Random graph dynamics

Random Graph Dynamics

Volumn , Issue , 2006, Pages 1-212

  Durrett, Rick ^a  

^a Department of Obstetrics Gynecology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84929729015     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1017/CBO9780511546594     Document Type: Book

References (186)

1. Aiello, W., Chung, F., and Lu, L. (2000) A random graph model for massive graphs. In Proceedings of Symposium on the Theory of Computing, pp. 171–180.
2. Aiello, W., Chung, F., and Lu, L. (2001) A random graph model for power law graphs. Exp. Math. 10, 53–66.
3. Aizenman, M., Chayes, J., Chayes, L., and Newman, C.M. (1988) Discontinuity of the magnetization in one-dimensional 1/|x − y|2 Ising and Potts models. J. Stat. Phys. 50, 1–40.
4. Albert, R., and Barabási, A.L. (2002) Statistical mechanics of complex systems. Rev. Mod. Phys. 74, 47–97.
5. Albert, R., Jeong, H., and Barabási, A.L. (1999) Diameter of the world-wide web. Nature 401, 130–131.
6. Albert, R., Jeong, H., and Barabási, A.L. (2000) Error and attack tolerance of complex networks. Nature 406, 378–382.
7. Aldous, D. (1982) Some inequalities for reversible Markov chains. J. Lond. Math. Soc. 25, 564–572.
8. Aldous, D. (1989) The Poisson Clumping Heuristic. Springer, New York.
9. Aldous, D. (1997) Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25, 812–854.
10. Aldous, D. (2001) Stochastic models and descriptive statistics for phylogenetic trees, from Yule to today. Stat. Sci. 16, 23–34.
11. Aldous, D., and Fill, J. (2003) Chapter 3. In Reversible Markov Chains. Part of unpublished book. http://www.stat.berkeley.edu/aldous/RWG/book.html.
12. Aldous, D., and Limic, V. (1998) The entrance boundary of the multiplicative coalescent. Electron. J. Probab. 3 (3), 59.
13. Aldous, D., Lovász, L., and Winkler, P. (1997) Mixing times for uniformly ergodic Markov chains. Stoch. Proc. Appl. 125, 408–420.
14. Almaas, E., Kulkarni, R.V., and Stroud, D. (2003) Scaling properties of random walks on small world networks. Phys. Rev. E 68, 056105.
15. Alon, N., and Spencer, J. (2002) The Probabilistic Method. John Wiley, New York.
16. Arratia, R., Goldstein, L., and Gordon, L. (1990) Poisson approximation and the Chen–Stein method. Stat. Sci. 5, 403–434.
17. Assmussen, S., and Kaplan, N. (1976) Branching random walks. I, II. Stoch. Proc. Appl. 4, 1–31.
18. Athreya, K.B., and Ney, P.E. (1972) Branching Process. Springer, New York.
19. Azuma, K. (1967) Weighted sums of certain dependent random variables. Toohoku Math J. 3, 357–367.
20. Balthrop, J., Forrest, S., Newman, M.E.J., and Wiliiamson, M.M. (2004) Technological networks and the spread of computer viruses. Science 304, 527–529.
21. Barabási, A.L. (2002) Linked. The New Science of Networks. Perseus, Cambridge.
22. Barabási, A.L., and Albert, R. (1999) Emergence of scaling in random network. Science 286, 509–512.
23. Barabási, A.L., Albert, R., and Jeong, H. (2000) Scale-free characteristics of random networks: The topology of the world-wide web. Physica A 281, 69–77.
24. Barabási, A.L., Jeong, H., Néda, Z., Ravasz, E., Schubert, A., and Vicsek, T. (2002) Evolution of the social network of scientific collaborations. Physica A 311, 590–614.
25. Barbour, A.D., and Reinert, G. (2001) Small worlds. Random Struct. Alg. 19, 54–74.
26. Barrat, A., and Weigt, M. (2000) On the properties of small-world networks. Eur. Phys. J. B 13, 547–560.
27. Barrez, D., Boucheron, S., and Fernandez de la Vega, W. (2000) On the fluctuations of the gaint component. Comb. Probab. Comput. 9, 287–304.
