1. Explosive Percolation: Defused and Reignited Henning Thomas (joint with Konstantinos Panagiotou, Reto Spöhel and Angelika Steger) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

2. Henning ThomasExplosive PercolationETH Zurich 2011 There is s.t. whp. Erdős-Rényi Random Graph Process 0n n Erdős-Rényi # steps L(. ) Notation L(G): size of the largest component in G G ER (0)G ER (2)G ER (1)G ER (4)G ER (3)G ER (5)

3. Henning ThomasExplosive PercolationETH Zurich 2011 Tree Process Erdős-Rényi 0n n # steps L(. ) Tree

4. Henning ThomasExplosive PercolationETH Zurich 2011 Explosive Percolation  Definition A process P exhibits explosive percolation if there exist constants d>0, and t c such that whp.  Alternatively A process P exhibits explosive percolation if f P is discontinuous. Erdős-Rényi Tree 0n n # steps L(. )

5. Henning ThomasExplosive PercolationETH Zurich 2011 Achlioptas Process Erdős-Rényi Tree 0n n # steps L(. )

6. Henning ThomasExplosive PercolationETH Zurich 2011 Achlioptas Process Erdős-Rényi Tree 0n n # steps L(. )

7. Henning ThomasExplosive PercolationETH Zurich 2011 Achlioptas Process Erdős-Rényi Tree 0n n # steps L(. )

8. Henning ThomasExplosive PercolationETH Zurich 2011 Achlioptas Process Erdős-Rényi Tree 0n n # steps L(. )

9. Henning ThomasExplosive PercolationETH Zurich 2011 Min-Product Rule. Always select the edge that minimizes the product of the component sizes of the endpoints. 2¢2 = 4 1¢3 = 3 Erdős-Rényi Tree Min-Product Achlioptas Process 0n n # steps L(. )

10. Henning ThomasExplosive PercolationETH Zurich 2011 Half-Restricted Process Erdős-Rényi Tree  Draw restricted vertex from n/2 vertices in smaller components  Draw unrestricted vertex from whole vertex set  Connect both vertices Min-Product 0n n # steps L(. ) G HR (0)G HR (1)

11. Henning ThomasExplosive PercolationETH Zurich 2011 Half-Restricted Process Erdős-Rényi Tree Min-Product  Draw restricted vertex from n/2 vertices in smaller components  Draw unrestricted vertex from whole vertex set  Connect both vertices 0n n # steps L(. ) G HR (1)G HR (2)

12. Henning ThomasExplosive PercolationETH Zurich 2011 Half-Restricted Process Erdős-Rényi Tree Min-Product 0n n # steps L(. )  Draw restricted vertex from n/2 vertices in smaller components  Draw unrestricted vertex from whole vertex set  Connect both vertices G HR (2)G HR (3)

13. Henning ThomasExplosive PercolationETH Zurich 2011 Half-Restricted Process Erdős-Rényi Tree Min-Product 0n n # steps L(. )  Draw restricted vertex from n/2 vertices in smaller components  Draw unrestricted vertex from whole vertex set  Connect both vertices G HR (3)G HR (4)

14. Henning ThomasExplosive PercolationETH Zurich 2011 Half-Restricted Process Erdős-Rényi Tree Min-Product Half-Restricted 0n n # steps L(. )  Draw restricted vertex from n/2 vertices in smaller components  Draw unrestricted vertex from whole vertex set  Connect both vertices G HR (4)G HR (5)

15. Henning ThomasExplosive PercolationETH Zurich 2011 Half-Restricted Process Erdős-Rényi Tree  Parameter, say  Draw restricted vertex from vertices in smaller components  Draw unrestricted vertex from whole vertex set  Connect both vertices Min-Product 0n n # steps L(. ) Half-Restricted 0.5 0.9

