The evolution of subcritical Achlioptas processes

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. Although the evolution of such `local' modifications of the Erd{\H o}s--R\'enyi random graph process has received considerable attention during the last decade, so far only rather simple rules are well understood. Indeed, the main focus has been on `bounded-size' rules, where all component sizes larger than some constant $B$ are treated the same way, and for more complex rules very few rigorous results are known. In this paper we study Achlioptas processes given by (unbounded) size rules such as the sum and product rules. Using a variant of the neighbourhood exploration process and branching process arguments we show that certain key statistics are tightly concentrated at least until the susceptibility (the expected size of the component containing a randomly chosen vertex) diverges. Our convergence result is most likely best possible for certain rules: in the later evolution the number of vertices in small components may not be concentrated. Furthermore, we believe that for a large class of rules the critical time where the susceptibility `blows up' coincides with the percolation threshold.

[1]  Oliver Riordan,et al.  Achlioptas processes are not always self-averaging. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Svante Janson.  Networking—Smoothly Does It , 2011, Science.

[3]  Oliver Riordan,et al.  Explosive Percolation Is Continuous , 2011, Science.

[4]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[5]  J. Spencer,et al.  Explosive Percolation in Random Networks , 2009, Science.

[6]  M. Aizenman,et al.  Sharpness of the phase transition in percolation models , 1987 .

[7]  Paul Erdös,et al.  The Giant Component 1960-1993 , 1993, Random Struct. Algorithms.

[8]  Alan M. Frieze,et al.  Ramsey games with giants , 2009, Random Struct. Algorithms.

[9]  N. Wormald.  The differential equation method for random graph processes and greedy algorithms , 1999 .

[10]  Amarjit Budhiraja,et al.  Bohman-Frieze processes at criticality and emergence of the giant component , 2011 .

[11]  O. Riordan,et al.  Achlioptas process phase transitions are continuous , 2011, 1102.5306.

[12]  N. Wormald.  Differential Equations for Random Processes and Random Graphs , 1995 .

[13]  Joel H. Spencer,et al.  Birth control for giants , 2007, Comb..

[14]  Alan M. Frieze,et al.  Avoiding a giant component , 2001, Random Struct. Algorithms.

[15]  Béla Bollobás,et al.  Handbook of large-scale random networks , 2008 .

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

[17]  Joel H. Spencer,et al.  The Bohman‐Frieze process near criticality , 2011, Random Struct. Algorithms.

[18]  Oliver Riordan,et al.  Convergence of Achlioptas Processes via Differential Equations with Unique Solutions , 2011, Combinatorics, Probability and Computing.

[19]  Oliver Riordan,et al.  Achlioptas processes can be nonconvergent , 2011 .

[20]  Colin McDiarmid,et al.  Surveys in Combinatorics, 1989: On the method of bounded differences , 1989 .

[21]  Svante Janson,et al.  Phase transitions for modified Erdős–Rényi processes , 2010, 1005.4494.

[22]  Richard M. Karp,et al.  The Transitive Closure of a Random Digraph , 1990, Random Struct. Algorithms.

[23]  Béla Bollobás,et al.  Random Graphs and Branching Processes , 2008 .

[24]  Béla Bollobás,et al.  The phase transition in inhomogeneous random graphs , 2007, Random Struct. Algorithms.

[25]  Xuan Wang,et al.  Aggregation models with limited choice and the multiplicative coalescent , 2015, Random Struct. Algorithms.

[26]  Tom Bohman,et al.  Creating a Giant Component , 2006, Comb. Probab. Comput..