Random networks with sublinear preferentia attachment: The giant component

Annals of Probability

Volumn 41, Issue 1, 2013, Pages 329-384

  Mörters, Peter ^a   Dereich, Steffen ^b  

^a UNIVERSITY OF MÜNSTER   (Germany)

^b University of Bath   (United Kingdom)

Author keywords

Barab si Albert model; Cluster; Dynamic random graph; Erdos R nyi model; Giant component; Multitype branching random walk; Nonlinear preferential attachment; Power law; Scale free network

Indexed keywords

EID: 84874991571     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AOP697     Document Type: Article

References (18)

1. BARABÁSI, A.-L. andALBERT, R. (1999). Emergence of scaling in random networks. Science 286 509-512. MR2091634
2. BENJAMINI, I. and SCHRAMM, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 13 pp. (electronic). MR1873300
3. BERGER, N., BORGS, C., CHAYES, J. T. and SABERI, A. (2009). A weak local limit for preferential attachment graphs. Preprint.
4. BOLLOBÁS, B., JANSON, S. and RIORDAN, O. (2005). The phase transition in the uniformly grown random graph has infinite order. Random Structures Algorithms 26 1-36. MR2116573
5. BOLLOBÁS, B., JANSON, S. andRIORDAN, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3-122. MR2337396
6. BOLLOBÁS, B. and RIORDAN, O. (2003). Robustness and vulnerability of scale-free random graphs. Internet Math. 1 1-35. MR2076725
7. BOLLOBÁS, B., RIORDAN, O., SPENCER, J. and TUSNÁDY, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18 279-290. MR1824277
8. DEREICH, S. andMÖRTERS, P. (2009). Random networks with sublinear preferential attachment: Degree evolutions Electron. J. Probab. 14 1222-1267. MR2511283
9. DOMMERS, S., VAN DER HOFSTAD, R. and HOOGHIEMSTRA, G. (2010). Diameters in preferential attachment models. J. Stat. Phys. 139 72-107. MR2602984
10. HARDY, R. and HARRIS, S. C. (2009). A spine approach to branching diffusions with applications to Lp-convergence of martingales. In Séminaire de Probabilités XLII. Lecture Notes in Math. 1979 281-330. Springer, Berlin. MR2599214
11. KATO, T. (1976). Perturbation Theory for Linear Operators, 2nd ed. Grundlehren der Mathematischen Wissenschaften 132. Springer, Berlin. MR0407617
12. KATO, T. (1982). Superconvexity of the spectral radius, and convexity of the spectral bound and the type. Math. Z. 180 265-273. MR0661703
13. KRAPIVSKY, P. L. and REDNER, S. (2001). Organization of growing random networks. Phys. Rev. E 63, paper 066123.
14. KYPRIANOU, A. E. and RAHIMZADEH SANI, A. (2001). Martingale convergence and the functional equation in the multi-type branching random walk. Bernoulli 7 593-604. MR1849369
15. PINSKY, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics 45. Cambridge Univ. Press, Cambridge. MR1326606
16. RAMM, A. G. (1983). Variational principles for eigenvalues of compact nonselfadjoint operators. II. J. Math. Anal. Appl. 91 30-38. MR0688529
17. SHEPP, L. A. (1989). Connectedness of certain random graphs. Israel J. Math. 67 23-33. MR1021358
18. VAN DER HOFSTAD, R. (2011). Random graphs and complex networks. Lecture notes. Available at http://www.win.tue.nl/~rhofstad/NotesRGCN2010.pdf.