28. Berger, N., Borgs, C., Chayes, J.T., and Saberi, A. (2005) On the spread of viruses on the internet. In Proceedings of the 16th Symposium on Discrete Algorithms.
29. Biggins, J.D. (1976) The first and last-birth problem for a multitype age-dependent branching process. Adv. Appl. Prob. 8, 446–459.
30. Biggins, J.D. (1979) Growth rates in the branching random walk. Z. fur. Wahrscheinlichkeitstheorie verw Gebiete 48, 17–34.
31. Bleher, P.M., Ruiz, J., and Zagrebnov, V.A. (1995) On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Stat. Phys. 79, 473–482.
32. Bollobás, B. (1988) The isoperimetric number for random regular graphs. Eur. J. Comb. 9, 241–244.
33. Bollobás, B. (2001) Random Graphs, 2nd ed. Cambridge University Press, Cambridge.
34. Bollobás, B. (2003) Mathematical results on scale-free random graphs. In Handbook of Graphs and Networks. Edited by S. Bornholdt and H.G. Schuster. Wiley-VCH, Weinheim, pp. 1–34.
35. Bollobás, B., Borgs, C., Chayes, J., and Riordan, O. (2003) Directed scale free graphs. In Proceedings of the 14th Symposium on Discrete Algorithms, pp. 132–139.
36. Bollobás, B., and Chung, F.R.K. (1988) The diameter of a cycle plus a random matching. SIAM J. Discrete Math. 1, 328–333.
37. Bollobás, B., Janson, S., and Riordan, O. (2005) The phase transition in the uniformly grown random graph has infinite order. Random Struct. Alg. 26, 1–36.
38. Bollobás, B., Janson, S., and Riordan, O. (2006) The phase transition in inhomogeneous random graphs. Random Struct. Alg., to appear.
39. Bollobás, B., and Riordan, O. (2004a) Shortest paths and load scaling in scale-free trees. Phys. Rev. E 69, 036114.
40. Bollobás, B., and Riordan, O. (2004b) The diameter of a scale-free random graph. Combinatorica 24, 5–34.
41. Bollobás, B., and Riordan, O. (2004c) Robustness and vulnerability of scale-free random graphs. Internet Math. 1, 1–35.
42. Bollobás, B., and Riordan, O. (2004d) Coupling scale-free and classical random graphs. Internet Math. 1, 215–225.
43. Bollobás, B., Riordan, O., Spencer, J., and Tusndáy, G. (2001) The degree sequence of a scale-free random graph process. Random Struct. Alg. 18, 279–290.
44. Borel, E. (1942) Sur l’empoi du théorème de Bernoulli pour faciliter le calcul d’une infinité de coefficients. Application au problème de l’attente á un guichet. C.R. Acad. Sci. Paris 214, 452–456.
45. Brown, M. (1983) Approximating IMRL distributions with applications to first passage times. Ann. Probab. 11, 419–427.
46. Callaway, D.S., Hopcroft, J.H., Kleinberg, J.M., Newman, M.E.J., and Strogatz, S.H. (2001) Are randomly grown graphs really random? Phys. Rev. E 64, paper 041902.
47. Callaway, D.S., Newman, M.E.J., Strogatz, S.H., and Watts, D.J. (2000) Network robustness and fragility: Percolation on random graphs. Phys. Rev. Lett. 85, 5468–5471.
48. Carlson, J.M., Chayes, J.T., Chayes, L., Sethan, J.P., and Thouless, D.J. (1989) Critical behavior of the Bethe lattice spin-glass. Europhys. Lett. 5, 355–360.
49. Castelano, C., Vilone, D., and Vespignani, A. (2003) Incomplete ordering of voter model on a small world network. Europhys. Lett. 63, 153–158.
