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How the k -core decomposition helps in understanding the Internet - - PowerPoint PPT Presentation

Apr 06, 2023 157 likes •749 views

k -core decomposition Applications How the k -core decomposition helps in understanding the Internet Topology J. Ignacio Alvarez-Hamelin Luca DallAsta Alain Barrat Alessandro Vespignani Laboratoire de Physique

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k-core decomposition Applications

How the k-core decomposition helps in understanding the Internet Topology

  • J. Ignacio Alvarez-Hamelin∗‡

Luca Dall’Asta∗ Alain Barrat∗ Alessandro Vespignani†∗

∗Laboratoire de Physique Théorique, Université Paris-Sud XI †School of Informatics and Department of Physics, Indiana University ‡CONICET and Departamento de Electrónica, Universidad de Buenos Aires

Worshop on Internet Topology :: San Diego :: May 10-12

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 2

k-core decomposition Applications

Outline

1

k-core decomposition Definition Examples Applicability of k-cores

2

Applications Properties analysis Temporal analysis Visualization

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 3

k-core decomposition Applications Definition Examples Applicability of k-cores

Outline

1

k-core decomposition Definition Examples Applicability of k-cores

2

Applications Properties analysis Temporal analysis Visualization

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 4

k-core decomposition Applications Definition Examples Applicability of k-cores

Outline

1

k-core decomposition Definition Examples Applicability of k-cores

2

Applications Properties analysis Temporal analysis Visualization

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Definition

Given G = {V, E} a undirected graph, where V is the vertices set and E is the edges set. Definition (Seidman, 1983 [4]) : A subgraph H = (C, E|C) induced by the set C ⊆ V is a k-core

  • r a core of order k iff ∀v ∈ C : degreeH(v) ≥ k, and H is the

maximum subgraph with this property. Then, a minimal degree k is imposed to the core of order k.

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Definition

Given G = {V, E} a undirected graph, where V is the vertices set and E is the edges set. Definition (Seidman, 1983 [4]) : A subgraph H = (C, E|C) induced by the set C ⊆ V is a k-core

  • r a core of order k iff ∀v ∈ C : degreeH(v) ≥ k, and H is the

maximum subgraph with this property. Then, a minimal degree k is imposed to the core of order k.

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 7

k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Definition

Given G = {V, E} a undirected graph, where V is the vertices set and E is the edges set. Definition (Seidman, 1983 [4]) : A subgraph H = (C, E|C) induced by the set C ⊆ V is a k-core

  • r a core of order k iff ∀v ∈ C : degreeH(v) ≥ k, and H is the

maximum subgraph with this property. Then, a minimal degree k is imposed to the core of order k.

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 8

k-core decomposition Applications Definition Examples Applicability of k-cores

Outline

1

k-core decomposition Definition Examples Applicability of k-cores

2

Applications Properties analysis Temporal analysis Visualization

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 9

k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Examples

tree : 1-core

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Examples

tree : 2-core ?

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Examples

tree : ✘✘✘

✘ ❳❳❳ ❳

2-core ?

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Examples

tree : 1-core cycle : 2-core

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Examples

tree : 1-core cycle : 2-core clique n : (n − 1)-core

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Examples

tree : 1-core cycle : 2-core clique n : (n − 1)-core

Remark : a k-connected graph is a k-core.

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Examples

A graph :

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Examples

A graph :

3−core 2−core 1−core

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

k-core decomposition

Examples

A graph :

3−core 2−core 1−core

Definition A vertex i has a shell index c, if it belongs to the c-core but not to (c + 1)-core.

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

Outline

1

k-core decomposition Definition Examples Applicability of k-cores

2

Applications Properties analysis Temporal analysis Visualization

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

Why the connectivity is interesting ?

k-core decomposition

The connectivity is mainly related to : robustness

faults, attacks

routing (to find a path between two vertices)

QoS, efficiency (of several parameters).

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 20

k-core decomposition Applications Definition Examples Applicability of k-cores

Why the connectivity is interesting ?

k-core decomposition

The connectivity is mainly related to : robustness

faults, attacks

routing (to find a path between two vertices)

QoS, efficiency (of several parameters).

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 21

k-core decomposition Applications Definition Examples Applicability of k-cores

Why the connectivity is interesting ?

k-core decomposition

The connectivity is mainly related to : robustness

faults, attacks

routing (to find a path between two vertices)

QoS, efficiency (of several parameters).

