Skip to main content
SpringerLink
Log in

Random Graphs, a Whirlwind Tour of

  • Reference work entry
Computational Complexity

Article Outline

Glossary

Introduction

Some Basic Graph Theory

Random Graphs in a Nutshell

Classical Random Graphs

Random Power Law Graphs

On‐Line Random Graphs

Remarks

Bibliography

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 1,500.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 1,399.99
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

Random graph:

a graph which is chosen from a certain probability distribution over the set of all graphs satisfying some given set of constraints. Statements about random graphs are thus statements about “typical” graphs for the given constraints.

Graph diameter:

the maximum distance between any pair of vertices in a graph. Here the distance between two vertices is the length of any shortest path joining the vertices.

Node degree distribution n k :

the number of vertices having degree k.

Power law distribution:

a node degree distribution which (at least approximately) obeys \( { n_k \propto k^{-\beta} } \) with fixed positive exponent β.

Erdős–Rényi random graph \( { {\mathcal G}(n,p) } \) :

a graph with n vertices, in which each possible edge between the \( { \binom n 2 } \) pairs of vertices is present with probability p. The fact that the edges are chosen independently makes it relatively easy to compute properties of \( { {\mathcal G}(n,p) } \).

Random graph model \( { {\mathcal G}({\mathbf w}) } \) :

for expected degree sequence \( { \mathbf w } \):] a model which generates random graphs which, on average, will have a prescribed degree sequence \( { {\mathbf w} = (w_1, w_2, \dots, w_n) } \).

On‐line random graph:

a graph which grows in size over time, according to given probabilistic rules, starting from a start graph G 0. One can make statements about these graphs in the limit of long time (and hence large vertex number n, as n approaches infinity).

Bibliography

  1. Abello J, Buchsbaum A, Westbrook J (1998) A functional approach to externalgraph algorithms. In: Proc. 6th European Symposium on Algorithms. Springer, Berlin, pp 332–343

    Google Scholar 

  2. Aiello W, Chung F, Lu L (2000) A random graph model for massive graphs. In:Proc. of the Thirty‐Second Annual ACM Symposium on Theoryof Computing. ACM Press, New York, pp 171–180

    Google Scholar 

  3. Albert R, Barabási A-L (2002) Statistical mechanics of complexnetworks. Rev Mod Phys 74:47–97

    Google Scholar 

  4. Barabási A-L, Albert R (1999) Emergence of scaling in randomnetworks. Science 286:509–512

    Google Scholar 

  5. Barabási A-L, Albert R, Jeong H (2000) Scale‐free characteristics ofrandom networks: the topology of the world‐wide web. Physica A 281:69–77

    Google Scholar 

  6. Bender EA, Canfield ER (1978) The asymptotic number of labeled graphs with givendegree sequences. J Comb Theory Ser A 24:296–307

    Article  MathSciNet  MATH  Google Scholar 

  7. Bollobás B (1984) The evolution of random graphs. Trans Amer Math Soc286:257–274

    Google Scholar 

  8. Bollobás B (1981) The diameter of random graphs. Trans Amer Math Soc267:41–52

    Google Scholar 

  9. Bollobás B (1985) Random graphs. Academic Press, New York,pp xvi+447

    Google Scholar 

  10. Bollobás B (1984) The evolution of sparse graphs. In: Graph theory andcombinatorics. Academic Press, London, New York, pp 35–57, (Cambridge 1983)

    Google Scholar 

  11. Broder A, Kumar R, Maghoul F, Raghavan P, Rajagopalan S, Stata R, Tomkins A,Wiener J (2000) Graph structure in the web. Proc. of the WWW9 Conference, Amsterdam, May 2000, Paper version appeared in Computer Networks 33,pp 309–320

    Google Scholar 

  12. Chung F, Lu L (2001) The diameter of sparse random graphs. Adv Appl Math26:257–279

    Article  MathSciNet  MATH  Google Scholar 

  13. Chung F, Lu L (2002) Connected components in random graphs with given expecteddegree sequences. Ann Comb 6:125–145

    Article  MathSciNet  MATH  Google Scholar 

  14. Chung F, Lu L (2002) The average distances in random graphs with givenexpected degrees. Proc Nat Acad Sci 99:15879–15882

    Article  MathSciNet  MATH  Google Scholar 

  15. Chung F, Lu L (2006) The volume of the giant component of a random graphwith given expected degrees. SIAM J Discret Math 20:395–411

    Article  MathSciNet  MATH  Google Scholar 

  16. Chung F, Lu L (2006) Complex graphs and networks. In: CBMS Lecture Series,vol 107. AMS Publications, Providence, pp vii+264

    Google Scholar 

  17. Chung F, Lu L, Dewey G, Galas DJ (2003) Duplication models for biologicalnetworks. J Comput Biol 10(5):677–687

    Article  Google Scholar 

  18. Chung F, Lu L, Vu V (2003) The spectra of random graphs with given expecteddegrees. Proc Nat Acad Sci 100(11):6313–6318

