Explosive Percolation : Defused and Reignited

1. Explosive Percolation:DefusedandReignited Henning Thomas (jointwith Konstantinos Panagiotou,Reto Spöheland Angelika Steger) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

2. Erdős-Rényi Random Graph Process n L( . ) GER(0) GER(2) GER(1) GER(4) GER(3) GER(5) Notation L(G): sizeofthelargestcomponent in G 0 # steps n Erdős-Rényi Thereis s.t. whp.

3. TreeProcess n L( . ) 0 # steps n Erdős-Rényi Tree

4. Explosive Percolation n • DefinitionA processP exhibitsexplosive percolationifthereexistconstantsd>0, andtc such thatwhp. • AlternativelyA processPexhibitsexplosive percolationiffPisdiscontinuous. L( . ) 0 # steps n Erdős-Rényi Tree

5. AchlioptasProcess n L( . ) 0 # steps n Erdős-Rényi Tree

6. AchlioptasProcess n L( . ) 0 # steps n Erdős-Rényi Tree

7. AchlioptasProcess n L( . ) 0 # steps n Erdős-Rényi Tree

8. AchlioptasProcess n L( . ) 0 # steps n Erdős-Rényi Tree

9. AchlioptasProcess n 2¢2 = 4 1¢3 = 3 L( . ) Min-ProductRule. Alwaysselecttheedgethatminimizestheproductofthecomponentsizesoftheendpoints. 0 # steps n Erdős-Rényi Tree Min-Product

10. Half-RestrictedProcess n L( . ) GHR(0) GHR(1) • Draw restrictedvertexfromn/2vertices in smallercomponents • Draw unrestrictedvertexfromwholevertexset • Connect bothvertices 0 # steps n Erdős-Rényi Tree Min-Product

11. Half-RestrictedProcess n L( . ) GHR(1) GHR(2) • Draw restrictedvertexfromn/2verticesin smallercomponents • Draw unrestrictedvertexfromwholevertexset • Connect bothvertices 0 # steps n Erdős-Rényi Tree Min-Product

12. Half-RestrictedProcess n L( . ) GHR(2) GHR(3) • Draw restrictedvertexfromn/2verticesin smallercomponents • Draw unrestrictedvertexfromwholevertexset • Connect bothvertices 0 # steps n Erdős-Rényi Tree Min-Product

13. Half-RestrictedProcess n L( . ) GHR(3) GHR(4) • Draw restrictedvertexfromn/2verticesin smallercomponents • Draw unrestrictedvertexfromwholevertexset • Connect bothvertices 0 # steps n Erdős-Rényi Tree Min-Product

14. Half-RestrictedProcess n L( . ) GHR(4) GHR(5) • Draw restrictedvertexfromn/2verticesin smallercomponents • Draw unrestrictedvertexfromwholevertexset • Connect bothvertices 0 # steps n Erdős-Rényi Tree Min-Product Half-Restricted

15. Introduction Summary • Erdős-RényiProcess Not Explosive • TreeProcess Explosive (d = 1) • Min-Product-Rule Explosive??? • Draw 2 edgesandkeeptheonethatminimizestheproductofthecomp.sizes • Half-RestrictedProcess Explosive??? • Connect a restrictedvertexwithan unrestrictedvertex Not Explosive Explosive Achlioptas, D’Souza, Spencer (2009) Theorem (Riordan, Warnke, 2011), simplified. NoAchlioptasProcesscanexhibitexplosive percolation. Theorem (Panagiotou, Spöhel, Steger, T., 2011), simplified. The Half-RestrictedProcessexhibitsexplosive percolation.

16. One Main Difference • In everyAchlioptasProcess: • Probabilitytoinsert an edgewithinSisat least • In Half-RestrictedProcess: • Probabilitytoinsert an edgewithinSis0aslongas

17. The Half-RestrictedProcess • DefineTCasthelast stepin whichtherestrictedvertexisdrawnfromcomponentsofsizesmallerthanlnln n. Theorem (Panagiotou, Spöhel, Steger, T., 2011) Foreveryε>0theHalf-RestrictedProcesswhp.satisfies and

18. (1) Observations • UptoTCchunkscannotbemerged. • Thereareatmostn/lnlnnchunks. Definitions • A1, A2, ...chunks in orderofappearance • E1, E2, ... eventsthatchunkAihassize in GHR(TC) “chunk”

19. (1) • In every step a chunk can grow byat most lnlnn. • For Ei to occur, chunk Ai needs to be“hit” by the unrestricted vertexat least times. • … • Technical details (essentiallyCoupon Collector concentration) • Union Bound: “chunk”

20. (2) • 2 parts:seta := n/(2 lnlnlnn) i) stepsTCtoTC + acollectenoughvertices in componentsofsizeat least lnlnn ii) stepsTC + a + 1toTC + 2abuild a giant on thesevertices

21. (2) i) stepsTCtoTC + a • Probabilitytoincreasethenumberofverticesin componentsofsize≥lnlnnisat least • Withina=θ(n/lnlnlnn) stepswehavebyChernoffwhp. a gainofΩ(n/lnlnlnn)vertices. atTC goalatTC + a restricted

22. (2) i) stepsTCtoTC + a • Probabilitytoincreasethenumberofverticesin componentsofsize≥lnlnnisat least • Withina=θ(n/lnlnlnn) stepswehavebyChernoffwhp. a gainofΩ(n/lnlnlnn)vertices. atTC + a restricted

23. (2) ii) stepsTC + a + 1toTC + 2a • Call stepsuccessfulifitconnectstwocomponents in U • Assumenocomponenthassize(1-ε)n/2. Then, atTC + a restricted