PREFERENTIAL ATTACHMENT GRAPHS ARE SOMEWHERE-DENSE Jan Dreier, - - PowerPoint PPT Presentation

PREFERENTIAL ATTACHMENT GRAPHS ARE SOMEWHERE-DENSE

Jan Dreier, Philipp Kuinke, Peter Rossmanith

TACO 2018

RWTH Aachen University

MOTIVATION

Sparsity

Star forests Bounded treedepth Bounded treewidth Excluding a minor Excluding a topological minor Bounded expansion Outerplanar Planar Bounded genus Linear forests Bounded degree Locally bounded treewidth Locally excluding a minor Forests

r r

∇

Locally bounded expansion Nowhere dense

∇ ∇

r

ω

Image by Felix Reidl

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Sparsity: r-shallow topological minor

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Sparsity: r-shallow topological minor

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Sparsity: r-shallow topological minor

G

▽ r : The set of all r-shallow topological minors of G.

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Sparsity: r-shallow topological minor

G

▽ r : The set of all r-shallow topological minors of G.

ω(G

▽ r) max

H∈G ▽ r ω(H)

(clique size)

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Sparsity

Definition (Nowhere-dense)

A graph class Gis nowhere-dense if there exists a function f , such that for all r and all G ∈ G, ω(G

▽ r) ≤ f (r).

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Sparsity

Definition (Nowhere-dense)

A graph class Gis nowhere-dense if there exists a function f , such that for all r and all G ∈ G, ω(G

▽ r) ≤ f (r). Definition (Somewhere-dense)

A graph class Gis somewhere-dense if for all functions f there exists an r and a G ∈ G, such that ω(G

▽ r) > f (r).

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Sparsity

Definition (Nowhere-dense)

A graph class Gis nowhere-dense if there exists a function f , such that for all r and all G ∈ G, ω(G

▽ r) ≤ f (r). Definition (Somewhere-dense)

A graph class Gis somewhere-dense if for all functions f there exists an r and a G ∈ G, such that ω(G

▽ r) > f (r).

Gis not nowhere-dense ⇔ Gis somewhere-dense

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Typical Properties of Complex Networks

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Typical Properties of Complex Networks

low diameter (small world property)

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Typical Properties of Complex Networks

low diameter (small world property) locally dense, but globally sparse

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Typical Properties of Complex Networks

low diameter (small world property) locally dense, but globally sparse heavy tail degree distribution

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Typical Properties of Complex Networks

low diameter (small world property) locally dense, but globally sparse heavy tail degree distribution clustering

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Typical Properties of Complex Networks

low diameter (small world property) locally dense, but globally sparse heavy tail degree distribution clustering community structure

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Typical Properties of Complex Networks

low diameter (small world property) locally dense, but globally sparse heavy tail degree distribution clustering community structure scale freeness

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Random Graphs

Random graphs with the goal of modeling real-world data: Mathematically analyzable Generation of infinite instances

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Sparse in the limit

Definition (a.a.s. nowhere-dense)

A random graph model Gis a.a.s. nowhere-dense if there exists a function f such that for all r lim

n→∞ P[ω(Gn

▽ r) ≤ f (r)] 1

where Gn is a random variable modeling a graph with n ver- tices randomly drawn from G.

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Sparse in the limit

Definition (a.a.s. somewhere-dense)

A random graph model Gis a.a.s. somewhere-dense if for all functions f there exists an r, such that lim

n→∞ P[ω(Gn

▽ r) > f (r)] 1

where Gn is a random variable modeling a graph with n ver- tices randomly drawn from G.

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Sparse in the limit (not as clear cut)

Assume you have a random graph on n vertices, such that it is with probability p complete and with probability 1 − p empty:

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Sparse in the limit (not as clear cut)

Assume you have a random graph on n vertices, such that it is with probability p complete and with probability 1 − p empty:

• 1. p 1/n

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Sparse in the limit (not as clear cut)

Assume you have a random graph on n vertices, such that it is with probability p complete and with probability 1 − p empty:

• 1. p 1/n

→ a.a.s. nowhere-dense

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Sparse in the limit (not as clear cut)

Assume you have a random graph on n vertices, such that it is with probability p complete and with probability 1 − p empty:

• 1. p 1/n

→ a.a.s. nowhere-dense

• 2. p 1 − 1/n

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Sparse in the limit (not as clear cut)