    Article  MathSciNet  MATH  Google Scholar 

  19. Conlon D A new upper bound for diagonal Ramsey numbers. Ann Math (inpress)

    Google Scholar 

  20. Dorogovtsev SN, Mendes JFF (2001) Scaling properties of scale‐freeevolving networks: continuous approach. Phys Rev E 63(056125):19

    Google Scholar 

  21. Erdős P (1947) Some remarks on the theory of graphs. Bull Amer Math Soc53:292–294

    Google Scholar 

  22. Erdős P, Rényi A (1959) On random graphs, vol I. Publ MathDebrecen 6:290–297

    Google Scholar 

  23. Erdős P, Rényi A (1960) On the evolution of randomgraphs. Magyar Tud Akad Mat Kutató Int Közl 5:17–61

    Google Scholar 

  24. Erdős P, Gallai T (1961) Gráfok előírt fokú pontokkal(Graphs with points of prescribed degrees, in Hungarian). Mat Lapok 11:264–274

    Google Scholar 

  25. Exoo G (1989) A lower bound for \( R(5,5) \). J Graph Theory13:97–98

    Google Scholar 

  26. Fabrikant A, Koutsoupias E, Papadimitriou CH (2002) Heuristically optimizedtrade‐offs: a new paradigm for power laws in the internet. In: Automata, languages and programming. Lecture Notes in Computer Science,vol 2380. Springer, Berlin, pp 110–122

    Chapter  Google Scholar 

  27. Faloutsos M, Faloutsos P, Faloutsos C (1999) On power‐law relationshipsof the Internet topology. In: Proc. of the ACM SIGCOM Conference. ACM Press, New York, pp 251–262

    Google Scholar 

  28. Friedman R, Hughes A (2001) Gene duplications and the structure ofeukaryotic genomes. Genome Res 11:373–381

    Article  Google Scholar 

  29. GrossmanJ, Ion P, De Castro R (2007) The Erdős numberproject. http://www.oakland.edu/enp

  30. Gu Z, Cavalcanti A, Chen F-C, Bouman P, Li W-H (2002) Extent of geneduplication in the genomes of drosophila, nematode, and yeast. Mol Biol Evol 19:256–262

    Article  Google Scholar 

  31. Janson S, Knuth DE, Łuczak T, Pittel B (1993) The birth of the giantcomponent. Rand Struct Algorithms 4:231–358

    Google Scholar 

  32. Kleinberg J (2000) The small‐world phenomenon: an algorithmicperspective. Proc. 32nd ACM Symposium on Theory ofComputing. ACM, New York, pp 163–170

    Google Scholar 

  33. Kleinberg J, Kumar R, Raphavan P, Rajagopalan S, Tomkins A (1999) The webas a graph: measurements, models and methods. In: Proc. of the International Conference on Combinatorics and Computing. Lecture Notes in ComputerScience, vol 1627. Springer, Berlin, pp 1–17

    Google Scholar 

  34. Kumar R, Raghavan P, Rajagopalan S, Tomkins A (1999) Trawling the web foremerging cyber communities. In: Proc. of the 8th World Wide Web Conference. Toronto, 1999

    Google Scholar 

  35. Łuczak T (1998) Random trees and random graphs. Rand Struct Algorithms13:485–500

    Google Scholar 

  36. McKay BD, Radziszowski SP (1994) Subgraph counting identities and Ramseynumbers. J Comb Theory (B) 61:125–132

    Article  MathSciNet  MATH  Google Scholar 

  37. Milgram S (1967) The small world problem. Psychol Today2:60–67

    Google Scholar 

  38. Molloy M,Reed B (1995) A critical point for random graphs with a given degree sequence. Rand StructAlgorithms 6:161–179

    Article  MathSciNet  MATH  Google Scholar 

  39. Molloy M, Reed B (1998) The size of the giant component of a random graphwith a given degree sequence. Combin Probab Comput 7:295–305

    Article  MathSciNet  MATH  Google Scholar 

  40. Ramsey FP (1930) On a problem of formal logic. Proc London Math Soc30:264–286

    Article  MathSciNet  Google Scholar 

  41. Simon HA (1955) On a class of skew distribution functions. Biometrika42:425–440

    MathSciNet  MATH  Google Scholar 

  42. Spencer J (1975) Ramsey's theorem—a new lower bound. J CombTheory (A) 18:108–115

    Article  MATH  Google Scholar 

  43. Stubbs L (2002) Genome comparison techniques. In: Galas D, McCormack S (eds)Genomic technologies: present and future. Caister Academic Press

    Google Scholar 

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

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of California, San Diego, USA

    Fan Chung

Authors
  1. Fan Chung
    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

© 2012 Springer-Verlag

About this entry

Cite this entry

Chung, F. (2012). Random Graphs, a Whirlwind Tour of. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_155

Download citation

Publish with us

Policies and ethics

Search

Navigation