Introduction to Random Graphs - SJTUlonghuan/teaching/CS499/19.pdfย ยท Properties of almost all...
Transcript of 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
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The ๐บ๐บ(๐๐,๐๐) model
Properties of almost all graphs
Phase transition3
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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
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Example: ๐ฎ๐ฎ(๐๐, 1/2)
Pr ๐พ๐พ = ๐๐ = ๐๐ โ 1๐๐
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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
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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.
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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
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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
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The ๐บ๐บ(๐๐,๐๐) model
Properties of almost all graphs
Phase transition12
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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
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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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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
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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
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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
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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
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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.
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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.
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The ๐บ๐บ(๐๐,๐๐) model
Properties of almost all graphs
Phase transition21
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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
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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)
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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)
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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)
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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.
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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
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
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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
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
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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
๐๐ ๐๐ ๐๐
๐ข๐ข
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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
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
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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
๐ธ๐ธ(๐ผ๐ผ๐๐๐๐2 ) = ๐ธ๐ธ(๐ผ๐ผ๐๐๐๐)
๏ฟฝ๐๐๐๐
๐ฌ๐ฌ ๐ผ๐ผ๐๐๐๐2 = ๐ธ๐ธ ๐ฅ๐ฅ โ 12๐๐2โ๐๐2
๐๐ ๐๐
๐ข๐ข
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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 โค ๐ฌ๐ฌ2(๐ฅ๐ฅ)(1 + ๐๐ 1 )
A graph almost surely has at least one badpair of vertices and thus diameter greaterthan two.