, Mehra V, Grassberger P (2003) Structure optimization in an off-lattice protein model. Phys Rev E 68(3):037703

    Article  Google Scholar 

  75. Kaden F (1982) Graphmetriken und Distanzgraphen. ZKI-Informationen, Akademie der Wissenschaften DDR 2(82):1–63

    Google Scholar 

  76. Kaden F (1983) Halbgeordnete Graphmengen und Graphmetriken. In: Proceedings of the conference graphs, hypergraphs, and applications DDR, pp 92–95

    Google Scholar 

  77. Kaden F (1986) Graphmetriken und Isometrieprobleme zugehöriger Distanzgraphen. ZKI-Informationen, Akademie der Wissenschaften DDR, pp 1–100

    MATH  Google Scholar 

  78. Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22:437–467

    Article  MathSciNet  Google Scholar 

  79. Kieffer J, Yang E (1997) Ergodic behavior of graph entropy. Electronic Research Announcements of the American Mathematical Society 3:11–16

    Article  MATH  MathSciNet  Google Scholar 

  80. Kondor RI, Lafferty J (2002) Diffusion kernels on graphs and other discrete input spaces. In: Machine learning: Proceedings of the 19th international conference, Morgan Kaufmann, San Mateo, CA

    Google Scholar 

  81. König D (1936) Theorie der endlichen und unendlichen Graphen. Chelsea, New York, NY

    Google Scholar 

  82. Körner J (1973) Coding of an information source having ambiguous alphabet and the entropy of graphs. Transactions of the 6th Prague conference on information theory, pp 411–425

    Google Scholar 

  83. Koschützki D, Lehmann KA, Peters L, Richter S, Tenfelde-Podehl D, Zlotkowski O (2005) Clustering. In: Brandes U, Erlebach T (eds) Centrality indices. Lecture notes in computer science. Springer, Berlin, pp 16–61

    Google Scholar 

  84. Kullback S (1959) Information theory and statistics. Wiley, New York, NY

    MATH  Google Scholar 

  85. Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22(1):79–86

    Article  MATH  MathSciNet  Google Scholar 

  86. Laubenbacher RC (2007) Modeling and simulation of biological networks. In: Proceedings of symposia in applied mathematics. American Mathematical Society, Providence, RI

    MATH  Google Scholar 

  87. Li M, Vitányi P (1997) An introduction to Kolmogorov complexity and its applications. Springer, Berlin

    MATH  Google Scholar 

  88. Mason O, Verwoerd M (2007) Graph theory and networks in biology. IET Syst Biol 1(2): 89–119

    Article  Google Scholar 

  89. Mehler A (2006) In search of a bridge between network analysis in computational linguistics and computational biology – a conceptual note. In: Proceedings of the 2006 international conference on bioinformatics & computational biology (BIOCOMP’06), 2006, Las Vegas, Nevada, USA, pp 496–500

    Google Scholar 

  90. Mehler A, Dehmer M, Gleim R (2005) Towards logical hypertext structure. a graph-theoretic perspective. In: Proceedings of I2CS’04. Lecture notes. Springer, Berlin, pp 136–150

    Google Scholar 

  91. Messmer BT, Bunke H (1998) A new algorithm for error-tolerant subgraph isomorphism detection. IEEE Trans Pattern Anal Mach Intell 20(5):493–504

    Article  Google Scholar 

  92. Mowshowitz A (1968) Entropy and the complexity of the graphs I: an index of the relative complexity of a graph. Bull Math Biophys 30:175–204

    Article  MATH  MathSciNet  Google Scholar 

  93. Mowshowitz A (1968) Entropy and the complexity of graphs II: the information content of digraphs and infinite graphs. Bull Math Biophys 30:225–240

    Article  MATH  MathSciNet  Google Scholar 

  94. Mowshowitz A (1968) Entropy and the complexity of graphs III: graphs with prescribed information content. Bull Math Biophys 30:387–414

    Article  MATH  MathSciNet  Google Scholar 

  95. Mowshowitz A (1968) Entropy and the complexity of graphs IV: entropy measures and graphical structure. Bull Math Biophys 30:533–546