16. Henning ThomasExplosive PercolationETH Zurich 2011 Half-Restricted Process Erdős-Rényi Tree  Parameter, say  Draw restricted vertex from vertices in smaller components  Draw unrestricted vertex from whole vertex set  Connect both vertices Min-Product 0n n # steps L(. ) Half-Restricted 0.5 0.9 0.25

17. Henning ThomasExplosive PercolationETH Zurich 2011 Introduction Summary  Erdős-Rényi ProcessNot Explosive  Tree ProcessExplosive (d = 1)  Min-Product-RuleExplosive???  Draw 2 edges and keep the one that minimizes the product of the comp. sizes  Half-Restricted ProcessExplosive???  Connect a restricted vertex with an unrestricted vertex Theorem (Riordan, Warnke, 2011), simplified. No Achlioptas Process can exhibit explosive percolation. Theorem (Panagiotou, Spöhel, Steger, T., 2011), simplified. The Half-Restricted Process exhibits explosive percolation. Not Explosive Explosive Achlioptas, D’Souza, Spencer (2009)

18. Henning ThomasExplosive PercolationETH Zurich 2011 One Main Difference  In every Achlioptas Process:  Probability to insert an edge within S is at least  In Half-Restricted Process:  Probability to insert an edge within S is 0 as long as

19. Henning ThomasExplosive PercolationETH Zurich 2011 Observation Achlioptas Processes

20. Henning ThomasExplosive PercolationETH Zurich 2011 Achlioptas Processes  As long as no comp. in S has size, the probability to connect two components in S is at least   Choose, then Observation „successful“

21. Henning ThomasExplosive PercolationETH Zurich 2011  Probability that a component in S survives t 2 n steps is at least Achlioptas Processes Observation 2

22. Henning ThomasExplosive PercolationETH Zurich 2011 “Powder Keg” Intuition

23. Henning ThomasExplosive PercolationETH Zurich 2011 The Half-Restricted Process  Define T C as the last step in which the restricted vertex is drawn from components of size smaller than ln ln n. Theorem (Panagiotou, Spöhel, Steger, T., 2011) For every ε>0 the Half-Restricted Process whp. satisfies (1) and (2)

24. Henning ThomasExplosive PercolationETH Zurich 2011 Observations  Up to T C chunks cannot be merged.  There are at most n/ln ln n chunks. Definitions  A 1, A 2,... chunks in order of appearance  E 1, E 2,... events that chunk A i has size in G HR (T C ) (1) “chunk”

25. Henning ThomasExplosive PercolationETH Zurich 2011 (1)  In every step a chunk can grow by at most ln ln n.  For E i to occur, chunk A i needs to be “hit” by the unrestricted vertex at least times.   ……  Technical details (essentially Coupon Collector concentration)  Union Bound: “chunk”

26. Henning ThomasExplosive PercolationETH Zurich 2011 (2)  2 parts: set a := n/(2 ln ln ln n) i) steps T C to T C + a collect enough vertices in components of size at least ln ln n ii) steps T C + a + 1 to T C + 2a build a giant on these vertices

27. Henning ThomasExplosive PercolationETH Zurich 2011 (2) i) steps T C to T C + a  Probability to increase the number of vertices in components of size ≥ln ln n is at least  Within a=θ(n/ln ln ln n) steps we have by Chernoff whp. a gain of Ω(n/ln ln ln n) vertices. at T C restricted goal at T C + a

28. Henning ThomasExplosive PercolationETH Zurich 2011 i) steps T C to T C + a  Probability to increase the number of vertices in components of size ≥ln ln n is at least  Within a=θ(n/ln ln ln n) steps we have by Chernoff whp. a gain of Ω(n/ln ln ln n) vertices. (2) at T C + a restricted

29. Henning ThomasExplosive PercolationETH Zurich 2011 (2) ii) steps T C + a + 1 to T C + 2a  Call step successful if it connects two components in U  Assume no component has size (1-ε)n/2. Then,  at T C + a restricted