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Generating Infinite Random Graphs

  • Csaba Biró (a1) and Udayan B. Darji (a1)

We define a growing model of random graphs. Given a sequence of non-negative integers {dn}n=0 with the property that dii, we construct a random graph on countably infinitely many vertices v0, v1… by the following process: vertex vi is connected to a subset of {v0, …, vi−1} of cardinality di chosen uniformly at random. We study the resulting probability space. In particular, we give a new characterization of random graphs, and we also give probabilistic methods for constructing infinite random trees.

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1.Barabási, A.-L. and Albert, R., Emergence of scaling in random networks, Science 286(5439) (1999), 509512.
2.Bollobás, B. and Riordan, O. M., The diameter of a scale-free random graph, Combinatorica 24(1) (2004), 534.
3.Bonato, A., The search for n-e.c. graphs, Contrib. Discrete Math. 4(1) (2009), 4053.
4.Bonato, A. and Janssen, J. C. M., Infinite limits and adjacency properties of a generalized copying model, Internet Math. 4(2–3) (2007), 199223.
5.Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. L., Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing, Adv. Math. 219(6) (2008), 18011851.
6.Brusss, F. T., A counterpart of the Borel-Cantelli lemma, J. Appl. Probab. 17(4) (1980), 10941101.
7.Cameron, P. J., The random graph revisited, European congress of mathematics, volume I (Barcelona, 2000), Progress in Mathematics, Volume 201, pp. 267274 (Birkhäuser, Basel, 2001).
8.Darji, U. B. and Mitchell, J. D., Approximation of automorphisms of the rationals and the random graph, J. Group Theory 14(3) (2011), 361388.
9.Diestel, R., Graph theory, 4th edition, Graduate Texts in Mathematics, Volume 173 (Springer, 2010).
10.Erdős, P. and Rényi, A., Asymmetric graphs, Acta Math. Acad. Sci. Hung. 14 (1963), 295315.
11.Janson, S. and Severini, S., An example of graph limit of growing sequences of random graphs, J. Comb. 4(1) (2013), 6780.
12.Kleinberg, R. D. and Kleinberg, J. M., Isomorphism and embedding problems for infinite limits of scale-free graphs, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2005, pp. 277286 (electronic).
13.Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A. and Upfal, E., Stochastic models for the web graph, 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000), pp. 5765 (IEEE Computer Society Press, Los Alamitos, CA, 2000).
14.Lovász, L. and Szegedy, B., Limits of dense graph sequences, J. Combin. Theory Ser. B 96 (2006), 933957.
15.Rado, R., Universal graphs and universal functions, Acta Arith. 9 (1964), 331340.
16.Spencer, J. H., The strange logic of random graphs, Algorithms and Combinatorics, Volume 22 (Springer, 2001).
17.Truss, J. K., The group of the countable universal graph, Math. Proc. Cambridge Philos. Soc. 98(2) (1985), 213245.
18.van der Hofstad, R., Percolation and random graphs, In New perspectives in stochastic geometry (eds Molchanov, I. S. and Kendall, W. S.), pp. 173247 (Oxford University Press, Oxford, 2010).
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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