Alexander Krot – Limits of Local Algorithms for Randomly Generated Constraint Satisfaction...

Models based on preferential attachmentAnalysis of PA-models

Conclusion

Local clustering coefficient in preferentialattachment graphs

Liudmila Ostroumova, Egor Samosvat, Alexander Krot

Moscow Institute of Physics and TechnologyMoscow State University

Yandex

June, 2014

Liudmila Ostroumova, Egor Samosvat, Alexander Krot Local clustering in preferential attachment graphs

Conclusion

1 Models based on the preferential attachment

2 Theoretical analysis of the general case

3 Conclusion

Conclusion

Properties of interestGeneralized preferential attachmentExamples

Degree distribution

Real-world networks often have the power law degreedistribution:

#{v : deg(v) = d}n

where 2 < γ < 3.

Conclusion

Clustering coefficient

Global clustering coefficient of a graph G:

C1(n) =3#(triangles in G)

#(pairs of adjacent edges in G).

Average local clustering coefficientT i is the number of edges between the neighbors of a vertex iP i2 is the number of pairs of neighbors

C(i) = T i

P i2is the local clustering coefficient for a vertex i

C2(n) = 1n

∑ni=1C(i) – average local clustering coefficient

Conclusion

Clustering coefficient

Global clustering coefficient of a graph G:

C1(n) =3#(triangles in G)

#(pairs of adjacent edges in G).

Average local clustering coefficientT i is the number of edges between the neighbors of a vertex iP i2 is the number of pairs of neighbors

C(i) = T i

P i2is the local clustering coefficient for a vertex i

C2(n) = 1n

∑ni=1C(i) – average local clustering coefficient

Conclusion

Preferential attachment

Idea of preferential attachment [Barabasi, Albert]:Start with a small graphAt every step we add new vertex with m edgesThe probability that a new vertex will be connected to a vertexi is proportional to the degree of i

Conclusion

PA-class of models

Start from an arbitrary graph Gn0m with n0 vertices and mn0

We make Gn+1m from Gnm by adding a new vertex n+ 1 with

m edgesPA-condition: the probability that the degree of a vertex iincreases by one equals

Adeg(i)

((deg(i))2

The probability of adding a multiple edge is O((deg(i))2

)2mA+B = m, 0 ≤ A ≤ 1

Conclusion

PA-class of models

edgesWe make Gn+1

m from Gnm by adding a new vertex n+ 1 withm edges

PA-condition: the probability that the degree of a vertex iincreases by one equals

Adeg(i)

((deg(i))2

)2mA+B = m, 0 ≤ A ≤ 1

Conclusion

PA-class of models

edgesWe make Gn+1

m from Gnm by adding a new vertex n+ 1 withm edgesPA-condition: the probability that the degree of a vertex iincreases by one equals

Adeg(i)

((deg(i))2

)2mA+B = m, 0 ≤ A ≤ 1

Conclusion

PA-class of models

edgesWe make Gn+1

Adeg(i)

((deg(i))2

2mA+B = m, 0 ≤ A ≤ 1

Conclusion

PA-class of models

edgesWe make Gn+1

Adeg(i)

((deg(i))2

)2mA+B = m, 0 ≤ A ≤ 1

Conclusion

T -subclass

Triangles property:

The probability that the degree of two vertices i and jincreases by one equals

(dni d

)Here eij is the number of edges between vertices i and j inGnm and D is a positive constant.

Conclusion

Bollobas–Riordan, Buckley–Osthus, Mori, etc.

Fix some positive number a – "initial attractiveness".(Bollobas–Riordan model: a = 1).Start with a graph with one vertex and m loops.At n-th step add one vertex with m edges.We add m edges one by one. The probability to add an edgen→ i at each step is proportional to indeg(i) + am.

Outdegree: mTriangles property: D = 0

PA-condition: A = 11+a

Degree distribution: Power law with γ = 2 + a

Global clustering: (logn)2

n (a = 1), lognn (a > 1)

Conclusion

Bollobas–Riordan, Buckley–Osthus, Mori, etc.

Fix some positive number a – "initial attractiveness".(Bollobas–Riordan model: a = 1).Start with a graph with one vertex and m loops.At n-th step add one vertex with m edges.We add m edges one by one. The probability to add an edgen→ i at each step is proportional to indeg(i) + am.