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 22

k-core decomposition Applications Definition Examples Applicability of k-cores

relation between connectivity and k-cores ?

k-core decomposition

k-connectivity ⇒ k-core

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

relation between connectivity and k-cores ?

k-core decomposition

k-connectivity ⇒ k-core

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

relation between connectivity and k-cores ?

k-core decomposition

k-connectivity ⇒ k-core k-core ⇒ k-connectivity ?

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Definition Examples Applicability of k-cores

relation between connectivity and k-cores ?

k-core decomposition

k-connectivity ⇒ k-core

✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ ❤

k-core ⇒ k-connectivity ?

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 26

k-core decomposition Applications Definition Examples Applicability of k-cores

relation between connectivity and k-cores ?

k-core decomposition

k-connectivity ⇒ k-core Some results on k-core and k-connectivity :

Bollobas [1] : a random graph with large < d > has a minimum degree k when n → ∞, and it is k-connected. Dimes project : study [2] the number of distinct paths as function of k-shell. k-edge-connectivity : we show a theorem (kmax-core has diameter 2, and sets of vertices in s-shell are connected with at least s edges to higher cores).

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 27

k-core decomposition Applications Definition Examples Applicability of k-cores

relation between connectivity and k-cores ?

k-core decomposition

k-connectivity ⇒ k-core Some results on k-core and k-connectivity :

Bollobas [1] : a random graph with large < d > has a minimum degree k when n → ∞, and it is k-connected. Dimes project : study [2] the number of distinct paths as function of k-shell. k-edge-connectivity : we show a theorem (kmax-core has diameter 2, and sets of vertices in s-shell are connected with at least s edges to higher cores).

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 28

k-core decomposition Applications Definition Examples Applicability of k-cores

relation between connectivity and k-cores ?

k-core decomposition

k-connectivity ⇒ k-core Some results on k-core and k-connectivity :

Bollobas [1] : a random graph with large < d > has a minimum degree k when n → ∞, and it is k-connected. Dimes project : study [2] the number of distinct paths as function of k-shell. k-edge-connectivity : we show a theorem (kmax-core has diameter 2, and sets of vertices in s-shell are connected with at least s edges to higher cores).

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

Outline

1

k-core decomposition Definition Examples Applicability of k-cores

2

Applications Properties analysis Temporal analysis Visualization

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

Outline

1

k-core decomposition Definition Examples Applicability of k-cores

2

Applications Properties analysis Temporal analysis Visualization

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

k-shell size vs k

1 10 100

k

10 10

1

10

2

10

3

10

4

|k-shell|

RV 04/2005 CAIDA 04/2005 DIMES 05/2005 INET-3.0

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

cumulative degree distribution

10

  • 3

10

  • 2

10

  • 1

10

P> (d)

1-core 4-core 7-core 10-core 14-core 1-core 4-core 7-core 10-core 14-core

10 10

1

10

2

d/<d>

10

  • 4

10

  • 3

10

  • 2

10

  • 1

P> (d)

1-core 5-core 10-core 15-core 20-core

10 10

1

10

2

10

3

d/<d>

1-core 2-core 3-core 4-core 5-core

RV 2005/04 CAIDA 2005/04 DIMES 2005/05 INET 3.0

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

cumulative degree distribution

10

  • 3

10

  • 2

10

  • 1

10

P> (d)

1-core 4-core 7-core 10-core 14-core 1-core 4-core 7-core 10-core 14-core

10 10

1

10

2

d/<d>

10

  • 4

10

  • 3

10

  • 2

10

  • 1

P> (d)

