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# Bounded-Size Rules: The Barely Subcritical Regime

## Abstract

Bounded-size rules (BSRs) are dynamic random graph processes which incorporate limited choice along with randomness in the evolution of the system. Typically one starts with the empty graph and at each stage two edges are chosen uniformly at random. One of the two edges is then placed into the system according to a decision rule based on the sizes of the components containing the four vertices. For bounded-size rules, all components of size greater than some fixed K ≥ 1 are accorded the same treatment. Writing BSR(t) for the state of the system with ⌊nt/2⌋ edges, Spencer and Wormald [26] proved that for such rules, there exists a (rule-dependent) critical time tc such that when t < tc the size of the largest component is of order log n, while for t > tc, the size of the largest component is of order n. In this work we obtain upper bounds (that hold with high probability) of order n log4n, on the size of the largest component, at time instants tn = tcn−γ, where γ ∈ (0,1/4). This result for the barely subcritical regime forms a key ingredient in the study undertaken in [4], of the asymptotic dynamic behaviour of the process describing the vector of component sizes and associated complexity of the components for such random graph models in the critical scaling window. The proof uses a coupling of BSR processes with a certain family of inhomogeneous random graphs with vertices in the type space $\mathbb{R}_+\times \mathcal{D}([0,\infty):\mathbb{N}_0)$ , where $\mathcal{D}([0,\infty):\mathbb{N}_0)$ is the Skorokhod D-space of functions that are right continuous and have left limits, with values in the space of non-negative integers $\mathbb{N}_0$ , equipped with the usual Skorokhod topology. The coupling construction also gives an alternative characterization (from the usual explosion time of the susceptibility function) of the critical time tc for the emergence of the giant component in terms of the operator norm of integral operators on certain L2 spaces.

## References

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[1]Achlioptas, D., D'Souza, R. M. and Spencer, J. (2009) Explosive percolation in random networks. Science 323 1453.
[2]Aldous, D. (1997) Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812854.
[3]Bhamidi, S., Budhiraja, A. and Wang, X. (2013) Aggregation models with limited choice and the multiplicative coalescent. Random Struct. Alg. http://dx.doi.org/10.1002/rsa.20493
[4]Bhamidi, S., Budhiraja, A. and Wang, X. (2013) The augmented multiplicative coalescent, bounded size rules and critical dynamics of random graphs. Probab Theory Rel. Fields, 164. http://dx.doi.org/10.1007/s00440-013-0540-x
[5]Bohman, T. and Frieze, A. (2001) Avoiding a giant component. Random Struct. Alg. 19 7585.
[6]Bohman, T., Frieze, A. and Wormald, N.C. (2004) Avoidance of a giant component in half the edge set of a random graph. Random Struct. Alg. 25 432449.
[7]Bollobás, B. (2001) Random Graphs, second edition, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.
[8]Bollobás, B., Janson, S. and Riordan, O. (2007) The phase transition in inhomogeneous random graphs. Random Struct. Alg. 31 3122.
[9]Brémaud, P. (1981) Point Processes and Queues, Springer.
[10]Ding, J., Lubetzky, E. and Peres, Y. (2012) Mixing time of near-critical random graphs. Ann. Probab. 40 9791008.
[11]Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 1761.
[12]Erdős, P. and Rényi, A. (1961) On the evolution of random graphs. Bull. Inst. Internat. Statist. 38 343347.
[13]van der Hofstad, R. and Nachmias, A. (2012) Hypercube percolation. arXiv:1201.3953. To appear in Metrika.
[14]Janson, S. (2008) The largest component in a subcritical random graph with a power law degree distribution. Ann. Appl. Probab. 18 16511668.
[15]Janson, S. and Luczak, M. J. (2008) Susceptibility in subcritical random graphs. J. Math. Phys. 49 125207.
[16]Janson, S. and Spencer, J. (2012) Phase transitions for modified Erdős–Rényi processes. Arkiv för Matematik 50 305329.
[17]Janson, S., Knuth, D., Luczak, T. and Pittel, B. (1994) The birth of the giant component (with an introduction by the editors). Random Struct. Alg. 4 231358.
[18]Kang, M., Perkins, W. and Spencer, J. (2013) The Bohman–Frieze process near criticality. Random Struct. Alg. 43 221250.
[19]Kim, J. H. (2006) Poisson cloning model for random graphs. In Proc. International Congress of Mathematicians, Vol. III, European Mathematical Society, pp. 873897.
[20]Liptser, R. S. and Shiryayev, A. N. (1989) Theory of Martingales, Vol. 49 of Mathematics and its Applications (Soviet Series), Kluwer Academic. Translated from the Russian by K. Dzjaparidze (Kacha Dzhaparidze).
[21]Norris, J. R. (1998) Markov Chains, Vol. 2 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.
[22]Pittel, B. G. (2008) On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase. Ann. Appl. Probab. 18 16361650.
[23]Riordan, O. and Warnke, L. (2012) Achlioptas process phase transitions are continuous. Ann. Appl. Probab. 22 14501464.
[24]Riordan, O. and Warnke, L. (2014) The evolution of subcritical Achlioptas processes. Random Struct. Alg. http://dx.doi.org/10.1002/rsa.20530
[25]Sen, S. (2013) On the largest component in the subcritical regime of the Bohman–Frieze process. arXiv:1307.2041
[26]Spencer, J. and Wormald, N.. (2007) Birth control for giants. Combinatorica 27 587628.

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