On a random graph with immigrating vertices: Emergence of the giant component

A randomly evolving graph, with vertices immigrating at rate n and each possible edge appearing at rate 1/n, is studied. The detailed picture of emergence of giant components with On2/3‘ vertices is shown to be the same as in the Erdős–Rényi graph process with the number of vertices fixed at n at the start. A major difference is that now the transition occurs about a time t = π/2, rather than t = 1. The proof has three ingredients. The size of the largest component in the subcritical phase is bounded by comparison with a certain multitype branching process. With this bound at hand, the growth of the sum-of-squares and sum-of-cubes of component sizes is shown, via martingale methods, to follow closely a solution of the Smoluchowsky-type equations. The approximation allows us to apply results of Aldous [Brownian excursions, critical random graphs and the multiplicative coalescent, Ann Probab 25 (1997), 812–854] on emergence of giant components in the multiplicative coalescent, i.e., a nonuniform random graph process. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17, 79–102, 2000

[1]  David Aldous,et al.  Brownian excursions, critical random graphs and the multiplicative coalescent , 1997 .

[2]  J. Norris,et al.  Smoluchowski's coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent , 1998, math/9801145.

[3]  Michael Capobianco,et al.  Graph evolution by stochastic additions of points and lines , 1983, Discret. Math..

[4]  David Aldous,et al.  The Entrance Boundary of the Multiplicative Coalescent , 1998 .

[5]  Tomasz Luczak.  Phase transition phenomena in random discrete structures , 1994, Discret. Math..

[6]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[7]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[8]  H. Poincaré,et al.  Percolation ? , 1982 .

[9]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[10]  B. Pittel.  A random graph with a subcritical number of edges , 1988 .

[11]  V. E. Stepanov.  Combinatorial Algebra and Random Graphs , 1969 .

[12]  Béla Bollobás,et al.  Random Graphs , 1985 .

[13]  V. F. Kolchin.  On the Behavior of a Random Graph Near a Critical Point , 1987 .

[14]  J. Spouge.  Solutions and critical times for the polydisperse coagulation equation when a(x,y)=A+B(x+y)+Cxy , 1983 .

[15]  David Aldous.  Emergence of the giant component in special Marcus-Lushnikov processes , 1998 .

[16]  Boris G. Pittel,et al.  On Tree Census and the Giant Component in Sparse Random Graphs , 1990, Random Struct. Algorithms.

[17]  V. E. Stepanov.  On Some Features of the Structure of a Random Graph Near a Critical Point , 1988 .

[18]  B. Pittel,et al.  The structure of a random graph at the point of the phase transition , 1994 .

[19]  Svante Janson,et al.  The Birth of the Giant Component , 1993, Random Struct. Algorithms.