Skip to main content
SpringerLink
Log in

Complex Networks and Graph Theory

  • Reference work entry
Encyclopedia of Complexity and Systems Science

Definition of the Subject

The basic network concept is very simple: objects, connected by relationships. Because of this simplicity, the concept turns up almost everywhereone looks. The study of networks (or equivalently, graphs), both theoretically and empirically, has taken off over the last ten years, and shows no signof slowing down. The field is highly interdisciplinary, having important applications to the Internet and the World Wide Web, to social networks, toepidemiology, to biology, and in many other areas. This introductory article serves as a reader's guide to the 13 articles in the Section of theEncyclopedia which is titled “Complex Networks and Graph Theory”. These articles will be discussed in the context of three broad themes:network structure; dynamics of network structure; and dynamical processes running over networks.

Introduction

In the past ten years or so, the study of graphs has exploded, leaving forever the peaceful sanctum of pure mathematics to become...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 3,499.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

Directed/Undirected graph :

A set of vertices connected by directed or undirected edges. A directed edge is one-way \( { (A\to B) } \), while an undirected edge is two‐way or symmetric: \( { A-B } \).

Network :

For our purposes, a network is defined identically to a graph: it is an abstract object composed of vertices (nodes) joined by (directed or undirected) edges (links). Hence we will use the terms ‘graph’ and ‘network’ interchangeably.

Graph topology:

The list of nodes i and edges \( { (i,j) } \) or \( { (i \to j) } \) defines the topology of the graph.

Graph structure:

There is no single agreed definition for what constitutes the “structure” of a graph. To the contrary: this question has been the object of a great deal of research—research which is still ongoing.

Node degree distribution :

One crude measure of a graph's structure. If n k is the number of nodes having degree k in a graph with N nodes, then the set of n k is the node degree distribution—which is also often expressed in terms of the frequencies \( { p_{k}=n_{k}/N } \).

Small‐worlds graph :

A “small‐worlds graph” has two properties: it has short path lengths (as is typical of random graphs)—so that the “world” of the network is truly “small”, in that every node is within a few (or not too many) hops of every other; and secondly, it has (like real social networks, and unlike random graphs) a significant degree of clustering—meaning that two neighbors of a node have a higher‐than‐random probability of also being linked to one another.

Graph visualization :

The problem of displaying a graph's topology (or part of it) in a 2D image, so as to give the viewer insight into the structure of the graph. We see that this is a hard problem, as it involves both the unsolved problem of what we mean by the structure of the graph, and also the combined technological/psychological problem of conveying useful information about a (possibly large) graph via a 2D (or quasi‐3D) layout. Clearly, the notion of a good graph visualization is dependent on the use to which the visualization is to be put—in other words, on the information which is to be conveyed.

Section:

Here, a ‘bookkeeping’ definition. This article introduces the reader to all of the other articles in the Section of the Encyclopedia which is titled “Complex Networks and Graph Theory”. Therefore, whenever the word ‘Section’ (with a large ‘S’) is used in this ‘roadmap’ article, the word refers to that Section of the Encyclopedia. To avoid confusion, the various subdivisions of this roadmap article will be called ‘parts’.

Bibliography

Primary Literature

  1. Adamic LA (1999) The Small World Web. In: Proc 3rd European Conf Research and Advanced Technology for Digital Libraries, ECDL, London, pp 443–452

    Google Scholar 

  2. Albert R, Jeong H, Barabasi AL (1999) Diameter of World-Wide Web. Nature 410:130–131

    ADS  Google Scholar 

  3. Albert R, Jeong H, Barabasi AL (2000) Error and attack tolerance of complex networks. Nature 406:378–382

    ADS  Google Scholar 

  4. Badii R, Politi A (1997) Complexity: Hierarchical Structures and Scaling in Physics. Cambridge Nonlinear Science Series, vol 6. Cambridge University Press, Cambridge

    Google Scholar 

  5. Barabasi AL, Albert R (1999) Emergence of Scaling in Random Networks. Science 286:509–512

    MathSciNet  ADS  Google Scholar 

  6. Bjelland J, Canright G, Engø-Monsen K, Remple VP (2008) Topographic Spreading Analysis of an Empirical Sex Workers’ Network. In: Ganguly N, Mukherjee A, Deutsch A (eds) Dynamics on and of Complex Networks. Birkhauser, Basel,also in http://delis.upb.de/paper/DELIS-TR-0634.pdf

