Abstract
Let D be a non-negative integer-valued random variable and let G = (V, E) be an infinite transitive finite-degree graph. Continuing the work of Deijfen and Meester (Adv Appl Probab 38:287–298) and Deijfen and Jonasson (Electron Comm Probab 11:336–346), we seek an Aut(G)-invariant random graph model with V as vertex set, iid degrees distributed as D and finite mean connections (i.e., the sum of the edge lengths in the graph metric of G of a given vertex has finite expectation). It is shown that if G has either polynomial growth or rapid growth, then such a random graph model exists if and only if \({\mathbb{E}[D\,R(D)] < \infty}\). Here R(n) is the smallest possible radius of a combinatorial ball containing more than n vertices. With rapid growth we mean that the number of vertices in a ball of radius n is of at least order exp(n c) for some c > 0. All known transitive graphs have either polynomial or rapid growth. It is believed that no other growth rates are possible. When G has rapid growth, the result holds also when the degrees form an arbitrary invariant process. A counter-example shows that this is not the case when G grows polynomially. For this case, we provide other, quite sharp, conditions under which the stronger statement does and does not hold respectively. Our work simplifies and generalizes the results for \({G\,=\,\mathbb {Z}}\) in [4] and proves, e.g., that with \({G\,=\,\mathbb {Z}^d}\), there exists an invariant model with finite mean connections if and only if \({\mathbb {E}[D^{(d+1)/d}] < \infty}\). When G has exponential growth, e.g., when G is a regular tree, the condition becomes \({\mathbb {E}[D\,\log\,D] < \infty}\).
Article PDF
Similar content being viewed by others
References
Bartholdi L.: The growth of Grigorchuk’s torsion group. Int. Math. Res. Notices 20, 1049–1054 (1998)
Baum E., Katz M.: Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 120, 108–123 (1965)
Benjamini I., Lyons R., Peres Y., Schramm O.: Group-invariant percolation on graphs. Geom. Funct. Anal. 9, 29–66 (1999)
Deijfen M., Jonasson J.: Stationary random graphs on \(\mathbb{Z}\) with prescribed iid degrees and finite mean connections. Electron. Comm. Probab. 11, 336–346 (2006)
Deijfen M., Meester R.: Generating stationary random graphs on \(\mathbb{Z}\) with prescribed iid degrees. Adv. Appl. Probab. 38, 287–298 (2006)
Grigorchuk R.: On the Milnor problem of group growth. Sov. Math. Dokl. 28, 23–26 (1983)
Hoffman C., Holroyd A., Peres Y.: A stable marriage of Poisson and Lebesgue. Ann. Probab. 34, 1241–1272 (2006)
Imrich W., Seifter N.: A survey on graphs with polynomial growth. Discrete Math. 95, 101–117 (1991)
Kaimanovich V., Vershik A.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11, 457–490 (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jonasson, J. Invariant random graphs with iid degrees in a general geography. Probab. Theory Relat. Fields 143, 643–656 (2009). https://doi.org/10.1007/s00440-008-0160-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-008-0160-z
Keywords
- Random graphs
- Degree distribution
- Automorphism
- Invariant model
- Mass-transport principle
- Unimodular graph
- Polynomial growth
- Intermediate growth
- Exponential growth