Outdegree: mTriangles property: D = 0

PA-condition: A = 11+a

Degree distribution: Power law with γ = 2 + a

Global clustering: (logn)2

n (a = 1), lognn (a > 1)

Conclusion

Random Apollonian networks

Outdegree: m = 3

Triangles property: D = 3

PA-condition: A = 12

Degree distribution: Power law with γ = 3

Average local clustering: constantGlobal clustering: tends to zero

Conclusion

Outdegree: m = 3

Conclusion

Outdegree: m = 3

Conclusion

Outdegree: m = 3

Conclusion

Outdegree: m = 3

Conclusion

Outdegree: m = 3

Conclusion

Outdegree: m = 3

Conclusion

Outdegree: m = 3

Conclusion

Polynomial model

Put m = 2p

Fix α, β, δ ≥ 0 and α+ β + δ = 1

Add a new vertex i with m edges. We add m edges in p stepsα – probability of an indegree preferential stepβ – probability of an edge preferential stepδ – probability of a random step

Edge preferential: cite two endpoints of a random edgeOutdegree: 2p

Triangles property: D = βp

PA-condition: A = α+ β2 .

Degree distribution: Power law with γ = 1 + 22α+β

Average local clustering: constantGlobal clustering: constant for A > 1/2 (γ > 3), tends tozero for A ≤ 1/2 (2 < γ ≤ 3)

Conclusion

Polynomial model

Put m = 2p

Fix α, β, δ ≥ 0 and α+ β + δ = 1

Conclusion

Polynomial model

Put m = 2p

Fix α, β, δ ≥ 0 and α+ β + δ = 1

Add a new vertex i with m edges. We add m edges in p steps

α – probability of an indegree preferential stepβ – probability of an edge preferential stepδ – probability of a random step

Conclusion

Polynomial model

Put m = 2p

Fix α, β, δ ≥ 0 and α+ β + δ = 1

Conclusion

Polynomial model

Put m = 2p

Fix α, β, δ ≥ 0 and α+ β + δ = 1

Edge preferential: cite two endpoints of a random edge

Outdegree: 2p

Conclusion

Polynomial model

Put m = 2p

Fix α, β, δ ≥ 0 and α+ β + δ = 1

Conclusion

Degree distributionClustering coefficientLocal clustering

Degree distribution

Let Nn(d) be the number of vertices with degree d in Gnm.

ExpectationFor every d ≥ m we have

ENn(d) = c(m, d)(n+O

c(m, d) =Γ(d+ B

)Γ(m+ B+1

(d+ B+A+1

)Γ(m+ B

) ∼ Γ(m+ B+1

)d−1−

AΓ(m+ B

)and Γ(x) is the gamma function.

Conclusion

Degree distribution

Let Nn(d) be the number of vertices with degree d in Gnm.

ExpectationFor every d ≥ m we have

ENn(d) = c(m, d)(n+O

c(m, d) =Γ(d+ B

)Γ(m+ B+1

(d+ B+A+1

)Γ(m+ B

) ∼ Γ(m+ B+1

)d−1−

AΓ(m+ B

)and Γ(x) is the gamma function.

Conclusion

Degree distribution

ConcentrationFor every d = d(n) we have

P(|Nn(d)− ENn(d)| ≥ d

√n log n

(n− logn

Therefore, for any δ > 0 there exists a function ϕ(n) = o(1) suchthat

limn→∞

P(∃ d ≤ n

A−δ4A+2 : |Nn(d)− ENn(d)| ≥ ϕ(n) ENn(d)

)= 0 .

Conclusion

Global clustering

Let P2(n) be the number of all path of length 2 in Gnm.

(1) If 2A < 1, then whp P2(n) ∼(

2m(A+B) + m(m−1)2

1−2A .

(2) If 2A = 1, then whp P2(n) ∝ n log(n) .

(3) If 2A > 1, then whp P2(n) ∝ n2A .

Triangles

Whp the number of triangles T (n) ∼ Dn .

Conclusion

Global clustering

Let P2(n) be the number of all path of length 2 in Gnm.

(1) If 2A < 1, then whp P2(n) ∼(

2m(A+B) + m(m−1)2

1−2A .

(2) If 2A = 1, then whp P2(n) ∝ n log(n) .

(3) If 2A > 1, then whp P2(n) ∝ n2A .

Triangles

Whp the number of triangles T (n) ∼ Dn .

Conclusion

Global clustering

(1) If 2A < 1 then whp C1(n) ∼ 3(1−2A)D(2m(A+B)+

m(m−1)2 )

(2) If 2A = 1 then whp C1(n) ∝ (log n)−1 .

(2) If 2A > 1 then whp C1(n) ∝ n1−2A .

Conclusion

Average local clustering

Average local clusteringWhp

C2(n) ≥1

∑i:deg(i)=m

C(i) ≥ 2cD

m(m+ 1).

Conclusion

Local clustering: definition

Let Tn(d) be the number of triangles on the vertices of degreed in Gn

Let Nn(d) be the number of vertices of degree d in Gnm.

