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Statistical properties VS Exact answer to questions
The ðºðº(ðð,ðð) model
Properties of almost all graphs
Phase transition3
ð®ð®(ðð,ðð) Model
⢠ð®ð®(ðð, ðð) Model [Erdï¿œÌï¿œðs and Rï¿œÌï¿œðnyi1960]: V = ðð is the number of vertices, and for
and different ð¢ð¢, ð£ð£ â ðð, Pr ð¢ð¢, ð£ð£ â ðžðž = ðð.
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⢠Example. If ðð = ðððð.
Then ð¬ð¬ deg ð£ð£ = ðððððð â 1 â ðð
ðð â ðð â 1
Example: ð®ð®(ðð, 1/2)
Pr ðŸðŸ = ðð = ðð â 1ðð
12
ðð 12
ððâðð
â ðððð
12
ðð 12
ððâðð= 1
2ðððððð
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ðŸðŸ = deg(ð£ð£)
ðžðž(ðŸðŸ) = ðð/2ðððððð(ðŸðŸ) =ðð/4
Independence!
Binomial Distribution
Recall: Central Limit TheoremNormal distribution (Gauss Distribution): ðð ⌠ðð ðð,ðð2 , with density function:
ðð ð¥ð¥ =12ðððð
ððâð¥ð¥âðð 2
2ðð2 , ââ < ð¥ð¥ < +â
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As long as {ðððð} is independent identically distributed with ðžðž ðððð = ðð, ð·ð· ðððð = ðð2, then âðð=1ðð ðððð can be approximated by normal distribution (ðððð,ðððð2)when ðð is large enough.
⢠ð®ð®(ðð, 1/2)
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ðð = ððððð = ðžðž ðŸðŸ =ðð2
,
ðð2 = ðð(ððâ²)2 = ðððððð(ðŸðŸ) = ðð/4
(CLT) Near the mean, the binomial distribution is well approximated by the normal distribution.
1ðð 2ðð
ððâððâðððð 2
2ðð2 =1ðððð/2
ððâððâðð/2 2
ðð/2
It works well when ðð = Î ðð .
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⢠ð®ð®(ðð, 1/2): for any ðð > 0, the degree of each vertex almost surely is within 1 ± ðð ðð
2.
Proof. As we can approximate the distribution by
1ðððð/2
ððâððâðð/2 2
ðð/2
ðð =ðð2
,ðð =ðð
2
ðð ± ðððð = ðð2
± ðð ðð2â
1 ± ðð ðð2
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⢠ð®ð®(ðð, ðð): for any ðð > 0, if ðð is Ω ln ðððððð2
, then the degree of each vertex almost surely is within 1 ± ðð ðððð.
Proof. Omitted
ð®ð®(ðð, ðð) Model: independent set and clique
Lemma. For all integers ðð, ðð with ðð ⥠ðð ⥠2;the probability that G â ð®ð® ðð, ðð has a set of ððindependent vertices is at most
Pr ðŒðŒ ðºðº ⥠ðð †ðððð 1 â ðð
ðð2
the probability that G â ð®ð® ðð, ðð has a set of ððclique is at most
Pr ðð ðºðº ⥠ðð †ðððð ðð
ðð2
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Lemma. The expected number of ðð âcycles in G â ð®ð® ðð, ðð is ðžðž ð¥ð¥ = ðð ðð
2ðððððð.
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Proof. The expectation of certain ðð vertices ð£ð£0, ð£ð£1,⯠, ð£ð£ððâ1, ð£ð£0 form a length ðð cycle is: ðððð
The possible ways to choose ðð vertices to form a cycle ð¶ð¶ is ðð ðð
2ðð.
The expectation of the number of all cycles:
ðð = ï¿œð¶ð¶
ððð¶ð¶ =ðð ðð
2ðððððð
The ðºðº(ðð,ðð) model
Properties of almost all graphs
Phase transition12
Properties of almost all graphs
⢠For a graph property ðð , when ðð â â, If the limit of the probability of ðºðº â ð®ð®(ðð, ðð)having the property tends to â 1: we say than the property holds for almost
all (almost every / almost surely) ðºðº â ð®ð® ðð, ðð .â 0: we say than the property holds for almost
no ðºðº â ð®ð® ðð, ðð .
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Proposition. For every constant ðð â (0,1)and every graph ð»ð», almost every ðºðº â ð®ð®(ðð, ðð)contains an induced copy of H.
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ðºðº
ð»ð»
Proposition. For every constant ðð â (0,1)and every graph ð»ð», almost every ðºðº â ð®ð®(ðð, ðð)contains an induced copy of H.