50. Chayes, J.T., Chayes, L., Sethan, J.P., and Thouless, D.J. (1986) A mena field spin glass with short-range interactions. Commun. Math. Phys. 106, 41–89.
51. Chung, F., and Lu, L. (2001) The diameter of sparse random graphs. Adv. Appl. Math. 26, 257–279.
52. Chung, F., and Lu, L. (2002a) Connected components in random graphs with a given degree expected sequence. Ann. Comb. 6, 125–145.
53. Chung, F., and Lu, L. (2002b) The average distance in a random graph with given expected degrees. Proc. Natl. Acad. Sci. 99, 15879–15882.
54. Chung, F., and Lu, L. (2003) The average distance in a random graph with given expected degrees. Internet Math. 1, 91–114.
55. Clifford, P., and Sudbury, A. (1973) A model for spatial conflict. Biometrika 60, 581–588.
56. Cohen, R., Erez, K., ben-Avraham, D., and Havlin, S. (2000) Resilience of the Internet to random breakdown. Phys. Rev. Lett. 85, 4626–4628.
57. Cohen, R., Erez, K., ben-Avraham, D., and Havlin, S. (2001) Breakdown of the Internet under intentional attack. Phys. Rev. Lett. 86, 3682–3685.
58. Cohen, R., and Havlin, S. (2003) Scale-free networks are ultra-small. Phys. Rev. Lett. 90, paper 058701.
59. Cooper, C., and Frieze, A. (2003a) A general model for web graphs. Random Struct. Alg. 22, 311–335.
60. Cooper, C., and Frieze, A. (2003b) The cover time of sparse random graphs. In Proceedings of 14th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 140–147.
61. Cox, J.T. (1989) Coalescing random walks and voter model consensus times on the torus Zd. Ann. Probab. 17, 1333–1366.
62. Cox, J.T., and Griffeath, D.S. (1986) Diffusive clustering in the two dimensional voter model. Ann. Probab. 14, 347–370.
63. Davies, P.L. (1978) The simple branching process: A note on convergence when the mean is infinite. J. Appl. Prob. 15, 466–480.
64. Dezsö, Z., and Barabási, A.L. (2002) Halting viruses in scale-free networks. Phys. Rev. E 65, 055103(R).
65. Diaconis, P., and Stroock, D. (1991) Geomteric bounds for the eigenvalues of Markov chains. Ann. Appl. Prob. 1, 36–61.
66. Dorogovstev, S.N., Goltsev, A.V., and Mendes, J.F.F. (2002) Ising model on networks with an arbitrary distribution of connections. Phys. Rev. E 66, paper 016104.
67. Dorogovstev, S.N., and Mendes, J.F.F. (2001) Comment on “Breakdown of the internet under intentional attack.” Phys. Rev. Lett. 87, 219801.
68. Dorogovstev, S.N., and Mendes, J.F.F. (2002) Evolution of networks. Adv. Phys. 51, 1079–1187.
69. Dorogovstev, S.N., Mendes, J.F.F., and Samukhin, A.N. (2000) Structure of growing networks with preferential linking. Phys. Rev. Lett. 85, 4633–4636.
70. Dorogovstev, S.N., Mendes, J.F.F., and Samukhin, A.N. (2001) Anamolous percolation of growing networks. Phys. Rev. E 64, Paper 066110.
71. Doyle, J., Alderson, D.L., Li, L., Low, S., Roughan, M., Shaulnov, S., Tanaka, R., and Willinger, W. (2005) The “robust yet fragile” nature of the internet. Proc. Natl. Acad. Sci. 102, 14497–14502.
72. Durrett, R. (1996) Stochastic Calculus: A Practical Introduction. CRC Press, Boca Raton, FL.
73. Durrett, R. (2003) Rigorous results for the CHKNS random graph model. In Proceedings of the Discrete Random Walks 2003. http://dmtcs.loria.fr/proceedings/dmACind.html.
