1. Poisson Cloning Model for Random Graphs
Jeong Han Kim
Yonsei University

2. ~ Random Graph G(n,p) - Emergence of Giant (connected) Component
- Component and Branching process
- Motivation for a new model
Poisson Cloning Model GPC(n,p)
- Definition
- Theorem: GPC (n,p) G(n,p)
Cut-Off Line Algorithm (COLA)
- Generating GPC (n,p)
and simultaneously solving the problem
New Results obtained using PCM and COLA ~

3. ( ) Random Graph G(n,p) Each of edges is in G(n,p) with probability p,
independently of all others
P=1 : complete graph
P=0 : empty graph ) n 2 (

4. ( ) Expected number of edges p For fixed graph G with m edges,
Pr[ G(n,p) =G ] = pm(1-p) -m . ) n 2 ( ) n 2 (

5. For example
Pr[ G(n,p) = G ]= p5(1-p)21-5 b a e g f c d

6. (Connected) Componenta b d c e f g

7. Component containing a vertexb a e f g c d

8. Component containing a vertex in G(n,p)
Pr[ Bin(n-1,p)= k ] = pk(1-p)n-1-k a a
Bin(n-1,p) ……
Bin(n-g1,p)
Bin(n-g1,p)
Bin(n-3,p)
Bin(n-3,p) )
n-1 k (

9. Emergence of Giant Component
Erdős & Rényi (’60,’61):
In G(n,p) with pn= λ for a constant λ
there is a giant component
if and only if λ> 1

10. Emergence of Giant Component
Erdős & Rényi (’60,’61):
For the size W=W(n,p) of a largest component of G(n,p),
< cλlog n if pn →λ <1
W =
θλ n if pn →λ >1,
where θλ is the positive solution for
1-θ-e-θ λ =0 {

11. the giant component emerges
Łuczak (’90)
In G(n,p) with pn = λ,
improving λ-1 >> n-1/3 log n due to
Bollobás (’84).
the giant component emerges
when λ-1 >> n-1/3

12. and for the size W’ of a second largest component
Łuczak (’90)
In G(n,p) with ε:= λ-1>> n-1/3
(λ=pn),
W = (2+O(ε)) ε n
and for the size W’ of a second
largest component
W’ = Θ ( ε-2log (ε3n)) << ε n

13. Łuczak (’90)
Theorem (supercritical region) Let λ:=pn= 1+ε with ε>>n-1/3. Then with probability 1-O((ε3n)-1/9),
|W(n,p) - θλ n| ≤ 0.2 n2/3.

14. Emergence of Giant Component
Erdős & Rényi (’60,’61):
In G(n,p) with pn= λ for a constant λ
Why 1 ?
There is a giant component
if and only if λ> 1.

15. Component containing a vertex in G(n,p)
Bin(n-1,p)
Bin(n-g1,p) ……
For λ=p(n-1)
Poi(λ)
Poi(λ)
Poi(λ) λk k!
Pr[ Poi(λ)= k ] = e-λ

16. Poisson Branching Process……
If λ<1, the process dies out with probability 1
If λ>1, the process survives forever with probability θλ (recall θλ is the positive solution for 1-θ-e-θ λ =0).

17. ~ Two Obstacles The difference between Bin(n-1, p) and Poi(λ).
Bin(n-g1, p)
Bin(n-g1,p) …… a
Poi(λ) …… ~
The difference between
Bin(n-1, p) and Poi(λ).
The drift : p(n-gi) keeps decreasing

18. Similar but more serious obstacles occur in the analyses of
The k-core problem of the random graph
(Pittel-Spencer-Wormald, …)
Giant strong component of the random directed graph (Karp,…)
Pure literal algorithm for the random satisfiability problem (Broder-Frieze-Upfal, …)
Unit clause algorithm for the random satisfiability problem (Chao-Franco,…)
Karp-Sipser Algorithm to find a large matching in the random graph (Karp-Sipser, …)

