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
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Adamic L, Huberman B (2000) Power-law distribution of the world wide web. Science 287:2115
Albert R, Barabasi AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47
Bakir GH, Hofmann T, Schölkopf B, Smola AJ, Taskar B, Vishwanathan SVN (eds) (2007) Predicting structured data. MIT Press, Cambridge, MA
Barabasi AL, Albert R (1999) Emergence of scaling in random networks. Science 206: 509–512
Bavelas A (1948) A mathematical model for group structure. Hum Organ 7:16–30
Bavelas A (1950) Communication patterns in task-oriented groups. J Acoust Soc Am 22: 725–730
Bellman R (1957) Dynamic programming. International Series. Princeton University Press, Princeton, NJ
Bonchev D (1979) Information indices for atoms and molecules. Match 7:65–113
Bonchev D (1983) Information theoretic indices for characterization of chemical structures. Research Studies Press, Chichester
Bonchev D (1995) Kolmogorov’s information, shannon’s entropy, and topological complexity of molecules. Bulg Chem Commun 28:567–582
Bonchev D (2003) Complexity in chemistry. Introduction and fundamentals. Taylor & Francis, London (Philadelphia, PA)
Bonchev D, Rouvray DH (2005) Complexity in chemistry, biology, and ecology. Mathematical and computational chemistry. Springer, Berlin
Bonchev D, Trinajstić N (1977) Information theory, distance matrix and molecular branching. J Chem Phys 67:4517–4533
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
Bonacich P (1972) Factoring and weighting approaches to status scores and clique identification. J Math Sociol 2:113–120
Bornholdt S, Schuster HG (2003) Handbook of graphs and networks: from the genome to the internet. Wiley, New York, NY
Bornholdt S, Schuster HG (eds) (2003) Handbook of graphs and networks: from the genome to the internet. Wiley, New York, NY
Brandes U (2001) A faster algorithm for betweenness centrality. J Math Sociol 25(2):163–177
Brandes U, Erlebach T (2005) Network analysis. Lecture notes in computer science. Springer, Berlin
Brandstädt A, Le VB, Sprinrand JP (1999) Graph classes. A survey. SIAM Monographs on Discrete Mathematics and Applications
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
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
Buckley F, Harary F (1990) Distance in graphs. Addison-Wesley, Reading, MA
Bunke H (1983) What is the distance between graphs? Bull EATCS 20:35–39
Bunke H (1997) On a relation between graph edit distance and maximum common subgraph. Pattern Recognit Lett 18(9):689–694
Bunke H (1998) A graph distance metric based on the maximum common subgraph. Pattern Recognit Lett 19(3):255–259
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
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
Carrière SJ, Kazman R (1997) Webquery: searching and visualizing the web through connectivity. Computer Networks and ISDN Systems 29(8–13):1257–1267
Cayley A (1857) On the theory of analytic forms called trees. Philos Mag 13:19–30
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
Chowdhury D, Stauffer D (2000) Principles of equilibrium statistical mechanics. Wiley-VCH, Weinheim
Clauset A, Newman MEJ, Moore C (2004) Finding community structure in very large networks. Phys Rev E 70:066111
Claussen JC (2007) Characterization of networks by the offdiagonal complexity. Physica A 365–373:321–354
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
Cook D, Holder LB (2007) Mining graph data. Wiley, New York, NY
Cormen T, Leiserson CE, Rivest RL, Leiserson C, Rivest R (2001) Introduction to algorithms. MIT Press, Cambridge, MA
Cristianini N, Shawe-Taylor J (2000) An introduction to support vector machines. Cambridge University Press, Cambridge
Dehmer M (2006) Strukturelle Analyse web-basierter Dokumente. Multimedia und Telekooperation. Deutscher Universitäts Verlag, Wiesbaden
Dehmer M (2008) A novel method for measuring the structural information content of networks. Cybern Syst 39:825–842
Dehmer M, Emmert-Streib F (2008) Structural information content of chemical networks. Zeitschrift für Naturforschung, Part A 63a:155–159
Deo N, Gupta P (2001) World wide web: a graph-theoretic perspective. Technical report, Department of Computer Science, University of Central Florida
Dickinson PJ, Bunke H, Dadej A, Kraetzl M (2004) Matching graphs with unique node labels. Pattern Anal Appl 7:243–266
Diestel R (2000) Graph theory. Springer, Berlin
Duch J, Arenas A (2005) Community detection in complex networks using extremal optimization. Phys Rev E, 72:027104
Emmert-Streib F (2007) The chronic fatigue syndrome: a comparative pathway analysis. J Comput Biol 14(7):961–972
Emmert-Streib F, Chen L, Storey J (2007) Functional annotation of genes in Saccharomyces cerevisiae based on joint betweenness. arXiv:0709.3291
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
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
Emmert-Streib F, Dehmer M (2007) Topolocial mappings between graphs, trees and generalized trees. Appl Math Comput 186(2):1326–1333
Emmert-Streib F, Dehmer M (eds) (2008) Analysis of microarray data: a network based approach. Wiley-VCH, Weinheim
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
Emmert-Streib F, Mushegian A (2007) A topological algorithm for identification of structural domains of proteins. BMC Bioinformatics 8:237
Erdös P, Rényi A (1959) On random graphs. Publicationes Mathematicae 6:290–297
Erdös P, Rényi A (1960) On the evolution of random graphs. Publications of Mathematical Institute of the Hungarian Academy of Sciences 5:17–61
Euler L (1736) Solutio problematis ad geometriam situs pertinentis. Comentarii Academiae Scientiarum Imperialis Petropolitanae 8:128–140
Even S (1979) Algorithms. Computer Science Press, Potomac, MD
Freeman LC (1977) A set of measures of centrality based on betweenness. Sociometry 40: 35–41
