introduction to random graphs - sjtulonghuan/teaching/cs499/19.pdfย ยท properties of almost all...
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Statistical properties VS Exact answer to questions
The ๐บ๐บ(๐๐,๐๐) model
Properties of almost all graphs
Phase transition3
๐ฎ๐ฎ(๐๐,๐๐) Model
โข ๐ฎ๐ฎ(๐๐, ๐๐) Model [Erd๏ฟฝฬ๏ฟฝ๐s and R๏ฟฝฬ๏ฟฝ๐nyi1960]: V = ๐๐ is the number of vertices, and for
and different ๐ข๐ข, ๐ฃ๐ฃ โ ๐๐, Pr ๐ข๐ข, ๐ฃ๐ฃ โ ๐ธ๐ธ = ๐๐.
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โข Example. If ๐๐ = ๐๐๐๐.
Then ๐ฌ๐ฌ deg ๐ฃ๐ฃ = ๐๐๐๐๐๐ โ 1 โ ๐๐
๐๐ โ ๐๐ โ 1
Example: ๐ฎ๐ฎ(๐๐, 1/2)
Pr ๐พ๐พ = ๐๐ = ๐๐ โ 1๐๐
12
๐๐ 12
๐๐โ๐๐
โ ๐๐๐๐
12
๐๐ 12
๐๐โ๐๐= 1
2๐๐๐๐๐๐
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๐พ๐พ = deg(๐ฃ๐ฃ)
๐ธ๐ธ(๐พ๐พ) = ๐๐/2๐๐๐๐๐๐(๐พ๐พ) =๐๐/4
Independence!
Binomial Distribution
Recall: Central Limit TheoremNormal distribution (Gauss Distribution): ๐๐ โผ ๐๐ ๐๐,๐๐2 , with density function:
๐๐ ๐ฅ๐ฅ =12๐๐๐๐
๐๐โ๐ฅ๐ฅโ๐๐ 2
2๐๐2 , โโ < ๐ฅ๐ฅ < +โ
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As long as {๐๐๐๐} is independent identically distributed with ๐ธ๐ธ ๐๐๐๐ = ๐๐, ๐ท๐ท ๐๐๐๐ = ๐๐2, then โ๐๐=1๐๐ ๐๐๐๐ can be approximated by normal distribution (๐๐๐๐,๐๐๐๐2)when ๐๐ is large enough.
โข ๐ฎ๐ฎ(๐๐, 1/2)
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๐๐ = ๐๐๐๐๐ = ๐ธ๐ธ ๐พ๐พ =๐๐2
,
๐๐2 = ๐๐(๐๐โฒ)2 = ๐๐๐๐๐๐(๐พ๐พ) = ๐๐/4
(CLT) Near the mean, the binomial distribution is well approximated by the normal distribution.
1๐๐ 2๐๐
๐๐โ๐๐โ๐๐๐๐ 2
2๐๐2 =1๐๐๐๐/2
๐๐โ๐๐โ๐๐/2 2
๐๐/2
It works well when ๐๐ = ฮ ๐๐ .
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โข ๐ฎ๐ฎ(๐๐, 1/2): for any ๐๐ > 0, the degree of each vertex almost surely is within 1 ยฑ ๐๐ ๐๐
2.
Proof. As we can approximate the distribution by
1๐๐๐๐/2
๐๐โ๐๐โ๐๐/2 2
๐๐/2
๐๐ =๐๐2
,๐๐ =๐๐
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๐๐ ยฑ ๐๐๐๐ = ๐๐2
ยฑ ๐๐ ๐๐2โ
1 ยฑ ๐๐ ๐๐2
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โข ๐ฎ๐ฎ(๐๐, ๐๐): for any ๐๐ > 0, if ๐๐ is ฮฉ ln ๐๐๐๐๐๐2
, then the degree of each vertex almost surely is within 1 ยฑ ๐๐ ๐๐๐๐.