  7. Brin S, Page L (1998) The anatomy of a large-scale hypertextual Web search engine. In: Proceedings of the seventh international conference on World Wide Web, pp 107–117

    Google Scholar 

  8. Broder A, Kumar R, Maghoul F, Raghavan P, Rajagopalan S, Stata S, Tomkins A, Wiener J (2000) Graph structure in the web. Comput Netw 33:309–320

    Google Scholar 

  9. Dorogovtsev SN, Mendes JFF (2003) Evolution of Networks: From Biological Nets to the Internet and WWW. Oxford University Press, Oxford

    Google Scholar 

  10. Erdős P (1947) Some remarks on the theory of graphs. Bull Amer Math Soc 53:292–294

    Google Scholar 

  11. Erdős P, Rényi A (1959) On random graphs, I. Publ Math Debrecen 6:290–297

    Google Scholar 

  12. Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proc Natl Acad Sci USA 99:8271–8276

    MathSciNet  Google Scholar 

  13. Kaplan D, Glass L (1995) Understanding Nonlinear Dynamics. Springer, New York

    Google Scholar 

  14. Kauffman S (1969) Homeostasis and differentiation in random genetic control networks. Nature 224:177–8

    ADS  Google Scholar 

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

    MathSciNet  Google Scholar 

  16. Kephart JO, White SR (1991) Directed-graph epidemiological models of computer viruses. In: Proceedings of the 1991 IEEE Computer Society Symposium on Research in Security and Privacy, pp 343–359

    Google Scholar 

  17. Kleinberg JM (1999) Authoritative sources in a hyperlinked environment. J ACM 46(5):604–632

    MathSciNet  Google Scholar 

  18. Milgram S (1967) The Small World Problem. Psychol Today 2:60–67

    Google Scholar 

  19. Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U (2002) Network Motifs: Simple Building Blocks of Complex Networks. Science 298:824–827

    ADS  Google Scholar 

  20. Newman MEJ, Forrest S, Balthrop J (2002) Email networks and the spread of computer viruses. Phys Rev E 66:35–101

    Google Scholar 

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

    ADS  Google Scholar 

  22. Page L, Brin S, Motwani R, Winograd T (1998) The PageRank citation ranking: Bringing order to the web. Technical report. Stanford University, Stanford

    Google Scholar 

  23. Pastor-Satorras R, Vespignani A (2001) Epidemic spreading in scale-free networks. Phys Rev Lett 86:3200–3203

    ADS  Google Scholar 

  24. Watts DJ (1999) Small Worlds: The Dynamics of Networks between Order and Randomness. Princeton Studies in Complexity. Princeton University Press, Princeton

    Google Scholar 

  25. Watts D, Strogatz S (1998) Collective dynamics of ‘small world’ networks. Nature 393:440–442

    ADS  Google Scholar 

Books and Reviews

  1. Albert R, Barabasi AL (2002) Statistical Mechanics of Complex Networks. Rev Mod Phys 74:47–97

    MathSciNet  ADS  Google Scholar 

  2. Bornholdt S, Schuster HG (2003) Handbook of Graphs and Networks: From the Genome to the Internet. Wiley-VCH, Berlin

    Google Scholar 

  3. Caldarelli G, Vespignani A (2007) Large Scale Structure and Dynamics of Complex Networks: From Information Technology to Finance and Natural Science. Cambridge University Press, Cambridge

    Google Scholar 

  4. Chung F, Lu L (2006) Complex Graphs and Networks, CBMS Regional Conference. Series in Mathematics, vol 107. AMS, Providence

    Google Scholar 

  5. da F Costa L, Osvaldo N, Oliveira J, Travierso G, Rodrigues FA, Paulino R, Boas V, Antiqueira L, Viana MP, da Rocha LEC Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications. Working Paper: http://arxiv.org/abs/0711.3199

  6. Kauffman SA (1993) The origins of order. Oxford University Press, Oxford

    Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

  8. Newman M, Barabasi A, Watts DJ (2006) The Structure and Dynamics of Networks. Princeton Studies in Complexity. Princeton University Press, Princeton

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Telenor R&I, Fornebu, Norway

    Geoffrey Canright

Authors
  1. Geoffrey Canright
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag

About this entry

Cite this entry

Canright, G. (2009). Complex Networks and Graph Theory. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_83

Download citation

Publish with us

Policies and ethics

Search

Navigation