    Article  MATH  MathSciNet  Google Scholar 

  96. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45: 167–256

    Article  MATH  MathSciNet  Google Scholar 

  97. Newman MEJ, Girvan M (2004) Finding and evaluating community structures in networks. Phys Rev E 69:026113

    Article  Google Scholar 

  98. Newman MEJ (2006) Modularity and community structure in networks. Proc Natl Acad Sci USA 103:8577–8582

    Article  Google Scholar 

  99. Pearl J (1998) Probabilistic reasoning in intelligent systems. Morgan Kaufmann, Los Altos, CA

    Google Scholar 

  100. Rashewsky N (1955) Life, information theory, and topology. Bull Math Biophys 17:229–235

    Article  MathSciNet  Google Scholar 

  101. Roberts F (1989) Applications of combinatorics and graph theory to the biological and social sciences series. IMA volumes in mathematics and its applications. Springer, Berlin

    Google Scholar 

  102. Rosvall M, Bergstrom CT (2007) An information-theoretic framework for resolving community structure in complex networks. In: Proc Natl Acad Sci USA 104(18):7327–31

    Article  Google Scholar 

  103. Sabidussi G (1966) The centrality index of a graph. Psychometrika 31:581–603

    Article  MATH  MathSciNet  Google Scholar 

  104. Scott F (2001) Social network analysis. Sage, Beverly Hills, CA

    Google Scholar 

  105. Shannon CE, Weaver W (1997) The mathematical theory of communication. University of Illinois Press, Champaign, IL

    Google Scholar 

  106. Simonyi G (2001) Perfect graphs and graph entropy. An updated survey. In: Ramirez-Alfonsin J, Reed B (eds) Perfect graphs. Wiley, New York, NY, pp 293–328

    Google Scholar 

  107. Skorobogatov VA, Dobrynin AA (1988) Metrical analysis of graphs. MATCH 23:105–155

    MATH  MathSciNet  Google Scholar 

  108. Sobik F (1982) Graphmetriken und Klassifikation strukturierter Objekte. ZKI-Informationen, Akademie der Wissenschaften DDR 2(82):63–122

    Google Scholar 

  109. Sobik F (1986) Modellierung von Vergleichsprozessen auf der Grundlage von Ähnlichkeitsmaßen für Graphen. ZKI-Informationen, Akademie der Wissenschaften DDR 4:104–144

    Google Scholar 

  110. Solé RV, Valverde S (2004) Information theory of complex networks: on evolution and architectural constraints. In: Lecture notes in physics, vol 650, pp 189–207

    Google Scholar 

  111. Temkin O, Zeigarnik AV, Bonchev D (1996) Chemical reaction networks. A graph-theoretical approach. CRC Press, West Palm Beach, FL

    Google Scholar 

  112. Trucco E (1956) A note on the information content of graphs. Bull Math Biol 18(2):129–135

    MathSciNet  Google Scholar 

  113. Ullmann JR (1976) An algorithm for subgraph isomorphism. J ACM 23(1):31–42

    Article  MathSciNet  Google Scholar 

  114. Wasserman S, Faust K (1994) Social network analysis: methods and applications. Structural analysis in the social sciences. Cambridge University Press, Cambridge

    Google Scholar 

  115. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393: 440–442

    Article  Google Scholar 

  116. Zachary W (1977) An information flow model for conflict and fission in small groups. J Anthropol Res 33:452–473

    Google Scholar 

  117. Zelinka B (1975) On a certain distance between isomorphism classes of graphs. Časopis pro p̆est. Mathematiky 100:371–373

    MATH  MathSciNet  Google Scholar 

  118. Zhang K, Statman R, Shasha D (1992) On the editing distance between unordered labeled trees. Inform Process Lett 42(3):133–139

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I would like to thank Matthias Dehmer for fruitful discussions.

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Emmert-Streib, F. (2011). A Brief Introduction to Complex Networks and Their Analysis. In: Dehmer, M. (eds) Structural Analysis of Complex Networks. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4789-6_1

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