Proof. Omitted
๐ฎ๐ฎ(๐๐, ๐๐) Model: independent set and clique
Lemma. For all integers ๐๐, ๐๐ with ๐๐ โฅ ๐๐ โฅ 2;the probability that G โ ๐ฎ๐ฎ ๐๐, ๐๐ has a set of ๐๐independent vertices is at most
Pr ๐ผ๐ผ ๐บ๐บ โฅ ๐๐ โค ๐๐๐๐ 1 โ ๐๐
๐๐2
the probability that G โ ๐ฎ๐ฎ ๐๐, ๐๐ has a set of ๐๐clique is at most
Pr ๐๐ ๐บ๐บ โฅ ๐๐ โค ๐๐๐๐ ๐๐
๐๐2
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Lemma. The expected number of ๐๐ โcycles in G โ ๐ฎ๐ฎ ๐๐, ๐๐ is ๐ธ๐ธ ๐ฅ๐ฅ = ๐๐ ๐๐
2๐๐๐๐๐๐.
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Proof. The expectation of certain ๐๐ vertices ๐ฃ๐ฃ0, ๐ฃ๐ฃ1,โฏ , ๐ฃ๐ฃ๐๐โ1, ๐ฃ๐ฃ0 form a length ๐๐ cycle is: ๐๐๐๐
The possible ways to choose ๐๐ vertices to form a cycle ๐ถ๐ถ is ๐๐ ๐๐
2๐๐.
The expectation of the number of all cycles:
๐๐ = ๏ฟฝ๐ถ๐ถ
๐๐๐ถ๐ถ =๐๐ ๐๐
2๐๐๐๐๐๐
The ๐บ๐บ(๐๐,๐๐) model
Properties of almost all graphs
Phase transition12
Properties of almost all graphs
โข For a graph property ๐๐ , when ๐๐ โ โ, If the limit of the probability of ๐บ๐บ โ ๐ฎ๐ฎ(๐๐, ๐๐)having the property tends to โ 1: we say than the property holds for almost
all (almost every / almost surely) ๐บ๐บ โ ๐ฎ๐ฎ ๐๐, ๐๐ .โ 0: we say than the property holds for almost
no ๐บ๐บ โ ๐ฎ๐ฎ ๐๐, ๐๐ .
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Proposition. For every constant ๐๐ โ (0,1)and every graph ๐ป๐ป, almost every ๐บ๐บ โ ๐ฎ๐ฎ(๐๐, ๐๐)contains an induced copy of H.
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๐บ๐บ
๐ป๐ป
Proposition. For every constant ๐๐ โ (0,1)and every graph ๐ป๐ป, almost every ๐บ๐บ โ ๐ฎ๐ฎ(๐๐, ๐๐)contains an induced copy of H.
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Proof. ๐๐ ๐บ๐บ = ๐ฃ๐ฃ0, ๐ฃ๐ฃ1, โฆ , ๐ฃ๐ฃ๐๐โ1 ,๐๐ = |๐ป๐ป|
Fix some ๐๐ โ ๐๐(๐บ๐บ)๐๐ ๏ผthen Pr ๐๐ โ ๐ป๐ป = ๐๐ > 0
๐๐ depends on ๐๐, ๐๐ not on ๐๐.There are ๐๐/๐๐ disjoint such ๐๐. The probability that none of the ๐บ๐บ[๐๐] is isomorphic to ๐ป๐ป is: = 1 โ ๐๐ ๐๐/๐๐
โ0
๐๐ โ โPr[ยฌ(๐ป๐ป โ ๐บ๐บ induced)]: โค 1 โ ๐๐ ๐๐/๐๐
Proposition. For every constant ๐๐ โ (0,1)and ๐๐, ๐๐ โ ๐๐, almost every graph ๐บ๐บ โ ๐ฎ๐ฎ(๐๐, ๐๐)has the property ๐๐๐๐,๐๐.
