explosive percolation: defused and reignited henning thomas (joint with konstantinos panagiotou,...

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)


Tree Process

Erdős-Rényi

0 n

n

# steps

L( . )

Tree


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( . )


Achlioptas Process

Erdős-RényiTree

0 n

n

# steps

L( . )


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( . )


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)



Erdős-RényiTree

Min-Product




0 n

n

# steps

L( . )

GHR(1)GHR(2)



Erdős-RényiTree

Min-Product

0 n

n

# steps

L( . )




GHR(2)GHR(3)



Erdős-RényiTree

Min-Product

0 n

n

# steps

L( . )




GHR(3)GHR(4)



Erdős-RényiTree

Min-ProductHalf-Restricted

0 n

n

# steps

L( . )




GHR(4)GHR(5)


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)


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


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)


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”


(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”


(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


(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


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


(2)

ii) steps TC + a + 1 to TC + 2a Call step successful if it connects two

components in U Assume no component has

size (1-ε)n/2. Then,

at TC + a

restricted

explosive percolation: defused and reignited henning thomas (joint with konstantinos panagiotou,...

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