## The Annals of Probability

### Random networks with sublinear preferential attachment: The giant component

#### Abstract

We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function $f$ of its current degree. We give a criterion for the existence of a giant component, which is both necessary and sufficient, and which becomes explicit when $f$ is linear. Otherwise it allows the derivation of explicit necessary and sufficient conditions, which are often fairly close. We give an explicit criterion to decide whether the giant component is robust under random removal of edges. We also determine asymptotically the size of the giant component and the empirical distribution of component sizes in terms of the survival probability and size distribution of a multitype branching random walk associated with $f$.

#### Article information

Source
Ann. Probab., Volume 41, Number 1 (2013), 329-384.

Dates
First available in Project Euclid: 23 January 2013

https://projecteuclid.org/euclid.aop/1358951989

Digital Object Identifier
doi:10.1214/11-AOP697

Mathematical Reviews number (MathSciNet)
MR3059201

Zentralblatt MATH identifier
1260.05143

Subjects
Secondary: 60C05: Combinatorial probability 90B15: Network models, stochastic

#### Citation

Dereich, Steffen; Mörters, Peter. Random networks with sublinear preferential attachment: The giant component. Ann. Probab. 41 (2013), no. 1, 329--384. doi:10.1214/11-AOP697. https://projecteuclid.org/euclid.aop/1358951989

#### References

• [1] Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 509–512.
• [2] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 13 pp. (electronic).
• [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.
• [5] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
• [6] Bollobás, B. and Riordan, O. (2003). Robustness and vulnerability of scale-free random graphs. Internet Math. 1 1–35.
• [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.
• [8] Dereich, S. and Mörters, P. (2009). Random networks with sublinear preferential attachment: Degree evolutions. Electron. J. Probab. 14 1222–1267.
• [9] Dommers, S., van der Hofstad, R. and Hooghiemstra, G. (2010). Diameters in preferential attachment models. J. Stat. Phys. 139 72–107.
• [10] Hardy, R. and Harris, S. C. (2009). A spine approach to branching diffusions with applications to ${L}^{p}$-convergence of martingales. In Séminaire de Probabilités XLII. Lecture Notes in Math. 1979 281–330. Springer, Berlin.
• [11] Kato, T. (1976). Perturbation Theory for Linear Operators, 2nd ed. Grundlehren der Mathematischen Wissenschaften 132. Springer, Berlin.
• [12] Kato, T. (1982). Superconvexity of the spectral radius, and convexity of the spectral bound and the type. Math. Z. 180 265–273.
• [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.
• [15] Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics 45. Cambridge Univ. Press, Cambridge.
• [16] Ramm, A. G. (1983). Variational principles for eigenvalues of compact nonselfadjoint operators. II. J. Math. Anal. Appl. 91 30–38.
• [17] Shepp, L. A. (1989). Connectedness of certain random graphs. Israel J. Math. 67 23–33.
• [18] van der Hofstad, R. (2011). Random graphs and complex networks. Lecture notes. Available at http://www.win.tue.nl/~rhofstad/NotesRGCN2010.pdf.