74. Durrett, R. (2004) Probability: Theory and Examples, 2nd ed. Brooks-Cole, Belmont, CA.
75. Durrett, R., and Jung, P. (2005) Two phase transitions for the contact on small worlds. arXiv math. PR/0501481.
76. Durrett, R., and Kesten, H. (1990) The critical parameter for connectedness of some random graphs. In A Tribute to P. Erdös. Edited by A. Baker, B. Bollobas, and A. Hajnal. Cambridge University Press, Cambridge, pp. 161–176.
77. Durrett, R., and Liu, X. (1988) The contact process on a finite set. Ann. Probab. 16, 1158–1173.
78. Ebel, H., Mielsch, L.I., and Bornholdt, S. (2002) Scale-free topology of email networks. Phys. Rev. E 66, paper 035103(R).
79. Evans, W., Kenyon, C., Peres, Y., and Schulman, L.J. (2000) Broadcasting on trees and the Ising model. Ann. Appl. Prob. 10, 410–433.
80. Faloutous, M., Faloutous, P., and Faloutous, C. (1999) On the power-law relationship of the internet topology. In Proceedings of SIGCOMM ’99, pp. 251–262.
81. Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley and Sons, New York.
82. Flajolet, P., and Odlyzko, A. (1990) Singularity analysis of generating functions. SIAM J. Discrete Math. 3, 216–240.
83. Fortuin, C.M., and Kasteleyn, P.W. (1972) On the random-cluster model I. Introduction and relation to other models. Physica 57, 536–564.
84. Fountoulakis, N., and Reed, B. (2006) Faster mixing and small bottlenecks. Probab. Theory Relat. Fields. Published online April 27, 2006.
85. Gitterman, M. (2000) Small-world phenomena in physics: The Ising model. J. Phys. A 33, 8373–8381.
86. Gkantsis, C., Mihail, M., and Saberi, A. (2003) Conductance and congestion in power law graphs. In Proceedings of the 2003 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, pp. 148–159.
87. Goh, K.I., Kahng, B., and Kim, D. (2001) Universal behavior of load distribution in scale-free networks. Phys. Rev. Lett. 87, paper 278701.
88. Goh, K.I., Oh, E., Jeong, H., Kahng, B., and Kim, D. (2002) Classification of scale-free networks. Proc. Natl. Acad. Sci. 99, 12583–12588.
89. Grimmett, G. (1995) The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab. 23, 1461–1510.
90. Grossman, J.W. (2000) The evolution of the mathematical research collaboration graph. Erdös Number Project web site. www.oakland.edu/˜grossman/trivia.html.
91. Häggström, O. (1998) Random cluster representations in the study of phase transitions. Markov Processes. Relat. Fields 4, 275–321.
92. Häggström, O. (2000) Markov random fields an percolation on general graphs. Adv. Appl. Prob. 32, 39–66.
93. Harris, T.E. (1974) Contact interactions on a lattice. Ann. Probab. 2, 969–988.
94. Hoeffding (1963) Probability inequalities for sums of bounded random variables. J. Am. Stat. Soc. 58, 13–30.
95. Holley, R.A. (1974) Remarks on the FKG inequalities. Commun. Math. Phys. 36, 227–231.
96. Holley, R., and Liggett, T. (1975) Ergodic theorems for weakly interacting systems and the voter model. Ann. Probab. 4, 195–228.
97. Hong, H., Kim, B.J., and Choi, M.Y. (2002) Comment on “Ising model on a small world network.” Phys. Rev. E. 66, paper 018101.
98. Ioffe, D. (1996) On the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37, 137–143.
99. Jacod, J., and Shiryaev, A.N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag, New York.
100. Janson, S. (1993) Multicyclic components in a random graph process. Random Struct. Alg. 4, 71–84.
101. Janson, S., Knuth, D.E., Luczak, T., and Pittel, B. (1993) The birth of the giant component. Random Struct. Alg. 4, 233–358.
102. Jeong, H., Néda, Z., and Barbási, A.L. (2001) Measuring preferential attachment in growing networks. arXiv:cond-mat/0104131 (0104162?).
