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Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

Part of: Graph theory Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Maria Deijfen  and
Ronald Meester
Maria Deijfen*
Affiliation:
Chalmers University of Technology
Ronald Meester*
Affiliation:
Vrije Universiteit Amsterdam
*
∗ Current address: DIAM, Mekelweg 4, 2628 CD Delft, The Netherlands. Email address: [email protected]
∗∗ Postal address: Divisie Wiskunde en Informatica, Faculteit der Exacte Wetenshappen, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. Email address: [email protected]
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Abstract

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Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

Keywords

Random graphdegree distributionstationary algorithmrandom walkfinitary isomorphism

MSC classification

Primary: 05C80: Random graphs 60G50: Sums of independent random variables; random walks
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 287 - 298
Copyright
Copyright © Applied Probability Trust 2006 

References

Arratia, R. (1981). Limiting point processes for rescalings of coalescing and annihilating random walks on Z d . Ann. Prob. 9, 909–936.Google Scholar
Bollobás, B. and Riordan, O. M. (2002). Mathematical results on scale-free random graphs. In Handbook of Graphs and Networks. From the Genome to the Internet, eds Bornholdt, S. and Schuster, H. G., Wiley-VCH, Berlin, pp. 1–34.Google Scholar
Britton, T., Deijfen, M. and Martin-Löf, A. (2005). Generating simple random graphs with prescribed degree distribution. To appear in J. Statist. Phys. Available at http://www.math.su.se/∼mia.Google Scholar
Chung, F. and Lu, L. (2002a). Connected components in random graphs with given expected degree sequences. Ann. Combinatorics 6, 125–145.Google Scholar
Chung, F. and Lu, L. (2002b). The average distances in random graphs with given expected degrees. Proc. Nat. Acad. Sci. USA 99, 15879–15882.Google Scholar
Dorogovtsev, S. N. and Mendes, J. F. F. (2003). Evolution of Networks. From Biological Nets to the Internet and WWW. Oxford University Press.Google Scholar
Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6, 290–297.Google Scholar
Holroyd, A. E. and Peres, Y. (2005). Extra heads and invariant allocations. Ann. Prob. 33, 31–52.Google Scholar
Mattera, M. (2003). Annihilating random walks and perfect matchings of planar graphs. Discrete Math. Theoret. Computer Sci. AC, 173–180.Google Scholar
Meshalkin, L. D. (1959). A case of isomorphisms of Bernoulli schemes. Dokl. Akad. Nauk. SSSR 128, 41–44.Google Scholar
Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6, 161–179.Google Scholar
Molloy, M. and Reed, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combinatorics Prob. Comput. 7, 295–305.Google Scholar
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, 026118.Google Scholar
Parry, W. (1979). An information obstruction to finite expected coding length. In Ergodic Theory (Proc. Conf., Oberwolfach, 1978; Lecture Notes Math. 729), eds Denker, M. and Jacobs, K., Springer, Berlin, pp. 163–168.Google Scholar
Schmidt, K. (1984). Invariants for finitary isomorphisms with finite expected coding lengths. Invent. Math. 76, 33–40.Google Scholar
Van der Hofstad, R., Hooghiemstra, G. and Znamenski, D. (2005). Random graphs with arbitrary i.i.d. degrees. Preprint. Available at http://www.win.tue.nl/∼rhofstad/research.html.Google Scholar