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Proof. ðð ðºðº = ð£ð£0, ð£ð£1, ⊠, ð£ð£ððâ1 ,ðð = |ð»ð»|
Fix some ðð â ðð(ðºðº)ðð ïŒthen Pr ðð â ð»ð» = ðð > 0
ðð depends on ðð, ðð not on ðð.There are ðð/ðð disjoint such ðð. The probability that none of the ðºðº[ðð] is isomorphic to ð»ð» is: = 1 â ðð ðð/ðð
â0
ðð â âPr[¬(ð»ð» â ðºðº induced)]: †1 â ðð ðð/ðð
Proposition. For every constant ðð â (0,1)and ðð, ðð â ðð, almost every graph ðºðº â ð®ð®(ðð, ðð)has the property ðððð,ðð.
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ðºðº
ðð †ðððð †ðð
ð£ð£
Proposition. For every constant ðð â (0,1)and ðð, ðð â ðð, almost every graph ðºðº â ð®ð®(ðð, ðð)has the property ðððð,ðð.
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Proof. Fix ðð,ðð and ð£ð£ â ðºðº â ðð âªðð , ðð = 1 â ðð,
The probability that ðððð,ðð holds for ð£ð£: ðð ðð ðð|ðð| ⥠ðððððððð
The probability thereâs no such ð£ð£ for chosen ðð,ðð:
= 1 â ðð ðð ðð ðð ððâ ðð â|ðð|†1 â ðððððððð ððâððâðð
The upper bound for the number of different choice of ðð,ðð: ðððð+ðð
The probability there exists some ðð,ðð without suitable ð£ð£:
†ðððð+ðð 1 â ðððððððð ððâððâðð ððââ0
Colouring⢠Vertex coloring: to ðºðº = (ðð,ðžðž), a vertex
coloring is a map ðð:ðð â ðð such that ðð ð£ð£ â ðð(ð€ð€) whenever ð£ð£ and ð€ð€ are adjacent.
⢠Chromatic number ðð(ð®ð®): the smallest size of ðð.
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ðð ðºðº = 3
Colouring
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⢠Some famous resultsïŒâ Whether ðð ðºðº = ðð is NP-complete.â Every Planar graph is 4-colourable.â [Grtï¿œÌï¿œðzsch 1959]Every Planar graph not
containing a triangle is 3-colourable.
⢠Vertex coloring: to ðºðº = (ðð,ðžðž), a vertex coloring is a map ðð:ðð â ðð such that ðð ð£ð£ â ðð(ð€ð€) whenever ð£ð£ and ð€ð€ are adjacent.
⢠Chromatic number ðð(ð®ð®): the smallest size of ðð.
Proposition. For every constant ðð â (0,1) and every ðð > 0, almost every graph ðºðº â ð®ð®(ðð, ðð) has chromatic number ðð ðºðº > log(1/ðð)
2+ððâ ðððððððððð
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Proof.Pr ðŒðŒ ðºðº ⥠ðð â€
ðððððð
ðð2 †ðððððð
ðð2
= ððððlog ððlog ðð+
12ðð(ððâ1) = ðð
ðð2 â 2log ðð
log(1/ðð)+ððâ1
Take ðð = 2 + ðð log ððlog(1/ðð)
then (*) tends to â with ðð.
The size of the maximum independent set in ðºðº:ðŒðŒ(ðºðº)
⎠Pr ðŒðŒ ðºðº ⥠ððððââ
0 â
⎠ðð ðºðº >ðððð
=log 1/ðð
2 + ððâ ðð
logðð
(*)
No ðð vertices can have the same colour.
The ðºðº(ðð,ðð) model
Properties of almost all graphs
Phase transition21
Phase transitionThe interesting thing about the ð®ð®(ðð, ðð)model is that even though edges are chosen independently, certain global properties of the graph emerge from the independent choice.
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Definition. If there exists a function ðð(ðð)such that
â when limððââ
ðð1(ðð)ðð(ðð)
= 0, ð®ð® ðð,ðð1 ðð almost surely does not have the property.
â when limððââ
ðð2(ðð)ðð(ðð)
= â, ð®ð® ðð, ðð2 ðð almost surely has the property.
Then we say phase transition occurs and ðð(ðð) is the threshold.
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Phase transition
Every increasing property has a threshold.
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Phase transition
First moment methodMarkovâs Inequality: Let ð¥ð¥ be a random variable that assumes only nonnegative values. Then for all ðð > 0
Pr ð¥ð¥ ⥠ðð â€ð¬ð¬[ð¥ð¥]ðð
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First moment method : for non-negative, integer valued variable ð¥ð¥
Pr ð¥ð¥ > 0 = Pr ð¥ð¥ ⥠1 †ð¬ð¬(ð¥ð¥)⎠Pr ð¥ð¥ = 0 = 1 â Pr ð¥ð¥ > 0 ⥠1 â ð¬ð¬(ð¥ð¥)
⢠If the expectation goes to 0: the property almost surely does not happen.