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๐บ๐บ
๐๐ โค ๐๐๐๐ โค ๐๐
๐ฃ๐ฃ
Proposition. For every constant ๐๐ โ (0,1)and ๐๐, ๐๐ โ ๐๐, almost every graph ๐บ๐บ โ ๐ฎ๐ฎ(๐๐, ๐๐)has the property ๐๐๐๐,๐๐.
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Proof. Fix ๐๐,๐๐ and ๐ฃ๐ฃ โ ๐บ๐บ โ ๐๐ โช๐๐ , ๐๐ = 1 โ ๐๐,
The probability that ๐๐๐๐,๐๐ holds for ๐ฃ๐ฃ: ๐๐ ๐๐ ๐๐|๐๐| โฅ ๐๐๐๐๐๐๐๐
The probability thereโs no such ๐ฃ๐ฃ for chosen ๐๐,๐๐:
= 1 โ ๐๐ ๐๐ ๐๐ ๐๐ ๐๐โ ๐๐ โ|๐๐|โค 1 โ ๐๐๐๐๐๐๐๐ ๐๐โ๐๐โ๐๐
The upper bound for the number of different choice of ๐๐,๐๐: ๐๐๐๐+๐๐
The probability there exists some ๐๐,๐๐ without suitable ๐ฃ๐ฃ:
โค ๐๐๐๐+๐๐ 1 โ ๐๐๐๐๐๐๐๐ ๐๐โ๐๐โ๐๐ ๐๐โโ0
Colouringโข Vertex coloring: to ๐บ๐บ = (๐๐,๐ธ๐ธ), a vertex
coloring is a map ๐๐:๐๐ โ ๐๐ such that ๐๐ ๐ฃ๐ฃ โ ๐๐(๐ค๐ค) whenever ๐ฃ๐ฃ and ๐ค๐ค are adjacent.
โข Chromatic number ๐๐(๐ฎ๐ฎ): the smallest size of ๐๐.
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๐๐ ๐บ๐บ = 3
Colouring
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โข Some famous results๏ผโ Whether ๐๐ ๐บ๐บ = ๐๐ is NP-complete.โ Every Planar graph is 4-colourable.โ [Grt๏ฟฝฬ๏ฟฝ๐zsch 1959]Every Planar graph not
containing a triangle is 3-colourable.
โข Vertex coloring: to ๐บ๐บ = (๐๐,๐ธ๐ธ), a vertex coloring is a map ๐๐:๐๐ โ ๐๐ such that ๐๐ ๐ฃ๐ฃ โ ๐๐(๐ค๐ค) whenever ๐ฃ๐ฃ and ๐ค๐ค are adjacent.
โข Chromatic number ๐๐(๐ฎ๐ฎ): the smallest size of ๐๐.
Proposition. For every constant ๐๐ โ (0,1) and every ๐๐ > 0, almost every graph ๐บ๐บ โ ๐ฎ๐ฎ(๐๐, ๐๐) has chromatic number ๐๐ ๐บ๐บ > log(1/๐๐)
2+๐๐โ ๐๐๐๐๐๐๐๐๐๐
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Proof.Pr ๐ผ๐ผ ๐บ๐บ โฅ ๐๐ โค
๐๐๐๐๐๐
๐๐2 โค ๐๐๐๐๐๐
๐๐2
= ๐๐๐๐log ๐๐log ๐๐+
12๐๐(๐๐โ1) = ๐๐
๐๐2 โ 2log ๐๐
log(1/๐๐)+๐๐โ1
Take ๐๐ = 2 + ๐๐ log ๐๐log(1/๐๐)
then (*) tends to โ with ๐๐.
The size of the maximum independent set in ๐บ๐บ:๐ผ๐ผ(๐บ๐บ)
โด Pr ๐ผ๐ผ ๐บ๐บ โฅ ๐๐๐๐โโ
0 โ
โด ๐๐ ๐บ๐บ >๐๐๐๐
=log 1/๐๐
2 + ๐๐โ ๐๐
log๐๐
(*)
No ๐๐ vertices can have the same colour.