Freeman LC (1979) Centrality in social networks: conceptual clarification. Soc Networks 1:215–239
Fujii JI, Yuki S (1997) Entropy and coding for graphs. Int J Math Stat Sci 6(1):63–77
Gagneur J, Krause R, Bouwmeester T, Casari G (2004) Modular decomposition of protein–protein interaction networks. Genome Biol 5:R57
Gärtner T, Flach PA, Wrobel S (2003) On graph kernels: hardness results and efficient alternatives. In: COLT, pp 129–143
Gernert D (1979) Measuring the similarity of complex structures by means of graph grammars. Bull EATCS 7:3–9
Gernert D (1981) Graph grammars which generate graphs with specified properties. Bull EATCS 13:13–20
Gleiser PM, Danon L (2003) Community structure in jazz. Advances in complex systems 6(4):565–574
Hage P, Harary F (1995) Eccentricity and centrality in networks. Soc Networks 17:57–63
Halin R (1989) Graphentheorie. Akademie Verlag, Berlin
Harary F (1959) Status and contrastatus. Sociometry 22:23–43
Harary F (1965) Structural models. An introduction to the theory of directed graphs. Wiley, NY
Harary F (1967) Graph theory and theoretical physics. Academic, New York, NY
Harary F (1969) Graph theory. Addison-Wesley, Reading, MA
Hastie T, Tibshirani R, Friedman JH (2001) The elements of statistical learning. Springer, Berlin
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
Hsu H-P, Mehra V, Grassberger P (2003) Structure optimization in an off-lattice protein model. Phys Rev E 68(3):037703
Kaden F (1982) Graphmetriken und Distanzgraphen. ZKI-Informationen, Akademie der Wissenschaften DDR 2(82):1–63
Kaden F (1983) Halbgeordnete Graphmengen und Graphmetriken. In: Proceedings of the conference graphs, hypergraphs, and applications DDR, pp 92–95
Kaden F (1986) Graphmetriken und Isometrieprobleme zugehöriger Distanzgraphen. ZKI-Informationen, Akademie der Wissenschaften DDR, pp 1–100
Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22:437–467
Kieffer J, Yang E (1997) Ergodic behavior of graph entropy. Electronic Research Announcements of the American Mathematical Society 3:11–16
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
König D (1936) Theorie der endlichen und unendlichen Graphen. Chelsea, New York, NY
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
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
Kullback S (1959) Information theory and statistics. Wiley, New York, NY
Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22(1):79–86
Laubenbacher RC (2007) Modeling and simulation of biological networks. In: Proceedings of symposia in applied mathematics. American Mathematical Society, Providence, RI
Li M, Vitányi P (1997) An introduction to Kolmogorov complexity and its applications. Springer, Berlin
Mason O, Verwoerd M (2007) Graph theory and networks in biology. IET Syst Biol 1(2): 89–119
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
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
Messmer BT, Bunke H (1998) A new algorithm for error-tolerant subgraph isomorphism detection. IEEE Trans Pattern Anal Mach Intell 20(5):493–504
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
Mowshowitz A (1968) Entropy and the complexity of graphs II: the information content of digraphs and infinite graphs. Bull Math Biophys 30:225–240
Mowshowitz A (1968) Entropy and the complexity of graphs III: graphs with prescribed information content. Bull Math Biophys 30:387–414
Mowshowitz A (1968) Entropy and the complexity of graphs IV: entropy measures and graphical structure. Bull Math Biophys 30:533–546
Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45: 167–256
Newman MEJ, Girvan M (2004) Finding and evaluating community structures in networks. Phys Rev E 69:026113
Newman MEJ (2006) Modularity and community structure in networks. Proc Natl Acad Sci USA 103:8577–8582
Pearl J (1998) Probabilistic reasoning in intelligent systems. Morgan Kaufmann, Los Altos, CA
Rashewsky N (1955) Life, information theory, and topology. Bull Math Biophys 17:229–235
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
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
Sabidussi G (1966) The centrality index of a graph. Psychometrika 31:581–603
Scott F (2001) Social network analysis. Sage, Beverly Hills, CA
Shannon CE, Weaver W (1997) The mathematical theory of communication. University of Illinois Press, Champaign, IL
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
Skorobogatov VA, Dobrynin AA (1988) Metrical analysis of graphs. MATCH 23:105–155
Sobik F (1982) Graphmetriken und Klassifikation strukturierter Objekte. ZKI-Informationen, Akademie der Wissenschaften DDR 2(82):63–122
Sobik F (1986) Modellierung von Vergleichsprozessen auf der Grundlage von Ähnlichkeitsmaßen für Graphen. ZKI-Informationen, Akademie der Wissenschaften DDR 4:104–144
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
Temkin O, Zeigarnik AV, Bonchev D (1996) Chemical reaction networks. A graph-theoretical approach. CRC Press, West Palm Beach, FL
Trucco E (1956) A note on the information content of graphs. Bull Math Biol 18(2):129–135
Ullmann JR (1976) An algorithm for subgraph isomorphism. J ACM 23(1):31–42
Wasserman S, Faust K (1994) Social network analysis: methods and applications. Structural analysis in the social sciences. Cambridge University Press, Cambridge
Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393: 440–442
Zachary W (1977) An information flow model for conflict and fission in small groups. J Anthropol Res 33:452–473
Zelinka B (1975) On a certain distance between isomorphism classes of graphs. Časopis pro p̆est. Mathematiky 100:371–373
Zhang K, Statman R, Shasha D (1992) On the editing distance between unordered labeled trees. Inform Process Lett 42(3):133–139
Acknowledgments
I would like to thank Matthias Dehmer for fruitful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4789-6_1
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4788-9
Online ISBN: 978-0-8176-4789-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)