Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Growth Models for Networks

  • Sergey N. Dorogovtsev
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_265-3

Definition of the Subject

For several decades, mathematicians, biologists, and physicists have studied growing networks and their models. In physics and physical chemistry, models of this kind were first proposed in the theory of polymers (Flory 1941; Stockmayer 1943/1944). In graph theory and its applications, e.g., to biology and to computer science, very simple growing networks – random recursive trees – were an issue of wide interest for many years, see, for example, early articles (Na and Rapoport 1970; Moon 1974). Growth models for networks have become, maybe, the hottest topic in statistical mechanics, graph theory, and multidisciplinary research after the work (Barabási and Albert 1999), where these models were used to explain universal complex structures of the Internet, the World Wide Web, and other real networks.

Introduction

Numerous natural and artificial networks have essentially more complex architectures than classical random graphs in graph theory. The classical random...

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

Notes

Acknowledgments

The author thanks A. V. Goltsev, J. F. F. Mendes, and A. N. Samukhin for numerous discussions. This work was partially supported by projects POCI: FAT/46241, MAT/46176, FIS/61665, and BIA-BCM/62662, and DYSONET.

Bibliography

Primary Literature

  1. Albert R, Barabási A-L, Jeong H (1999) Mean-field theory for scale-free random networks. Phys Rev A 272:173–187Google Scholar
  2. Andrade JS Jr, Hermann HJ, Andrade RFS, da Silva LR (2005) Apollonian networks. Phys Rev Lett 94:018702CrossRefADSGoogle Scholar
  3. Barabási A-L, Albert R (1999) Emergence of scaling in complex networks. Science 286:509–512MathSciNetCrossRefADSGoogle Scholar
  4. Barabási A-L, Ravasz E, Vicsek T (2001) Deterministic scale-free networks. J Phys A 299:559–564zbMATHGoogle Scholar
  5. Barabási A-L, Jeong H, Neda Z, Ravasz E, Schubert A, Vicsek T (2002) Evolution of the social network of scientific collaborations. J Phys A 311:590–614zbMATHGoogle Scholar
  6. Barrat A, Barthélemy M, Pastor-Satorras R, Vespignani A (2004a) The architecture of complex weighted networks. Proc Natl Acad Sci 101:3747–3752CrossRefADSGoogle Scholar
  7. Barrat A, Barthélemy M, Vespignani A (2004b) Weighted evolving networks: coupling topology and weights dynamics. Phys Rev Lett 92:228701CrossRefADSGoogle Scholar
  8. Bauke H, Sherrington D (2007) Topological phase transition in complex networks. arXiv:0831Google Scholar
  9. Bauke H, Moore C, Rouquier JB, Sherrington D (2011) Topological phase transition in a network model with preferential attachment and node removal. Eur Phys J B 83:519–524Google Scholar
  10. Ben-Naim E, Krapivsky PL (2007) Addition-deletion networks. J Phys A 40:8607–8619MathSciNetCrossRefADSzbMATHGoogle Scholar
  11. Berger N, Bollobás B, Borgs C, Chayes J, Riordan O (2003) Degree distribution of the FKP network model, Lecture notes in computer science. Springer, Berlin, pp 725–738Google Scholar
  12. Bialas P, Burda Z, Jurkiewicz J, Krzywicki A (2003) Tree networks with causal structure. Phys Rev E 67:066106CrossRefADSGoogle Scholar
  13. Bianconi G (2005) Emergence of weight-topology correlations in complex scale-free networks. Europhys Lett 71:1029–1035CrossRefADSGoogle Scholar
  14. Bianconi G, Barabási A-L (2001a) Competition and multiscaling in evolving networks. Europhys Lett 54:436–442CrossRefADSGoogle Scholar
  15. Bianconi G, Barabási A-L (2001b) Bose-Einstein condensation in complex networks. Phys Rev Lett 86:5632–5635CrossRefADSGoogle Scholar
  16. Bianconi G, Capocci A (2003) Number of loops of size h in growing scale-free networks. Phys Rev Lett 90:078701CrossRefADSGoogle Scholar
  17. Bollobás B, Riordan O (2003) Mathematical results on scale-free graphs. In: Bornholdt S, Schuster HG (eds) Handbook of graphs and networks. Wiley, Weinheim, pp 1–34Google Scholar
  18. Bollobás B, Riordan OM (2004a) The diameter of a scale-free random graph. Combinatorica 24:5–34MathSciNetCrossRefzbMATHGoogle Scholar
  19. Bollobás B, Riordan OM (2004b) Shortest paths and load scaling in scale-free trees. Phys Rev E 69:036114CrossRefADSGoogle Scholar
  20. Callaway DS, Hopcroft JE, Kleinberg JM, Newman MEJ, Strogatz SH (2001) Are randomly grown graphs really random? Phys Rev E 64:041902CrossRefADSGoogle Scholar
  21. Cohen R, Havlin S (2003) Scale-free networks are ultra-small. Phys Rev Lett 90:058701CrossRefADSGoogle Scholar
  22. Colizza V, Flammini A, Maritan A, Vespignani A (2005) Characterization and modeling of protein-protein interaction networks. J Phys A 352:1–27Google Scholar
  23. Coulomb S, Bauer M (2003) Asymmetric evolving random networks. Eur Phys J B 35:377–389CrossRefADSGoogle Scholar
  24. D’Souza RM, Borgs C, Chayes JT, Berger N, Kleinberg RD (2007) Emergence of tempered preferential attachment from optimization. Proc Natl Acad Sci 104:6112–6117CrossRefADSGoogle Scholar
  25. Dobrow RP (1996) On the distribution of distances in recursive trees. J Appl Probab 33:749–757MathSciNetCrossRefzbMATHGoogle Scholar
  26. Dorogovtsev S, Mendes JFF (2000) Evolution of networks with aging of sites. Phys Rev E 62:1842–1845CrossRefADSGoogle Scholar
  27. Dorogovtsev SN, Mendes JFF (2001) Effect of the accelerating growth of communications networks on their structure. Phys Rev E 63:025101(R)CrossRefADSGoogle Scholar
  28. Dorogovtsev SN, Mendes JFF (2003) Evolution of networks: from biological nets to the internet and WWW. Oxford University Press, OxfordCrossRefGoogle Scholar