Local clustering coefficient:

C(d) =Tn(d)

Nn(d)d(d−1)

Conclusion

Local clustering: number of triangles

Tn(d) is the number of triangles on the vertices of degree d inGnm

Expectation

ETn(d) = f(d)(n+O

wheref(d) ∼ d−1/A

Conclusion

Idea of the proof

The following recurrent formula holds:

ETn+1(d) = ETn(d)− ETn(d) ·[Ad+B

+ETn(d− 1) ·[A(d− 1) +B

((d− 1)2

j:j is a neighbor ofa vertex of degree d− 1

((d− 1) · dj

Next we prove that ETn(d) = f(d)(n+O(d2+1A )) where

f(d) ∼ d−1A .

Conclusion

Idea of the proof

The following recurrent formula holds:

ETn+1(d) = ETn(d)− ETn(d) ·[Ad+B

+ETn(d− 1) ·[A(d− 1) +B

((d− 1)2

j:j is a neighbor ofa vertex of degree d− 1

((d− 1) · dj

Next we prove that ETn(d) = f(d)(n+O(d2+1A )) where

f(d) ∼ d−1A .

Conclusion

Number of triangles

ConcentrationFor every d = d(n) we have

P(|Tn(d)− ETn(d)| ≥ d2

√n log n

(n− logn

Therefore, for any δ > 0 there exists a function ϕ(n) = o(1) suchthat

limn→∞

P(∃ d ≤ n

A−δ4A+2 : |Tn(d)− ETn(d)| ≥ ϕ(n) ETn(d)

)= 0 .

Conclusion

Idea of the proof

Azuma, Hoeffding

Let (Xi)ni=0 be a martingale such that |Xi −Xi−1| ≤ ci for any

1 ≤ i ≤ n. Then

P (|Xn −X0| ≥ x) ≤ 2e− x2

2∑ni=1

for any x > 0.

Xi(d) = E(Tn(d) | Gim), i = 0, . . . , n.Note that X0(d) = ETn(d) and Xn(d) = Tn(d).Xn(d) is a martingale.For any i = 0, . . . , n− 1: |Xi+1(d)−Xi(d)| ≤Md2, whereM > 0 is some constant.

Conclusion

Idea of the proof

Azuma, Hoeffding

Let (Xi)ni=0 be a martingale such that |Xi −Xi−1| ≤ ci for any

1 ≤ i ≤ n. Then

P (|Xn −X0| ≥ x) ≤ 2e− x2

2∑ni=1

for any x > 0.

Xi(d) = E(Tn(d) | Gim), i = 0, . . . , n.Note that X0(d) = ETn(d) and Xn(d) = Tn(d).Xn(d) is a martingale.For any i = 0, . . . , n− 1: |Xi+1(d)−Xi(d)| ≤Md2, whereM > 0 is some constant.

Conclusion

Idea of the proof

Fix 0 ≤ i ≤ n− 1 and some graph Gim.∣∣E (Tn(d) | Gi+1m

)− E

(Tn(d) | Gim

)∣∣ ≤≤ max

Gi+1m ⊃Gim

{E(Tn(d) | Gi+1

)}− minGi+1m ⊃Gim

{E(Tn(d) | Gi+1

Gi+1m = arg max E(Tn(d) | Gi+1

m ),Gi+1m = arg min E(Tn(d) | Gi+1

For i+ 1 ≤ t ≤ n put δit(d) = E(Tt(d) | Gi+1m )− E(Tt(d) | Gi+1

δit+1(d) ≤ δit(d)

(1− Ad+B

+δit(d− 1)

(A(d− 1) +B

Conclusion

Idea of the proof

)− E

(Tn(d) | Gim

)∣∣ ≤≤ max

Gi+1m ⊃Gim

{E(Tn(d) | Gi+1

(1− Ad+B

+δit(d− 1)

(A(d− 1) +B

Conclusion

Idea of the proof

)− E

(Tn(d) | Gim

)∣∣ ≤≤ max

Gi+1m ⊃Gim

{E(Tn(d) | Gi+1

(1− Ad+B

+δit(d− 1)

(A(d− 1) +B

Conclusion

Idea of the proof

)− E

(Tn(d) | Gim

)∣∣ ≤≤ max

Gi+1m ⊃Gim

{E(Tn(d) | Gi+1

(1− Ad+B

+δit(d− 1)

(A(d− 1) +B

Conclusion

Local clustering

C(d) =Tn(d)

Nn(d)d(d−1)

∼ d−1/A · d1+1/A · d−2 ∼ 1

Conclusion

Generalized preferential attachment models:

Power law degree distribution with any exponent γ > 2

Constant global clustering coefficient only for γ > 3

Constant average local clustering coefficient

Local clustering C(d) ∼ 1d

Conclusion

Thank You!Questions?

Alexander Krot – Limits of Local Algorithms for Randomly Generated Constraint Satisfaction...

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