103. Jespersen, S., Sokolov, I.M., and Blumen, A. (2000) Relaxation properties of small world networks. Phys. Rev. E 62, 4405–4408.
104. Jones, J.H., and Handcock, M.S. (2003) An assessment of preferential attachment as a mechanism of human sexual network formation. Proc. R. Soc. Lond. B 270, 1123–1128.
105. Kalikow, S., andWeiss, B. (1988) When are random graphs connected? Israel J. Math. 62, 257–268.
106. Kendall, D.G. (1956) Deterministic and stochastic epidemics in closed populations. In Proceedings of the 3rd Berekely Symposium on Statistical Probability, Vol. 4, pp. 149–165.
107. Kesten, H. (1980) Percolation Theory for Mathematicians. Birkhäuser, Boston.
108. Kleinberg, J., Kumar, R., Raghavan, P., Rajagopalan, S., and Tomkins, A. (1999) The web as a graph: Measurement, models, and methods. In Proceedings of the International Conference on Combinatorics and Computing, pp. 1–18.
109. Krapivsky, P.L., Redner, S., and Leyvraz, F. (2000) Connectivity of growing random networks. Phys. Rev. Lett. 85, 4629–4632.
110. Krapivsky, P.L., and Redner, S. (2001) Organization of growing random networks. Phys. Rev. E 63, paper 066123.
111. Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., and Upfal E. (2000) Stochastic models with the web graph. In Proceedings of Foundations of Computer Science, pp. 57–65.
112. Lahtinen, J., Kertesz, J., and Kaski, K. (2001) Scaling of random spreading in small world networks. Phys. Rev. E 64, paper 057105.
113. Lieberman, E., Hauert, C., and Nowak, M.A. (2005) Evolutionary dynamics on graphs. Nature 433, 312–316.
114. Liljeros, F., Edling, C.R., and Amaral, L.A.N. (2003) Sexual networks: Implications for the transmission of sexually transmitted infections. Microbes Infect. 3, 189–196.
115. Liljeros, F., Edling, C.R., Amaral, L.A.N., Stanley, H.E., and Aberg, Y. (2001) The web of human sexual contacts. Nature 411, 907–908.
116. Liggett, T.M. (1996). Multiple transition points for the contact process on the binary tree. Ann. Probab. 24, 1675–1710.
117. Liggett, T.M. (1999) Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, New York.
118. Lloyd, A.L., and May, R.M. (2001) How viruses spread among computers and people. Science 292, 1316.
119. Lloyd-Smith, J.O., Schreiber, S.E., Kopp, P.E., and Getz, W.M. (2005) Superspreading and the effect of individual variation on disease emergence. Nature 438, 355–359.
120. Lovász, L., and Kannan, R. (1999) Faster mixing via average conductance. In Symposium on the Theory of Computing 1999, pp. 282–287.
121. Luczak, T., Pittel, B., and Weirman, J.C. (1994) The structure of a random graph at the point of phase transition. Trans. AMS 341, 721–748.
122. Lyons, R. (1989) The Ising model and percolation on trees and tree-like graphs. Commun. Math. Phys. 125, 337–353.
123. Lyons, R. (1990) Random walks and percolation on trees. Ann. Probab. 18, 931–958.
124. Lyons, R. (2000) Phase transitions on nonamenable graphs. J. Math. Phys. 41, 1099–1126.
125. Madras, N., and Schinazi, R.B. (1992) Branching random walks on trees. Stoch. Proc. Appl. 42, 255–267.
126. Martin-Löf, A. (1986) Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265–282.
127. Martin-Löf, A. (1988) The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Prob. 35, 671–682.
128. Maruyama, T. (1970) The effective number of alleles in a subdivided population. Theor. Pop. Biol. 1, 273–306.
129. May, R.M., and Anderson, R.M. (1988) The transmission dynamics of human immunodeficiency virus (HIV). Phil. Trans. R. Soc. B 321, 565–607.