19. ~ The First Obstacle The difference between Bin(n-1, p) and Poi(λ).
More generally, is there a random
graph model Gnew(n,p), in which all
degrees are i.i.d Poi(λ) and
Gnew(n,p) G(n,p)?
NO: The sum of degrees must be even.
YES in an asymptotic sense ~

20. Is it possible to define
a random graph model
in which all degrees
are i.i.d Poi(λ)?
YES
HOW?
Just Do It!

21. Poisson Cloning Model GPC(n,p): Definition
Take i.i.d Poisson random variables d(v)'s, v in V, with mean λ= p(n-1).
For each vertex v in V, take d(v) copies, or clones, of v.
If Σ d(v) is even, generate a uniform random perfect matching on all clones and then contract clones of the same vertex.

22. V1 V2 V3 V4 V5 V6 V7 V8 V9
RandomTwo

23. If Σd(v) is odd, generate a perfect matching excluding a clone
If Σd(v) is odd, generate a perfect matching excluding a clone. The excluded clone induces a loop.
(When d(v)=d for all v, this is the configuration model for random regular graphs due to B. Bolloás ('84).)

24. G(n,p) GPC(n,p) ~ Theorem (K) If pn = O(1), then, for
any event A regarding G(n,p),
Pr [A] ≥ c1PrPC [A] - e-Ω(pn^2)
and
Pr [A] ≤ c2 PrPC [A]1/2 + e-Ω(pn^2)
In particular,
Pr [A] → 0 iff PrPC [A] → 0 and
Pr [A] → 1 iff PrPC [A] → 1

25. The Second Obstacle The drift : p(n-gi) keeps decreasing We introduce
the cut-off line algorithm
(COLA)

26. Poisson λ-cell
For each clone w, assign a uniform random (real) number between
0 and λ, independently of all others.
A clone is larger than another clone if so are the assigned numbers.

27. λ V1 V2 V3 V4 V5 V6 V7 V8 V9
Assign numbers
Poisson λ-Cell

28. A way to generate the random perfect matching
Take two largest clones and match them. Repeat this for the remaining unmatched clones

29. λ V1 V2 V3 V4 V5 V6 V7 V8 V9
Match Largest two

30. Cut-Off Line Algorithm (COLA)
Choose the first unmatched clone
according to a certain selection rule
independently of assigned numbers
and
match it to the largest
unmatched clone

31. λ V1 V2 V3 V4 V5 V6 V7 V8 V9
Arbitrary and largest Take the first clone according to a certain selection rule. Then move the clock. a

32. λ´ V1 V2 V3 V4 V5 V6 V7 V8 V9
And find the largest clone.

33. Cut-Off Line Algorithm (COLA)
Initially, the cut-off value Λ = λ. The cut-off line is the vertical line containing (Λ,0).
After one step, the cut-off value Λ = λ´. The cut-off line is the vertical line containing (Λ,0).
Notice that λ´ is the largest number
among N-1 independent uniform
random numbers between 0 and λ.

34. Cut-Off Line Algorithm for Component

35. λ V1 V2 V3 V4 V5 V6 V7 V8 V9
Select a light clone.

36. λ' V1 V2 V3 V4 V5 V6 V7 V8 V9
Match it to the largest

37. λ' V1 V2 V3 V4 V5 V6 V7 V8 V9

38. λ" V1 V2 V3 V4 V5 V6 V7 V8 V9

39. New Results

40. λ V1 V2 V3 V4 V5 V6 V7 V8 V9
Heavy and light clones

41. k= λ V1 V2 V3 V4 V5 V6 V7 V8 V9
Select a light clone.

42. λ' V1 V2 V3 V4 V5 V6 V7 V8 V9

43. λ' V1 V2 V3 V4 V5 V6 V7 V8 V9

44. λ" V1 V2 V3 V4 V5 V6 V7 V8 V9

45. λ" V1 V2 V3 V4 V5 V6 V7 V8 V9

46. λ" V1 V2 V3 V4 V5 V6 V7 V8 V9

47. Number of light clones k=3, λ=3.3 k=3, λ=3.4

48. k=3, λ=3.35