The ๐บ๐บ(๐๐,๐๐) model
Properties of almost all graphs
Phase transition21
Phase transitionThe interesting thing about the ๐ฎ๐ฎ(๐๐, ๐๐)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 ๐๐(๐๐)such that
โ when lim๐๐โโ
๐๐1(๐๐)๐๐(๐๐)
= 0, ๐ฎ๐ฎ ๐๐,๐๐1 ๐๐ almost surely does not have the property.
โ when lim๐๐โโ
๐๐2(๐๐)๐๐(๐๐)
= โ, ๐ฎ๐ฎ ๐๐, ๐๐2 ๐๐ almost surely has the property.
Then we say phase transition occurs and ๐๐(๐๐) is the threshold.
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Phase transition
Every increasing property has a threshold.
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Phase transition
First moment methodMarkovโs Inequality: Let ๐ฅ๐ฅ be a random variable that assumes only nonnegative values. Then for all ๐๐ > 0
Pr ๐ฅ๐ฅ โฅ ๐๐ โค๐ฌ๐ฌ[๐ฅ๐ฅ]๐๐
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First moment method : for non-negative, integer valued variable ๐ฅ๐ฅ
Pr ๐ฅ๐ฅ > 0 = Pr ๐ฅ๐ฅ โฅ 1 โค ๐ฌ๐ฌ(๐ฅ๐ฅ)โด Pr ๐ฅ๐ฅ = 0 = 1 โ Pr ๐ฅ๐ฅ > 0 โฅ 1 โ ๐ฌ๐ฌ(๐ฅ๐ฅ)
โข If the expectation goes to 0: the property almost surely does not happen.
โข If the expectation does not goes to 0:e.g. Expectation = 1
๐๐ร ๐๐2 + ๐๐โ1
๐๐ร 0 = ๐๐
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 ๐ฅ๐ฅ
Pr ๐ฅ๐ฅ > 0 = Pr ๐ฅ๐ฅ โฅ 1 โค ๐ฌ๐ฌ(๐ฅ๐ฅ)โด Pr ๐ฅ๐ฅ = 0 = 1 โ Pr ๐ฅ๐ฅ > 0 โฅ 1 โ ๐ฌ๐ฌ(๐ฅ๐ฅ)
Chebyshevโs Inequality
โข For any ๐๐ > 0,
Pr ๐๐ โ ๐ธ๐ธ ๐๐ โฅ ๐๐ โค๐๐๐๐๐๐[๐๐]๐๐2
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Second moment methodTheorem. Let ๐ฅ๐ฅ(๐๐) be a random variable with ๐ฌ๐ฌ ๐ฅ๐ฅ > 0. If
๐๐๐๐๐๐ ๐ฅ๐ฅ = ๐๐ ๐ฌ๐ฌ2 ๐ฅ๐ฅThen ๐ฅ๐ฅ is almost surely greater than zero.
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Proof. If ๐ฌ๐ฌ ๐ฅ๐ฅ > 0, then for ๐ฅ๐ฅ โค 0, Pr ๐ฅ๐ฅ โค 0 โค Pr ๐ฅ๐ฅ โ ๐ฌ๐ฌ ๐ฅ๐ฅ โฅ ๐ฌ๐ฌ(๐ฅ๐ฅ)
โค๐๐๐๐๐๐ ๐ฅ๐ฅ๐ธ๐ธ2 ๐ฅ๐ฅ
โ 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)
โข Diameter: the maximum length of the shortest path between a pair of nodes.
โข Theorem: The property that ๐ฎ๐ฎ(๐๐, ๐๐) has diameter two has a sharp threshold at ๐๐ =
2 ln ๐๐๐๐
.
