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Understanding Explosive Percolation in Erdős-Rényi Models

This paper explores the phenomenon of explosive percolation in Erdős-Rényi random graph processes. We define a process as exhibiting explosive percolation if it has discontinuity in the size of the largest component. Through our analysis of various processes, including tree processes and the Achlioptas process, we demonstrate the critical conditions under which explosive percolation occurs. Our findings highlight the importance of selection rules, like the Min-Product Rule and Half-Restricted Process, in influencing the percolative behavior of graphs in stochastic models.

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Understanding Explosive Percolation in Erdős-Rényi Models

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  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

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