⢠If the expectation does not goes to 0:e.g. Expectation = 1
ððà ðð2 + ððâ1
ððà 0 = ðð
i.e., a vanishingly small fraction of the sample contribute a lot to the expectation.
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First moment method : for non-negative , integer valued variable ð¥ð¥
Pr ð¥ð¥ > 0 = Pr ð¥ð¥ ⥠1 †ð¬ð¬(ð¥ð¥)⎠Pr ð¥ð¥ = 0 = 1 â Pr ð¥ð¥ > 0 ⥠1 â ð¬ð¬(ð¥ð¥)
Chebyshevâs Inequality
⢠For any ðð > 0,
Pr ðð â ðžðž ðð ⥠ðð â€ðððððð[ðð]ðð2
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Second moment methodTheorem. Let ð¥ð¥(ðð) be a random variable with ð¬ð¬ ð¥ð¥ > 0. If
ðððððð ð¥ð¥ = ðð ð¬ð¬2 ð¥ð¥Then ð¥ð¥ is almost surely greater than zero.
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Proof. If ð¬ð¬ ð¥ð¥ > 0, then for ð¥ð¥ †0, Pr ð¥ð¥ †0 †Pr ð¥ð¥ â ð¬ð¬ ð¥ð¥ ⥠ð¬ð¬(ð¥ð¥)
â€ðððððð ð¥ð¥ðžðž2 ð¥ð¥
â 0
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Example : Threshold for graph diameter two (two degrees of separation)
Example : Threshold for graph diameter two (two degrees of separation)
⢠Diameter: the maximum length of the shortest path between a pair of nodes.
⢠Theorem: The property that ð®ð®(ðð, ðð) has diameter two has a sharp threshold at ðð =
2 ln ðððð
.
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Example : Threshold for graph diameter two (two degrees of separation)
Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
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Proof. For any two different vertices ðð < ðð, ðŒðŒðððð = ï¿œ1 i, j âðžðž, no other vertex is adjacent to both ðð ðððððð ðð
0 ðððððððððð€ð€ðððððð
ð¥ð¥ = ï¿œðð<ðð
ðŒðŒðððð If ð¬ð¬ ð¥ð¥ððââ
0, then for large ðð, almost surely the diameter is at most two.
Example : Threshold for graph diameter two (two degrees of separation)
Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
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Proof. For any two different vertices ðð < ðð, ðŒðŒðððð = ï¿œ1 i, j âðžðž, no other vertex is adjacent to both ðð ðððððð ðð
0 ðððððððððð€ð€ðððððð
ð¬ð¬ ð¥ð¥ =ðð2
1 â ðð 1 â ðð2 ððâ2
Take ðð = ðð ln ðððð
, ðžðž(ð¥ð¥) â ðð2
21 â ðð ln ðð
ðð1 â ðð2 ln ðð
ðð
ðð
ð¥ð¥ = ï¿œðð<ðð
ðŒðŒðððð
â ðð2
2ððâðð2 ln ðð =
12ðð2âðð2
Example : Threshold for graph diameter two (two degrees of separation)
Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
33
Proof. For any two different vertices ðð < ðð, ðŒðŒðððð = ï¿œ1 i, j âðžðž, no other vertex is adjacent to both ðð ðððððð ðð
0 ðððððððððð€ð€ðððððð
ð¬ð¬ ð¥ð¥ =ðð2
1 â ðð 1 â ðð2 ððâ2
Take ðð = ðð ln ðððð
, ðð > 2, limððââ
ð¬ð¬ ð¥ð¥ = 0
ð¥ð¥ = ï¿œðð<ðð
ðŒðŒðððð
For large ðð, almost surely the diameter is at most two.
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Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
⢠Take ðð = ðð ln ðððð
, ðð > 2, limððââ
ð¬ð¬ ð¥ð¥ = 0
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Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
⢠Take ðð = ðð ln ðððð
, ðð < 2,
ð¬ð¬ ð¥ð¥2 = ð¬ð¬ ï¿œðð<ðð
ðŒðŒðððð2 If ðððððð ð¥ð¥ = ðð ð¬ð¬2 ð¥ð¥ , then for large ðð,
almost surely the diameter will be larger than two.