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Example : Threshold for graph diameter two (two degrees of separation)
Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
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Proof. For any two different vertices ๐๐ < ๐๐, ๐ผ๐ผ๐๐๐๐ = ๏ฟฝ1 i, j โ๐ธ๐ธ, no other vertex is adjacent to both ๐๐ ๐๐๐๐๐๐ ๐๐
0 ๐๐๐๐๐๐๐๐๐๐ค๐ค๐๐๐๐๐๐
๐ฅ๐ฅ = ๏ฟฝ๐๐<๐๐
๐ผ๐ผ๐๐๐๐ If ๐ฌ๐ฌ ๐ฅ๐ฅ๐๐โโ
0, then for large ๐๐, almost surely the diameter is at most two.
Example : Threshold for graph diameter two (two degrees of separation)
Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
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Proof. For any two different vertices ๐๐ < ๐๐, ๐ผ๐ผ๐๐๐๐ = ๏ฟฝ1 i, j โ๐ธ๐ธ, no other vertex is adjacent to both ๐๐ ๐๐๐๐๐๐ ๐๐
0 ๐๐๐๐๐๐๐๐๐๐ค๐ค๐๐๐๐๐๐
๐ฌ๐ฌ ๐ฅ๐ฅ =๐๐2
1 โ ๐๐ 1 โ ๐๐2 ๐๐โ2
Take ๐๐ = ๐๐ ln ๐๐๐๐
, ๐ธ๐ธ(๐ฅ๐ฅ) โ ๐๐2
21 โ ๐๐ ln ๐๐
๐๐1 โ ๐๐2 ln ๐๐
๐๐
๐๐
๐ฅ๐ฅ = ๏ฟฝ๐๐<๐๐
๐ผ๐ผ๐๐๐๐
โ ๐๐2
2๐๐โ๐๐2 ln ๐๐ =
12๐๐2โ๐๐2
Example : Threshold for graph diameter two (two degrees of separation)
Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
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Proof. For any two different vertices ๐๐ < ๐๐, ๐ผ๐ผ๐๐๐๐ = ๏ฟฝ1 i, j โ๐ธ๐ธ, no other vertex is adjacent to both ๐๐ ๐๐๐๐๐๐ ๐๐
0 ๐๐๐๐๐๐๐๐๐๐ค๐ค๐๐๐๐๐๐
๐ฌ๐ฌ ๐ฅ๐ฅ =๐๐2
1 โ ๐๐ 1 โ ๐๐2 ๐๐โ2
Take ๐๐ = ๐๐ ln ๐๐๐๐
, ๐๐ > 2, lim๐๐โโ
๐ฌ๐ฌ ๐ฅ๐ฅ = 0
๐ฅ๐ฅ = ๏ฟฝ๐๐<๐๐
๐ผ๐ผ๐๐๐๐
For large ๐๐, almost surely the diameter is at most two.
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Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
โข Take ๐๐ = ๐๐ ln ๐๐๐๐
, ๐๐ > 2, lim๐๐โโ
๐ฌ๐ฌ ๐ฅ๐ฅ = 0
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Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
โข Take ๐๐ = ๐๐ ln ๐๐๐๐
, ๐๐ < 2,
๐ฌ๐ฌ ๐ฅ๐ฅ2 = ๐ฌ๐ฌ ๏ฟฝ๐๐<๐๐
๐ผ๐ผ๐๐๐๐2 If ๐๐๐๐๐๐ ๐ฅ๐ฅ = ๐๐ ๐ฌ๐ฌ2 ๐ฅ๐ฅ , then for large ๐๐,
almost surely the diameter will be larger than two.