Assume you have a random graph on n vertices, such that it is with probability p complete and with probability 1 − p empty:

• 1. p 1/n

→ a.a.s. nowhere-dense

• 2. p 1 − 1/n

→ a.a.s. somewhere-dense

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Sparse in the limit (not as clear cut)

Assume you have a random graph on n vertices, such that it is with probability p complete and with probability 1 − p empty:

• 1. p 1/n

→ a.a.s. nowhere-dense

• 2. p 1 − 1/n

→ a.a.s. somewhere-dense

• 3. p 1/2

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Sparse in the limit (not as clear cut)

Assume you have a random graph on n vertices, such that it is with probability p complete and with probability 1 − p empty:

• 1. p 1/n

→ a.a.s. nowhere-dense

• 2. p 1 − 1/n

→ a.a.s. somewhere-dense

• 3. p 1/2

→ Neither!

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Preferential attachment graphs

“the rich get richer”, “preferential attachment”, “Barabási–Albert graphs” Start with some small fixed graph. Add vertices. Connect them to m vertices with a probability proportional to their degrees. Interesting properties: power law degree distribution scale free

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Preferential attachment graphs

m 2, n 100:

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Preferential attachment graphs

m 2, n 100: E[dn

m(vi)] ∼ m

• n/i

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TAIL BOUNDS

Degree Concentrations

Tail bounds exists for number of vertices with degree d. [Bollobás et al. 2001]

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Degree Concentrations

Tail bounds exists for number of vertices with degree d. [Bollobás et al. 2001] Via Martingales + Azuma-Hoeffding inequality

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Degree Concentrations

Tail bounds exists for number of vertices with degree d. [Bollobás et al. 2001] Via Martingales + Azuma-Hoeffding inequality Does not work for large d (i.e. order √n)

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Degree Concentrations

Tail bounds exists for number of vertices with degree d. [Bollobás et al. 2001] Via Martingales + Azuma-Hoeffding inequality Does not work for large d (i.e. order √n) But we need high degree vertices!

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Concentration of a single vertex

No vertex is sharply concentrated!

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Concentration of a single vertex

No vertex is sharply concentrated! P[dn

1 (vt) 1] 14

Concentration of a single vertex

No vertex is sharply concentrated! P[dn

1 (vt) 1] n

• it

(1 −

1 2i − 1)

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Concentration of a single vertex

No vertex is sharply concentrated! P[dn

1 (vt) 1] n

• it

(1 −

1 2i − 1) ≥ 1 n

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Concentration of a single vertex

No vertex is sharply concentrated! P[dn

1 (vt) 1] n

• it

(1 −

1 2i − 1) ≥ 1 n We can not hope for general exponential bounds.

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Concentration of a single vertex

p k 0.02 0.01 250 500

Distribution of d10000

1

(v1).

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Concentration of a single vertex

p k 0.02 0.01 250 500

Distribution of d10000

1

(v1) conditioned under d100

1 (v1) 18. 15

Concentration of a single vertex

p k 0.02 0.01 250 500

Distribution of d10000

1

(v1) conditioned under d1000

1

(v1) 56.

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The rich stay rich

Theorem

Let 0 < ε ≤ 1/40, t, m, n ∈ N, t > 1

ε6 and S ⊆ {v1, . . . , vt}. Then

P

• (1 − ε)
• n

t dt

m(S) < dn m(S) < (1 + ε)

• n

t dt

m(S) for all n ≥ t

• dt

m(S)

• ≥ 1 − ln(15t)eε−O(1)dt

m(S).

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The rich stay rich

Theorem (The approximate version)

Let ε ≥ 0, t, m, n ∈ N, and S ⊆ {v1, . . . , vt}: P

• (1 − ε) E[dn

m(S)] < dn m(S) < (1 + ε) E[dn m(S)]

• dt

m(S)

• ≥ 1 − e−εdt

m(S)

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The rich stay rich

Theorem (The approximate version)

Let ε ≥ 0, t, m, n ∈ N, and S ⊆ {v1, . . . , vt}: P

• (1 − ε) E[dn

m(S)] < dn m(S) < (1 + ε) E[dn m(S)]

• dt

m(S)

• ≥ 1 − e−εdt

m(S)

The rich stay rich

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The rich stay rich

Theorem (The approximate version)