130. Mihail, M., Papadimitrou, C., and Saberi, A. (2004) On certain connectivity properties of the Internet topology. In Proceedings of 44th Annual IEEE Symposium on Foundation of Computer Science 2003, pp. 28–35.
131. Milgram, S. (1967) The small world problem. Psychology Today, May 1967, pp. 60–67.
132. Mitzenmacher, M. (2004) A brief history of generative models for power law and lognormal distributions. Internet Math. 1, 226–251.
133. Molloy, M., and Reed, B. (1995) A critical point for random graphs with a given degree sequence. Random Struct. Alg. 6, 161–179.
134. Molloy, M., and Reed, B. (1998) The size of the giant component of a random graph with a given degree sequence. Comb. Probab. Comput. 7, 295–305.
135. Monasson, R. (1999) Diffusion, localization and dispersion relations on “small world” lattices. Eur. Phys. J. 12, 555–567.
136. Moore, C., and Newman, M.E.J. (2000) Exact solution of bond percolation on small-world networks. Phys. Rev. E 62, 7059–7064.
137. Moreno, Y., Pastor-Satorras, R., and Vespigniani, A. (2002) Epidemic outbreaks in complex heterogeneous networks. Eur. Phys. J. 26, 521–529.
138. Mori, T.F. (2002) On random trees. Studia Sci. Math. Hungarica 39, 143–155.
139. Mori, T.F. (2005) The maximum degree of the Barabási–Albert random tree. Comb. Probab. Comput. 14, 339–348.
140. Morris, B., and Peres, Y. (2003) Evolving sets and mixing. In Symposium on the Theory of Computing 2003, pp. 279–286.
141. Mountford, T.S. (1999) Existence of a constant for finite system extinction. J. Stat. Phys. 96, 1331–1341.
142. Nachmias, A., and Peres, Y. (2005) The critical random graph with martingales. arXiv:math.PR/0512201.
143. Newman, M.E.J. (2000) Models of the small world. J. Stat. Phys. 101, 819–841.
144. Newman, M.E.J. (2001a) The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. USA 98, 404–409.
145. Newman, M.E.J. (2001b) Scientific collaboration networks. I and II. Phys. Rev. E 64, paper 016131, 016132.
146. Newman, M.E.J. (2001c) Clustering and preferential attachment in growing networks. Phys. Rev. E 64, paper 025102(R).
147. Newman, M.E.J. (2002) Spread of epidemic disease on networks. Phys. Rev. E 66, paper 016128.
148. Newman, M.E.J. (2003) The structure and function of complex networks. SIAM Rev. 45, 167–256.
149. Newman, M.E.J., Barabási, A.L., and Watts, D.J. (2006) The Structure and Dynamics of Networks. Princeton University Press, Princeton, NJ.
150. Newman, M.E.J., Moore, C., and Watts, D.J. (2000) Mean-field solution of the small-world network model. Phys. Rev. Lett. 84, 3201–3204.
151. Newman, M.E.J., Strogatz, S.H., and Watts, D.J. (2001) Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, paper 026118.
152. Newman, M.E.J., Strogatz, S.H., and Watts, D.J. (2002) Random graph models of social networks. Proc. Natl. Acad. Sci. 99, 2566–2572.
153. Newman, M.E.J., and Watts, D.J. (1999) Renormalization group analysis of the small-world network model. Phys. Lett. A 263, 341–346.
154. Noh, J.D., and Reiger, H. (2004) Random walks on complex networks. Phys. Rev. Lett. 92, 118701.
155. Oliveira, R., and Spencer, J. (2005) Connectivity transitions in networks with super-linear preferential attachment. arXiv:math.PR/0510466.
156. Pandit, S.A., and Amritkar, R.E. (2001) Random spread on the family of small world networks. Phys. Rev. E 63, paper 041104.
157. Pastor-Satorras, R., and Vespigniani, A. (2001a) Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203.
158. Pastor-Satorras, R., and Vespigniani, A. (2001b) Epidemic dynamics and endemic states in complex networks. Phys. Rev. E 63, 066117.
