### Joint vertex degrees in the inhomogeneous random graph model G(n, {pij})

#### Abstract

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

#### Article information

Source
Adv. in Appl. Probab., Volume 44, Number 1 (2012), 139-165.

Dates
First available in Project Euclid: 8 March 2012

https://projecteuclid.org/euclid.aap/1331216648

Digital Object Identifier
doi:10.1239/aap/1331216648

Mathematical Reviews number (MathSciNet)
MR2951550

#### Citation

Lin, Kaisheng; Reinert, Gesine. Joint vertex degrees in the inhomogeneous random graph model G ( n , { p ij }). Adv. in Appl. Probab. 44 (2012), no. 1, 139--165. doi:10.1239/aap/1331216648. https://projecteuclid.org/euclid.aap/1331216648

#### References

• Barabasi, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.
• Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford University Press.
• Bartroff, J. and Goldstein, L. (2009). A Berry-Esseen bound with applications to the number of multinomial cells of given occupancy and the number of graph vertices of given degree. Preprint.
• Bollobás, B. (2001). Random Graphs. Cambridge University Press.
• Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3–122.
• Daudin, J-J., Picard, F. and Robin, S. (2008). A mixture model for random graphs. Statist. Comput. 18, 173–183.
• Dorogovtsev, S. N. and Mendes, J. F. F. (2003). Evolution of Networks. Oxford University Press.
• Erdös, P. and Rényi, A. (1959). On random graphs. Publ. Math. Debrecen 6, 290–297.
• Goldstein, L. and Rinott, Y. (1996). Multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Prob. 33, 1–17.
• Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge University Press.
• Lin, K. (2008). Motif counts, clustering coefficients, and vertex degrees in models of random networks. Doctoral Thesis, University of Oxford.
• McKay, B. D. and Wormald, N. C. (1997). The degree sequence of a random graph. I. The models. Random Structures Algorithms 11, 97–117.
• 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.
• Nowicki, K. and Snijders, T. A. B. (2001). Estimation and prediction for stochastic blockstructures. J. Amer. Statist. Assoc. 96, 1077–1087.
• Rinott, Y. and Rotar, V. (1996). A multivariate CLT for local dependence with $n^{-1/2}\log n$ rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56, 333–350.
• Solow, A. R., Costello, C. J. and Ward, M. (2003). Testing the power law model for discrete size data. Amer. Nat. 162, 685–689.
• Stein, C. (1986). Approximate Computation of Expectations (Inst. Math. Statist. Lecture Notes–-Monogr. Ser. 7). Institute of Mathematical Statistics, Hayward, CA.
• Stumpf, M. P. H., Wiuf, C. and May, R. M. (2005). Subnets of scale-free networks are not scale-free: sampling properties of networks. Proc. Nat. Acad. Sci. USA 12, 4221–4224.