1-core 5-core 10-core 15-core 20-core

10 10

1

10

2

10

3

d/<d>

1-core 2-core 3-core 4-core 5-core

RV 2005/04 CAIDA 2005/04 DIMES 2005/05 INET 3.0

  • Self-similarity of the k-cores

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

average neighbor degree vs degree

10

  • 1

10

dnn (d)/<dnn>

1-core 4-core 7-core 10-core 14-core 1-core 4-core 7-core 10-core 14-core

10 10

1

10

2

d/<d>

10

  • 1

10

dnn (d)/<dnn>

1-core 5-core 10-core 15-core 20-core

10 10

1

10

2

10

3

d/<d>

1-core 2-core 3-core 4-core 5-core

RV 2005/04 CAIDA 2005/04 DIMES 2005/05 INET 3.0

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

cluster coefficient vs degree

10

  • 2

10

  • 1

10

cc (d)/<cc>

1-core 4-core 7-core 10-core 14-core 1-core 4-core 7-core 10-core 14-core

10 10

1

10

2

d/<d>

10

  • 2

10

  • 1

10

cc (d)/<cc>

1-core 5-core 10-core 15-core 20-core

10 10

1

10

2

10

3

d/<d>

1-core 2-core 3-core 4-core 5-core

RV 2005/04 CAIDA 2005/04 DIMES 2005/05 INET 3.0

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

average betweenness vs shell index

1 10 10

3

10

4

10

5

10

6

10

7

10

8

<g>

1 10 10

2

10

3

10

4

10

5

10

6

10

7

10 10

1

10

2

shell index

10

3

10

4

10

5

10

6

10

7

<g>

1 10

shell index

10

3

10

4

10

5

10

6

RV DIMES INET3.0 CAIDA

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

average betweenness vs shell index

1 10 10

3

10

4

10

5

10

6

10

7

10

8

<g>

1 10 10

2

10

3

10

4

10

5

10

6

10

7

10 10

1

10

2

shell index

10

3

10

4

10

5

10

6

10

7

<g>

1 10

shell index

10

3

10

4

10

5

10

6

RV DIMES INET3.0 CAIDA

betweenness : O(n2 log n) k-core : O(e)

k-core decomposition establish a hierarchy in the network :

  • it can identify the vertex’s centrality.

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 38

k-core decomposition Applications Properties analysis Temporal analysis Visualization

average betweenness vs shell index

1 10 10

3

10

4

10

5

10

6

10

7

10

8

<g>

1 10 10

2

10

3

10

4

10

5

10

6

10

7

10 10

1

10

2

shell index

10

3

10

4

10

5

10

6

10

7

<g>

1 10

shell index

10

3

10

4

10

5

10

6

RV DIMES INET3.0 CAIDA

betweenness : O(n2 log n) k-core : O(e)

k-core decomposition establish a hierarchy in the network :

  • it can identify the vertex’s centrality.

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 39

k-core decomposition Applications Properties analysis Temporal analysis Visualization

Outline

1

k-core decomposition Definition Examples Applicability of k-cores

2

Applications Properties analysis Temporal analysis Visualization

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

AS Oregon R. V. : april -> may 2005

1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 25 5 10 15 20 25

04/2005 -> 05/2005

05/2005 04/2005

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

AS maps of Oregon R. V.

1e-05 1e-04 0.001 0.01 0.1 1 2 4 6 8 10 12 14 16 18 5 10 15 20

2001 -> 2002

2002 2001 1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 5 10 15 20

2002 -> 2003

2003 2002 1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 5 10 15 20 25

2003 -> 2004

2004 2003 1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 25 5 10 15 20 25

2004 -> 2005

2005 2004

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

AS maps of CAIDA

1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 25 5 10 15 20 25 30

2001 -> 2002

2002 2001 1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 25 30 5 10 15 20 25

2002 -> 2003

2003 2002 1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 25 5 10 15 20 25

2003 -> 2004

2004 2003 1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 25 5 10 15 20 25

2004 -> 2005

2005 2004

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

AS maps : 2004->2005

1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 25 5 10 15 20 25

2004 -> 2005

2005 2004 1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 25 5 10 15 20 25

2004 -> 2005

2005 2004

Oregon_RV CAIDA

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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k-core decomposition Applications Properties analysis Temporal analysis Visualization

Outline

1

k-core decomposition Definition Examples Applicability of k-cores

2

Applications Properties analysis Temporal analysis Visualization

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 45

k-core decomposition Applications Properties analysis Temporal analysis Visualization

Network fingerprints

CAIDA AS map

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 46

k-core decomposition Applications Properties analysis Temporal analysis Visualization

Network fingerprints

CAIDA AS map a portion of www.***.fr

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 47

k-core decomposition Applications Properties analysis Temporal analysis Visualization

Network fingerprints

CAIDA AS map CAIDA Router level map

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 48

k-core decomposition Applications Properties analysis Temporal analysis Visualization