  29. Dorogovtsev S, Mendes JFF (2005) Evolving weighted scale-free networks. AIP Conf Proc 776:29–36MathSciNetCrossRefADSGoogle Scholar
  30. Dorogovtsev SN, Mendes JFF, Samukhin AN (2000) Exact solution of the Barabási-Albert model. Phys Rev Lett 85:4633–4636CrossRefADSGoogle Scholar
  31. Dorogovtsev SN, Mendes JFF, Samukhin AN (2001a) Size-dependent degree distribution of a scale-free growing network. Phys Rev E 63:062101CrossRefADSGoogle Scholar
  32. Dorogovtsev SN, Mendes JFF, Samukhin AN (2001b) Anomalous percolation properties of growing networks. Phys Rev E 64:066110CrossRefADSGoogle Scholar
  33. Dorogovtsev SN, Goltsev AV, Mendes JFF (2002) Pseudofractal scale-free web. Phys Rev E 65:066122CrossRefADSGoogle Scholar
  34. Erdös P, Rényi A (1959) On random graphs. Publ Math Debr 6:290–297zbMATHGoogle Scholar
  35. Ergun G, Rodgers GJ (2002) Growing random networks with fitness. Phys A 303:261–272CrossRefGoogle Scholar
  36. Fabrikant A, Koutsoupias E, Papadimitriou CH (2002) Heuristically optimized trade-offs: a new paradigm for power laws in the internet, Lecture notes in computer science. Springer, Berlin, pp 110–122Google Scholar
  37. Ferrer I, Cancho R, Sole RV (2003) Optimization in complex networks. In: Pastor-Satorras R, Rubi M, Diaz-Guilera A (eds) Statistical mechanics of complex networks. Springer, Berlin, pp 114–125; cond-mat/0111222Google Scholar
  38. Flory PJ (1941) Molecular size distribution in three-dimensional polymers: I, II, III. J Am Chem Soc 63:83–100Google Scholar
  39. Goh KI, Kahng B, Kim D (2001) Universal behavior of load distribution in scale-free networks. Phys Rev Lett 87:278701–278704CrossRefADSGoogle Scholar
  40. Goh KI, Oh E, Jeong H, Kahng B, Kim D (2002) Classification of scale-free networks. Proc Natl Acad Sci 99:12583–12588MathSciNetCrossRefADSzbMATHGoogle Scholar
  41. Jung S, Kim S, Kahng B (2002) Geometric fractal growth model for scale-free networks. Phys Rev E 65:056101CrossRefADSGoogle Scholar
  42. Kim J, Krapivsky PL, Kahng B, Redner S (2002) Infinite-order percolation and giant fluctuations in a protein interaction network. Phys Rev E 66:055101CrossRefADSGoogle Scholar
  43. Kim BJ, Trusina A, Minnhagen P, Sneppen K (2005) Self organized scale-free networks from merging and regeneration. Eur Phys J B 43:369–372CrossRefADSGoogle Scholar
  44. Kleinberg JM, Kumar R, Raghavan P, Rajagopalan S, Tomkins AS (1999) The web as a graph: measurements, models and methods, Lecture notes in computer science. Springer, Berlin, pp 1–17Google Scholar
  45. Klemm K, Eguíluz VM (2002) Growing networks with small-world behavior. Phys Rev E 65:057102CrossRefADSGoogle Scholar
  46. Krapivsky PL, Derrida B (2004) Universal properties of growing networks. Phys A 340:714–724MathSciNetCrossRefGoogle Scholar
  47. Krapivsky PL, Redner S (2001) Organization of growing random networks. Phys Rev E 63:066123CrossRefADSGoogle Scholar
  48. Krapivsky PL, Redner S (2002) Finiteness and fluctuations in growing networks. J Phys A 35:9517–9534MathSciNetCrossRefADSzbMATHGoogle Scholar
  49. Krapivsky PL, Redner S, Leyvraz F (2000) Connectivity of growing random networks. Phys Rev Lett 85:4629–4632CrossRefADSGoogle Scholar
  50. Lancaster D (2002) Cluster growth in two growing network models. J Phys A 35:1179–1194MathSciNetCrossRefADSzbMATHGoogle Scholar
  51. Leskovec J, Kleinberg J, Faloutsos C (2007) Laws of graph evolution: densification and shrinking diameters. ACM TKDD (1)2 physics/0603229Google Scholar
  52. Mahmoud H (1991) Limiting distributions for path lengths in recursive trees. Probab Eng Inf Sci 5:53–59CrossRefzbMATHGoogle Scholar
  53. Moon JW (1974) The distance between nodes in recursive trees. Lond Math Soc Lect Notes Ser 13:125–132Google Scholar
  54. Na HS, Rapoport A (1970) Distribution of nodes of a tree by degree. Math Biosci 6:313–329MathSciNetCrossRefzbMATHGoogle Scholar
  55. Newman MEJ (2002) Assortative mixing in networks. Phys Rev Lett 89:208701CrossRefADSGoogle Scholar
  56. Pastor-Satorras R, Vazquez A, Vespignani A (2001) Dynamical and correlation properties of the Internet. Phys Rev Lett 87:258701–258704CrossRefADSGoogle Scholar
  57. Price de Solla DJ (1976) A general theory of bibliometric and other cumulative advantage processes. J Am Soc Inf Sci 27:292–306CrossRefGoogle Scholar
  58. Ramasco JJ, Dorogovtsev SN, Pastor-Satorras R (2004) Self-organization of collaboration networks. Phys Rev E 70:036106CrossRefADSGoogle Scholar
  59. Simon HA (1955) On a class of skew distribution functions. Biometrica 42:425–440CrossRefzbMATHGoogle Scholar
  60. Stockmayer WH (1943/1944) Theory of molecular size distribution and gel formation in branched chain polymers. J Chem Phys 11:45–55; 12:125–134Google Scholar
  61. Szabó G, Alava M, Kertész J (2002) Shortest paths and load scaling in scale-free trees. Phys Rev E 66:026101CrossRefADSGoogle Scholar
  62. Szymański J (1987) On a nonuniform random recursive trees. Ann Discret Math 33:297–306Google Scholar
  63. Vázquez A, Pastor-Satorras R, Vespignani A (2002) Large-scale topological and dynamical properties of the Internet. Phys Rev E 65:066130CrossRefADSGoogle Scholar
  64. Waclaw B, Sokolov IM (2007) Finite size effects in Barabási-Albert growing networks. Phys Rev E 75:056114MathSciNetCrossRefADSGoogle Scholar
  65. Willinger W, Govindan R, Jamin S, Paxson V, Shenker S (2002) Scaling phenomena in the internet: critically examining criticality. Proc Natl Acad Sci 99:2573–2580CrossRefADSGoogle Scholar
  66. Yule GU (1925) A mathematical theory of evolution based on the conclusions of Dr. JC Willis. Philos Trans R Soc Lond B 213:21–87CrossRefADSGoogle Scholar

