Will Perkins NYU joint work with Joel Spencer and Mihyun Kang ...

The Bohman-Frieze
Process
Rutgers, 2/22/2011
Will Perkins
NYU
joint work with Joel Spencer and Mihyun Kang

The Erdős–Rényi Random
Graph Process
• Begin with an empty graph on n vertices
• At each step, add a uniformly random edge to the graph
• The graph ‘evolves’ from empty to full

Random Graph Evolution

2/10/2011
The Erdős-Rényi Phase Transition
2.0
• At step
1.5
cn
2 , c1, |C1| = Θ(n)
|C1|
n
1.0 n
2/
Achlioptas Processes
1
0.5
The2 Erdős-Rényi · Edges Ph
t = An Achlioptas Process:
0.0 2 · Edgesn
0.0 0.5 1.0 1.5 2.0
t =
|C1|
0 1
n
•
|C1|
Size of smaller components
Achlioptas Processes n
• Inside the critical window: t =1+λn
• Structure Dimitris of components
Achlioptas asked: “Can the power of two choices in a random graph
process delay the critical point?”
An Achlioptas Process:
2 · Edges
t = −1/3
Erdős-Rényi time: t =
• At each step, random select t
• Add one of the two edges to
• Each rule corresponds to a ra
e1
#ofEdges
. n/2
G(t) ∼ G(n, t
n )
The Erdős-Rényi
Erdős-Rényi time: t = #ofE
n/
G(t) ∼ G(n, t
n )
optas Processes
Achlioptas asked: “Can the power of two choices in a random graph
elay the critical point?”
chlioptas Process:
Wil
Dimitris Achlioptas slope = 2 asked: “Can th
process delay the critical point?”
• Begin with an empty graph o
Achlioptas Processes

Achlioptas Processes
Can we delay the giant component by
allowing two choices at each step?
• Start with empty graph on n vertices
• At each step, select two random edges
• Add one of the edges according to some rule

Achlioptas Processes
• Start with empty graph on n vertices
• At each step, select two random edges
• Add one of the edges according to some rule

Achlioptas Processes
• Start with empty graph on n vertices
• At each step, select two random edges
• Add one of the edges according to some rule

Achlioptas Processes
• Start with empty graph on n vertices
• At each step, select two random edges
• Add one of the edges according to some rule

Achlioptas Processes
• Start with empty graph on n vertices
• At each step, select two random edges
• Add one of the edges according to some rule

Achlioptas Processes
• Start with empty graph on n vertices
• At each step, select two random edges
• Add one of the edges according to some rule

Achlioptas Processes
• Start with empty graph on n vertices
• At each step, select two random edges
• Add one of the edges according to some rule

as Processes
The Bohman-Frieze
2 · Edges
t = t =1+λn −1/3
Yes we can!
n
Process
t =1+λn −1/3
e1
hlioptas Processes
The Rule:
e2
itris Achlioptas asked: “Can the power of two choices in a r
• If the first edge would join
two isolated vertices, add it
cess delay the critical point?”
An Achlioptas
•
Process:
Otherwise, add the second
optas asked: “Can the power of two choices in a random graph
e1
e2

The Bohman-Frieze
The Rule:
Process
• If the first edge would join
two isolated vertices, add it
• Otherwise, add the second

as Processes
2 · Edges
t = t =1+λn
n
−1/3
The Bohman-Frieze
t =1+λn −1/3
e1
hlioptas Processes
The Rule:
Process
e2
itris Achlioptas asked: “Can the power of two choices in a r
• If the first edge would join
two isolated vertices, add it
cess delay the critical point?”
An Achlioptas
•
Process:
Otherwise, add the second close enough!
optas asked: “Can the power of two choices in a random graph
e1
e2
Shorthand for a larger
class of processes:
‘Bounded-Size Rules’
Not really the process
from their result, but

m quantities in the random graph process are
their expectation. The expectation is a deter-
system of ODE’s.
olated vertices at step j of the Bohman-Frieze
1.0
1(j)
n
= x1(t)+o(1)
2
0.8
0.6
0.4
0.2
0.0 0.5 1.0 1.5 2.0
t

Susceptibility
Average component size of a
randomly chosen vertex

Susceptibility
Average component size of a
randomly chosen vertex

Susceptibility
Average component size of a
randomly chosen vertex

Susceptibility
Average component size of a
randomly chosen vertex

Breadth-First
Search of C(v):
Why Susceptibility?
v

With susceptibility
we can ‘skip ahead’:
Why Susceptibility?
v

With susceptibility
we can ‘skip ahead’:
Why Susceptibility?
v
Offspring
distributed like
component size of a
randomly chosen
vertex

With susceptibility
we can ‘skip ahead’:
Why Susceptibility?
v
Offspring
distributed like
component size of a
randomly chosen
vertex
Mean = Susceptibility
2nd moment also
follows an ODE

Size of Second-Largest
Component Not a monotone
property

Other Rules?
• Bounded-size rules are in Erdős–Rényi Universality Class:
same qualitative behavior, different constants
• Is different behavior possible?
• In terms of growth of giant, ?
• Candidate processes: Least Product Rule / Least Sum Rule

|C1|
Other
2 · Edges
n Rules?
t =
n
t =1+λn −1/3
• Bounded-size rules are in Erdős–Rényi Universality Class:
2 · Edges
same qualitative t = behavior, different constants
• Is different behavior n possible?
• In terms of growth of giant, ?
• Candidate processes: Least Product Rule / Least Sum Rule
t =1+λn −1/3
e1
Numerical Evidence presented in ‘Explosive Percolation in
Random Networks’ (Achlioptas/D’Souza/Spencer ’09)
chlioptas Processes
e2
e1
e2

Product Rule
• Where is the critical point?
• Is the phase transition continuous?
• How big is the second-largest super-critical component?

Product Rule
• Where is the critical point?
• Is the phase transition continuous?
• How big is the second-largest super-critical component?
? infinite
slope?

Thank You!