159. Pastor-Satorras, R., and Vespigniani, A. (2002) Epidemic dynamics in finite size scale-free networks. Phys. Rev. E 65, 035108(R).
160. Pekalski, A. (2001) Ising model on a small world network. Phys. Rev. E 64, paper 057104.
161. Pemantle, R. (1992) The contact process on trees. Ann. Probab. 20, 2089–2116.
162. Pittel, B. (1990) On tree census and the giant component in sparse random graphs. Random Struct. Alg. 1, 311–342.
163. Pittel, B. (1994) Note on the heights of random recursive trees and random m-ary search trees. Random Struct. Alg. 5, 337–348.
164. Pittel, B. (2001) On the largest component of the random graph at a near critical stage. J. Comb. Theory B 82, 237–269.
165. Preston, C.J. (1974) Gibbs States on Countable Sets. Cambridge University Press, Cambridge.
166. Redner, S. (1998) How popular is your paper? An empirical study of the citation distribution. Eur. Phys. J. B 4, 131–134.
167. Riordan, O. (2004) The small giant component in scale free random graphs. Comb. Probab. Comput., to appear.
168. Saloff-Coste, L. (1996) Lectures on finite Markov chains. Ecole d’eté de probabilités de Saint-Flour XXVI. Springer Lecture Notes in Math, Vol. 1665.
169. Shepp, L.A. (1989) Connectedness of certain random graphs. Israel J. Math. 67, 23–33.
170. Simon, H.A. (1955) On a class of skew distribution functions. Biometrika 42, 425–440.
171. Sinclair, A., and Jerrum, M. (1989) Approximate counting, uniform generation and rapidly mixing Markov chains. Inf. Comput. 82, 93–133.
172. Slatkin, M. (1981) Fixation probabilities and fixation times in a subdivided population. Evolution 35, 477–488.
173. Sood, V., and Redner, S. (2005) Voter model on heterogeneous graphs. Phys. Rev. Lett. 94, paper 178701.
174. Stacey, A.M. (1996) The existence of an intermediate phase for the contact process on trees. Ann. Probab. 24, 1711–1726.
175. Suchecki, K., Eguu´ıluz, V.M., and Miguel, M.S. (2005) Conservation laws for the voter model in complex networks. Europhys. Lett. 69, 228–234.
176. Szabo, G., Alava, M., and Kertesz, J. (2002) Shortest paths and load scaling in scale-free networks. Phys. Rev. E 66, 026101.
177. Szymanski, J. (1987) Random Graphs ’85. Math Studies, Vol. 144, North-Holland, Amsterdam.
178. Tanner, J.C. (1961) A derivation of the Borel distribution. Biometrika 48, 222–224.
179. van der Hofstad, R., Hooghiemstra, G., and Van Mieghem, P. (2004a) Distance in random graphs with finite variance degrees. arXiv:math.PR/0407092. Random Struct. Alg. 27, 76–123.
180. van der Hofstad, R., Hooghiemstra, G., and Znamenski, D. (2004b) Distances in random graphs with infinite mean degrees. arXiv:math.PR/0407091.
181. van der Hofstad, R., Hooghiemstra, G., and Znamenski, D. (2005) Distances in random graphs with finite mean and infinite variance degrees. arXiv:math.PR/0502581.
182. Watts, D.J. (2003) Six Degrees. W.W. Norton, New York.
183. Watts, D.J., and Strogatz, S.H. (1998) Collective dynamics of ‘small-world’ networks. Nature 393, 440–442.
184. Wormald, N.C. (1995) Differential equations for random processes and random graphs. Ann. Appl. Prob. 5, 1217–1235.
185. Yule, G.U. (1925) A mathematical theory of evolution, based on the conclusions of Dr. J.C. Willis. Phil. Trans. R. Soc. Lond. B 213, 21–87.
186. Zhang, Y. (1991) A power law for connectedness of some random graphs at the critical point. Random Struct. Alg. 1, 101–119.