Let ε ≥ 0, t, m, n ∈ N, and S ⊆ {v1, . . . , vt}: P

• (1 − ε) E[dn

m(S)] < dn m(S) < (1 + ε) E[dn m(S)]

• dt

m(S)

• ≥ 1 − e−εdt

m(S)

The rich stay rich At first there is chaos

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The rich stay rich

Theorem (The approximate version)

Let ε ≥ 0, t, m, n ∈ N, and S ⊆ {v1, . . . , vt}: P

• (1 − ε) E[dn

m(S)] < dn m(S) < (1 + ε) E[dn m(S)]

• dt

m(S)

• ≥ 1 − e−εdt

m(S)

The rich stay rich At first there is chaos If we have information for t we can better predict n > t

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Proving the theorem

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Proving the theorem

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Proving the theorem

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SOMEWHERE-DENSE

Large cliques

Theorem

Gn

m contains a.a.s. a one-subdivided clique of size ∼ log(n). 19

Large cliques

Theorem

Gn

m contains a.a.s. a one-subdivided clique of size ∼ log(n).

Corollary

Gn

m is a.a.s. somewhere-dense for m ≥ 2. 19

How we get principals

k sets of vertices

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How we get principals

k sets of vertices If a set of s vertices has degree d one vertex has to have degree at least d/s.

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How we get principals

k sets of vertices If a set of s vertices has degree d one vertex has to have degree at least d/s. → Ensure with tail bounds it also has high degree in the future.

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Building cliques

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Building cliques

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Building cliques

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Building cliques

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Building cliques

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Building cliques

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Building cliques

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Building cliques

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Building cliques

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Building cliques

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Building cliques

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Building cliques

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Connecting principals: Why we need √ i

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Connecting principals: Why we need √ i

Step i:

• √

i red

• √

i blue remainder black Two balls drawn, success if red and blue

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Connecting principals: Why we need √ i

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Connecting principals: Why we need √ i

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Connecting principals: Why we need √ i

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Connecting principals: Why we need √ i

1 −

∞

• i10
• 1 − 2

√

i i

2

1

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Connecting principals: Why we need √ i

1 −

∞

• i10
• 1 − 2

√

i/ log(i) i

2 1

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CONCLUSION

Conclusion

Tail bounds for vertices where we know an earlier degree

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Conclusion

Tail bounds for vertices where we know an earlier degree Tail bounds could be used to prove further structure

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Conclusion

Tail bounds for vertices where we know an earlier degree Tail bounds could be used to prove further structure Preferential Attachment graphs are a.a.s. somewhere-dense

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Conclusion

Tail bounds for vertices where we know an earlier degree Tail bounds could be used to prove further structure Preferential Attachment graphs are a.a.s. somewhere-dense FO-model checking algorithm not directly applicable

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Conclusion

Tail bounds for vertices where we know an earlier degree Tail bounds could be used to prove further structure Preferential Attachment graphs are a.a.s. somewhere-dense FO-model checking algorithm not directly applicable What about more general PA-graphs with δ parameter?

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Conclusion

Tail bounds for vertices where we know an earlier degree Tail bounds could be used to prove further structure Preferential Attachment graphs are a.a.s. somewhere-dense FO-model checking algorithm not directly applicable What about more general PA-graphs with δ parameter?

• δ 0: Our model

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Conclusion

Tail bounds for vertices where we know an earlier degree Tail bounds could be used to prove further structure Preferential Attachment graphs are a.a.s. somewhere-dense FO-model checking algorithm not directly applicable What about more general PA-graphs with δ parameter?

• δ 0: Our model
• δ ∞: Uniform attachment

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References

Barabási, Albert-László and Réka Albert (1999). “Emergence of scaling in random networks”. In: Science 286.5439, pp. 509–512. Béla Bollobás Oliver Riordan, Joel Spencer and Gábor Tusnády (2001). “The Degree Sequence of a Scale-free Random Graph Process”. In: Random Struct. Algorithms 18.3, pp. 279–290. issn: 1042-9832. Martin Grohe, Stephan Kreutzer and Sebastian Siebertz (2017). “Deciding First-Order Properties of Nowhere Dense Graphs”. In: Journal of the ACM 64.3, p. 17. Nešetřil, Jaroslav and Patrice Ossona de Mendez (2012). Sparsity. Springer. Van Der Hofstad, Remco (2016). Random graphs and complex networks.

• Vol. 1. Cambridge University Press.

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