Network fingerprints

CAIDA AS map Dimes AS map

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 49

k-core decomposition Applications

Summary

Using k-core decomposition, we can determine :

a hierarchical and self-similar structure centrality connectivity

Comparative analysis of different maps Fingerprints (Visualization) http ://xavier.informatics.indiana.edu/lanet-vi/ http ://arxiv.org/abs/cs.NI/0504107/

(Large scale networks fingerprinting and visualization using the k-core decomposition, Advances in Neural Information Processing Systems 18, 2006),

and http ://arxiv.org/abs/cs.NI/0511007/

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 50

k-core decomposition Applications

Summary

Using k-core decomposition, we can determine :

a hierarchical and self-similar structure centrality connectivity

Comparative analysis of different maps Fingerprints (Visualization) http ://xavier.informatics.indiana.edu/lanet-vi/ http ://arxiv.org/abs/cs.NI/0504107/

(Large scale networks fingerprinting and visualization using the k-core decomposition, Advances in Neural Information Processing Systems 18, 2006),

and http ://arxiv.org/abs/cs.NI/0511007/

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

slide-51
SLIDE 51

k-core decomposition Applications

Summary

Using k-core decomposition, we can determine :

a hierarchical and self-similar structure centrality connectivity

Comparative analysis of different maps Fingerprints (Visualization) http ://xavier.informatics.indiana.edu/lanet-vi/ http ://arxiv.org/abs/cs.NI/0504107/

(Large scale networks fingerprinting and visualization using the k-core decomposition, Advances in Neural Information Processing Systems 18, 2006),

and http ://arxiv.org/abs/cs.NI/0511007/

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

slide-52
SLIDE 52

k-core decomposition Applications

Summary

Using k-core decomposition, we can determine :

a hierarchical and self-similar structure centrality connectivity

Comparative analysis of different maps Fingerprints (Visualization) http ://xavier.informatics.indiana.edu/lanet-vi/ http ://arxiv.org/abs/cs.NI/0504107/

(Large scale networks fingerprinting and visualization using the k-core decomposition, Advances in Neural Information Processing Systems 18, 2006),

and http ://arxiv.org/abs/cs.NI/0511007/

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

slide-53
SLIDE 53

k-core decomposition Applications

Summary

Using k-core decomposition, we can determine :

a hierarchical and self-similar structure centrality connectivity

Comparative analysis of different maps Fingerprints (Visualization) http ://xavier.informatics.indiana.edu/lanet-vi/ http ://arxiv.org/abs/cs.NI/0504107/

(Large scale networks fingerprinting and visualization using the k-core decomposition, Advances in Neural Information Processing Systems 18, 2006),

and http ://arxiv.org/abs/cs.NI/0511007/

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

slide-54
SLIDE 54

k-core decomposition Applications

Summary

Using k-core decomposition, we can determine :

a hierarchical and self-similar structure centrality connectivity

Comparative analysis of different maps Fingerprints (Visualization) http ://xavier.informatics.indiana.edu/lanet-vi/ http ://arxiv.org/abs/cs.NI/0504107/

(Large scale networks fingerprinting and visualization using the k-core decomposition, Advances in Neural Information Processing Systems 18, 2006),

and http ://arxiv.org/abs/cs.NI/0511007/

J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 55

k-core decomposition Applications

  • B. Bollobasr.

Graph Theory and Combinatorics, conference in honor of Paul Erdos, Academic Press, 35–57, 1983. Shai Carmi, Shlomo Havlin1, Scott Kirkpatrick, Yuval Shavitt and Eran Shir. MEDUSA – New Model of Internet Topology Using k-shell Decomposition arXiv, cond-mat/0601240, January 2006.

  • V. Batagelj and M. Zaversnik.

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SLIDE 56

k-core decomposition Applications

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Heuristically Optimized Trade-Offs : A New Paradigm for Power Laws in the Internet. LNCS, 2380 :110–, Jun 2002. J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...

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SLIDE 57

k-core decomposition Applications

k-edge-connectivity

kmax-core is k-edge-connected (or diameter 2) cluster in s-shell is s-edge-connected and it has at least s edges towards (s + 1)-core. then, the graph is k-edge-connected

−1)−shell

k

(

−core

max

k

k

(

k

( −2)−shell −3)−shell J.I.Alvarez-Hamelin :: ISMA 2006 WIT :: CAIDA :: May 10–12 How the k-core decomposition helps in understanding ...