Books and Reviews

  1. Albert R, Barabási A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97CrossRefADSzbMATHGoogle Scholar
  2. Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D-U (2006) Complex networks: structure and dynamics. Phys Rep 424:175–308MathSciNetCrossRefADSGoogle Scholar
  3. Caldarelli G (2007) Scale-free networks: complex webs in nature and technology, Oxford finance series. Oxford University Press, OxfordCrossRefGoogle Scholar
  4. Cohen R, Havlin S, Ben-Avraham D (2003) Structural properties of scale free networks. In: Bornholdt S, Schuster HG (eds) Handbook of graphs and networks. Wiley, Weinheim, pp 85–110Google Scholar
  5. Dorogovtsev SN (2010) Lectures on Complex networks. Oxford University Press, OxfordGoogle Scholar
  6. Dorogovtsev SN, Mendes JFF (2002) Evolution of networks. Adv Phys 51:1079–1187CrossRefADSGoogle Scholar
  7. Dorogovtsev SN, Goltsev AV, Mendes JFF (2008) Critical phenomena in complex networks. Rev Mod Phys 80(3):arXiv:00100 80:1275–1335Google Scholar
  8. Durrett R (2006) Random graph dynamics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  9. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167–256MathSciNetCrossRefADSzbMATHGoogle Scholar
  10. Pastor-Satorras R, Vespignani A (2004) Evolution and structure of the internet: a statistical physics approach. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  11. Smythe RT, Mahmoud HM (1995) A survey of recursive trees. Theory Probab Math Stat 51:1–27MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Physics & I3NUniversity of AveiroAveiroPortugal
  2. 2.A. F. Ioffe Physico-Technical InstituteSt. PetersburgRussia