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Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
โข Take ๐๐ = ๐๐ ln ๐๐๐๐
, ๐๐ < 2
๐ฌ๐ฌ ๐ฅ๐ฅ2 = ๐ฌ๐ฌ ๏ฟฝ๐๐<๐๐
๐ผ๐ผ๐๐๐๐2
= ๐ฌ๐ฌ ๏ฟฝ๐๐<๐๐
๐ผ๐ผ๐๐๐๐๏ฟฝ๐๐<๐๐
๐ผ๐ผ๐๐๐๐ = ๐ฌ๐ฌ ๏ฟฝ๐๐<๐๐๐๐<๐๐
๐ผ๐ผ๐๐๐๐ ๐ผ๐ผ๐๐๐๐ = ๏ฟฝ๐๐<๐๐๐๐<๐๐
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐
๐๐ = | ๐๐, ๐๐, ๐๐, ๐๐ |
๐ฌ๐ฌ ๐ฅ๐ฅ2 = ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=4
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ + ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=3
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ + ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=2
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐2
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Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
โข Take ๐๐ = ๐๐ ln ๐๐๐๐
, ๐๐ < 2
๐ฌ๐ฌ ๐ฅ๐ฅ2 = ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=4
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ + ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=3
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ + ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=2
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐2
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ โค 1 โ ๐๐2 2 ๐๐โ4 โค 1 โ ๐๐2ln๐๐๐๐
2๐๐
1 + ๐๐ 1 โค ๐๐โ2๐๐2(1 + ๐๐(1))
๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=4
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ โค ๐๐4โ2๐๐2(1 + ๐๐(1))๐๐ ๐๐ ๐๐ ๐๐
๐ข๐ข
38
Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
โข Take ๐๐ = ๐๐ ln ๐๐๐๐
, ๐๐ < 2
๐ฌ๐ฌ ๐ฅ๐ฅ2 = ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=4
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ + ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=3
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ + ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=2
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐2
๐๐ ๐๐ ๐๐
๐ข๐ข
39
Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
โข Take ๐๐ = ๐๐ ln ๐๐๐๐
, ๐๐ < 2
๐ฌ๐ฌ ๐ฅ๐ฅ2 = ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=4
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ + ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=3
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ + ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=2
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐2
Pr ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ = 1 โค 1 โ ๐๐ + ๐๐ 1 โ ๐๐ 2 = 1 โ 2๐๐2 + ๐๐3 โ 1 โ 2๐๐2
๏ฟฝ๐๐<๐๐๐๐<๐๐
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ โค ๐๐3โ2๐๐2
๐๐ ๐๐ ๐๐
๐ข๐ข
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ โค 1 โ 2๐๐2 ๐๐โ3 = 1 โ2๐๐2 ln๐๐
๐๐
๐๐โ3
โ ๐๐โ2๐๐2 ln ๐๐ = ๐๐โ2๐๐2
40
Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
โข Take ๐๐ = ๐๐ ln ๐๐๐๐
, ๐๐ < 2
๐ฌ๐ฌ ๐ฅ๐ฅ2 = ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=4
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ + ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=3
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐๐ผ๐ผ๐๐๐๐ + ๏ฟฝ๐๐<๐๐๐๐<๐๐๐๐=2
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐2
๐ธ๐ธ(๐ผ๐ผ๐๐๐๐2 ) = ๐ธ๐ธ(๐ผ๐ผ๐๐๐๐)
๏ฟฝ๐๐๐๐
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐2 = ๐ธ๐ธ ๐ฅ๐ฅ โ 12๐๐2โ๐๐2
๐๐ ๐๐
๐ข๐ข
41
Theorem. The property that ๐ฎ๐ฎ(๐๐,๐๐) has diameter
two has a sharp threshold at ๐๐ = 2 ln ๐๐๐๐
โข Take ๐๐ = ๐๐ ln ๐๐๐๐
, ๐๐ < 2
๐ฌ๐ฌ ๐ฅ๐ฅ2 โค ๐ฌ๐ฌ2(๐ฅ๐ฅ)(1 + ๐๐ 1 )
A graph almost surely has at least one badpair of vertices and thus diameter greaterthan two.