Explosive Percolation: Defused and Reignited Henning Thomas (joint with Konstantinos Panagiotou,...
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Explosive Percolation:Defused and ReignitedHenning Thomas(joint with Konstantinos Panagiotou,Reto Spöhel and Angelika Steger)
Henning Thomas Explosive Percolation ETH Zurich 2011
There is s.t. whp.
Erdős-Rényi Random Graph Process
0 n
n
Erdős-Rényi
# steps
L( . )
NotationL(G): size of the largest component in G
GER(0)GER(2)GER(1)GER(4)GER(3)GER(5)
Henning Thomas Explosive Percolation ETH Zurich 2011
Tree Process
Erdős-Rényi
0 n
n
# steps
L( . )
Tree
Henning Thomas Explosive Percolation ETH Zurich 2011
Explosive Percolation
DefinitionA process P exhibits explosive percolation if there exist constantsd>0, and tc such that whp.
AlternativelyA process P exhibits explosive percolation if fP
is discontinuous.
Erdős-RényiTree
0 n
n
# steps
L( . )
Henning Thomas Explosive Percolation ETH Zurich 2011
Achlioptas Process
Erdős-RényiTree
0 n
n
# steps
L( . )
Henning Thomas Explosive Percolation ETH Zurich 2011
Achlioptas Process
Erdős-RényiTree
0 n
n
# steps
L( . )
Henning Thomas Explosive Percolation ETH Zurich 2011
Achlioptas Process
Erdős-RényiTree
0 n
n
# steps
L( . )
Henning Thomas Explosive Percolation ETH Zurich 2011
Achlioptas Process
Erdős-RényiTree
0 n
n
# steps
L( . )
Henning Thomas Explosive Percolation ETH Zurich 2011
Min-Product Rule.Always select the edge that minimizes the product of the component sizes of the endpoints.
2¢2 = 41¢3 = 3
Erdős-RényiTree
Min-Product
Achlioptas Process
0 n
n
# steps
L( . )
Henning Thomas Explosive Percolation ETH Zurich 2011
Half-Restricted Process
Erdős-RényiTree
Draw restricted vertex from n/2 vertices in smaller components
Draw unrestricted vertex from whole vertex set
Connect both vertices
Min-Product
0 n
n
# steps
L( . )
GHR(0)GHR(1)
Henning Thomas Explosive Percolation ETH Zurich 2011
Half-Restricted Process
Erdős-RényiTree
Min-Product
Draw restricted vertex from n/2 vertices in smaller components
Draw unrestricted vertex from whole vertex set
Connect both vertices
0 n
n
# steps
L( . )
GHR(1)GHR(2)
Henning Thomas Explosive Percolation ETH Zurich 2011
Half-Restricted Process
Erdős-RényiTree
Min-Product
0 n
n
# steps
L( . )
Draw restricted vertex from n/2 vertices in smaller components
Draw unrestricted vertex from whole vertex set
Connect both vertices
GHR(2)GHR(3)
Henning Thomas Explosive Percolation ETH Zurich 2011
Half-Restricted Process
Erdős-RényiTree
Min-Product
0 n
n
# steps
L( . )
Draw restricted vertex from n/2 vertices in smaller components
Draw unrestricted vertex from whole vertex set
Connect both vertices
GHR(3)GHR(4)
Henning Thomas Explosive Percolation ETH Zurich 2011
Half-Restricted Process
Erdős-RényiTree
Min-ProductHalf-Restricted
0 n
n
# steps
L( . )
Draw restricted vertex from n/2 vertices in smaller components
Draw unrestricted vertex from whole vertex set
Connect both vertices
GHR(4)GHR(5)
Henning Thomas Explosive Percolation ETH Zurich 2011
Introduction Summary
Erdős-Rényi Process Not Explosive Tree Process Explosive (d =
1) Min-Product-Rule
Explosive??? Draw 2 edges and keep the one that
minimizes the product of the comp. sizes Half-Restricted Process
Explosive??? 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)
Henning Thomas Explosive Percolation ETH 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
Henning Thomas Explosive Percolation ETH Zurich 2011
The Half-Restricted Process
Define TC 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)
Henning Thomas Explosive Percolation ETH Zurich 2011
Observations Up to TC chunks cannot be merged. There are at most n/ln ln n chunks.
Definitions A1, A2, ... chunks in order of
appearance E1, E2, ... events that chunk Ai
has size in GHR(TC)
(1)
“chunk”
Henning Thomas Explosive Percolation ETH Zurich 2011
(1)
In every step a chunk can grow byat most ln ln n.
For Ei to occur, chunk Ai needs to be“hit” by the unrestricted vertexat least times.
… Technical details (essentially
Coupon Collector concentration)
Union Bound:
“chunk”
Henning Thomas Explosive Percolation ETH Zurich 2011
(2)
2 parts:set a := n/(2 ln ln ln n)
i) steps TC to TC + a collect enough vertices in components of size at least ln ln n
ii) steps TC + a + 1 to TC + 2a build a giant on these vertices
Henning Thomas Explosive Percolation ETH Zurich 2011
(2)
i) steps TC to TC + 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) stepswe have by Chernoff whp. a gainof Ω(n/ln ln ln n) vertices.
at TC
restricted
goal atTC + a
Henning Thomas Explosive Percolation ETH Zurich 2011
i) steps TC to TC + 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) stepswe have by Chernoff whp. a gainof Ω(n/ln ln ln n) vertices.
(2)
at TC + a
restricted