36
Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
⢠Take ðð = ðð ln ðððð
, ðð < 2
ð¬ð¬ ð¥ð¥2 = ð¬ð¬ ï¿œðð<ðð
ðŒðŒðððð2
= ð¬ð¬ ï¿œðð<ðð
ðŒðŒððððï¿œðð<ðð
ðŒðŒðððð = ð¬ð¬ ï¿œðð<ðððð<ðð
ðŒðŒðððð ðŒðŒðððð = ï¿œðð<ðððð<ðð
ð¬ð¬ ðŒðŒðððððŒðŒðððð
ðð = | ðð, ðð, ðð, ðð |
ð¬ð¬ ð¥ð¥2 = ï¿œðð<ðððð<ðððð=4
ð¬ð¬ ðŒðŒðððððŒðŒðððð + ï¿œðð<ðððð<ðððð=3
ð¬ð¬ ðŒðŒðððððŒðŒðððð + ï¿œðð<ðððð<ðððð=2
ð¬ð¬ ðŒðŒðððð2
37
Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
⢠Take ðð = ðð ln ðððð
, ðð < 2
ð¬ð¬ ð¥ð¥2 = ï¿œðð<ðððð<ðððð=4
ð¬ð¬ ðŒðŒðððððŒðŒðððð + ï¿œðð<ðððð<ðððð=3
ð¬ð¬ ðŒðŒðððððŒðŒðððð + ï¿œðð<ðððð<ðððð=2
ð¬ð¬ ðŒðŒðððð2
ð¬ð¬ ðŒðŒðððððŒðŒðððð †1 â ðð2 2 ððâ4 †1 â ðð2lnðððð
2ðð
1 + ðð 1 †ððâ2ðð2(1 + ðð(1))
ï¿œðð<ðððð<ðððð=4
ð¬ð¬ ðŒðŒðððððŒðŒðððð †ðð4â2ðð2(1 + ðð(1))ðð ðð ðð ðð
ð¢ð¢
38
Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
⢠Take ðð = ðð ln ðððð
, ðð < 2
ð¬ð¬ ð¥ð¥2 = ï¿œðð<ðððð<ðððð=4
ð¬ð¬ ðŒðŒðððððŒðŒðððð + ï¿œðð<ðððð<ðððð=3
ð¬ð¬ ðŒðŒðððððŒðŒðððð + ï¿œðð<ðððð<ðððð=2
ð¬ð¬ ðŒðŒðððð2
ðð ðð ðð
ð¢ð¢
39
Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
⢠Take ðð = ðð ln ðððð
, ðð < 2
ð¬ð¬ ð¥ð¥2 = ï¿œðð<ðððð<ðððð=4
ð¬ð¬ ðŒðŒðððððŒðŒðððð + ï¿œðð<ðððð<ðððð=3
ð¬ð¬ ðŒðŒðððððŒðŒðððð + ï¿œðð<ðððð<ðððð=2
ð¬ð¬ ðŒðŒðððð2
Pr ðŒðŒðððððŒðŒðððð = 1 †1 â ðð + ðð 1 â ðð 2 = 1 â 2ðð2 + ðð3 â 1 â 2ðð2
ï¿œðð<ðððð<ðð
ð¬ð¬ ðŒðŒðððððŒðŒðððð †ðð3â2ðð2
ðð ðð ðð
ð¢ð¢
ð¬ð¬ ðŒðŒðððððŒðŒðððð †1 â 2ðð2 ððâ3 = 1 â2ðð2 lnðð
ðð
ððâ3
â ððâ2ðð2 ln ðð = ððâ2ðð2
40
Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
⢠Take ðð = ðð ln ðððð
, ðð < 2
ð¬ð¬ ð¥ð¥2 = ï¿œðð<ðððð<ðððð=4
ð¬ð¬ ðŒðŒðððððŒðŒðððð + ï¿œðð<ðððð<ðððð=3
ð¬ð¬ ðŒðŒðððððŒðŒðððð + ï¿œðð<ðððð<ðððð=2
ð¬ð¬ ðŒðŒðððð2
ðžðž(ðŒðŒðððð2 ) = ðžðž(ðŒðŒðððð)
ï¿œðððð
ð¬ð¬ ðŒðŒðððð2 = ðžðž ð¥ð¥ â 12ðð2âðð2
ðð ðð
ð¢ð¢
41
Theorem. The property that ð®ð®(ðð,ðð) has diameter
two has a sharp threshold at ðð = 2 ln ðððð
⢠Take ðð = ðð ln ðððð
, ðð < 2
ð¬ð¬ ð¥ð¥2 †ð¬ð¬2(ð¥ð¥)(1 + ðð 1 )
A graph almost surely has at least one badpair of vertices and thus diameter greaterthan two.
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