k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Analysis and visualization of large scalenetworks using the k -core decomposition

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

European Conference on Complex SystemsECCS’05 :: Paris :: 14-18 November 2005

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Plan

1 k -core decomposition

2 A k -core analysis of AS and IR Internet graphs

3 Network fingerprints

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Plan

1 k -core decomposition

2 A k -core analysis of AS and IR Internet graphs

3 Network fingerprints

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionIntroduction

Motivation : Analysis of complex networks,in particular their hierarchy.

The hierarchy is related to the role that vertices play, i.e. thecentrality and connectivity patterns. The connectivity is mainlyrelated to :

robustnessfaults,attacks

routing (to find a path between two vertices)QoS,efficiency (of a determined parameter).

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionIntroduction

Motivation : Analysis of complex networks,in particular their hierarchy.

The hierarchy is related to the role that vertices play, i.e. thecentrality and connectivity patterns. The connectivity is mainlyrelated to :

robustnessfaults,attacks

routing (to find a path between two vertices)QoS,efficiency (of a determined parameter).

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionIntroduction

Motivation : Analysis of complex networks,in particular their hierarchy.

The hierarchy is related to the role that vertices play, i.e. thecentrality and connectivity patterns. The connectivity is mainlyrelated to :

robustnessfaults,attacks

routing (to find a path between two vertices)QoS,efficiency (of a determined parameter).

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionDefinition

Given G = {V , E} a undirected graph, where V is the verticesset and E is the edges set.

Definition [Batagelj and Zaversnik, 2002] :

A subgraph H = (C, E |C) induced by the set C ⊆ V is a k -coreor a core of order k iff ∀v ∈ C : degree H(v) ≥ k , and H is themaximum subgraph with this property.

Then, a minimal degree k is imposed to the core of order k .

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionDefinition

Given G = {V , E} a undirected graph, where V is the verticesset and E is the edges set.

Definition [Batagelj and Zaversnik, 2002] :

A subgraph H = (C, E |C) induced by the set C ⊆ V is a k -coreor a core of order k iff ∀v ∈ C : degree H(v) ≥ k , and H is themaximum subgraph with this property.

Then, a minimal degree k is imposed to the core of order k .

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionDefinition

Given G = {V , E} a undirected graph, where V is the verticesset and E is the edges set.

Definition [Batagelj and Zaversnik, 2002] :

A subgraph H = (C, E |C) induced by the set C ⊆ V is a k -coreor a core of order k iff ∀v ∈ C : degree H(v) ≥ k , and H is themaximum subgraph with this property.

Then, a minimal degree k is imposed to the core of order k .

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionExamples

tree : 1-core

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionExamples

tree : 2-core ?

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionExamples

tree : ����XXXX2-core ?

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionExamples

tree : 1-core cycle : 2-core

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionExamples

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

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionExamples

A graph :

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionExamples

A graph :

3−core

2−core

1−core

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionExamples

A graph :

3−core

2−core

1−core

Definition

A vertex i has coreness c, or its shell index is c, if it belongs tothe c-core but not to (c + 1)-core.

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionComplexity of the k -core decomposition

G is represented by its vertex list, where each one has aneighbors list.

1 Make an ordered array of lists, where each list iscomposed by the vertices with the same degree :O(n), where n is the number of vertices.

2 Compute each k -core starting by the minimum degreekmin, until any vertex remains in the graph :O(e), where e is the number of edges.

In general, the complexity is O(n + e)⇒ it is very useful for large networks .

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionComplexity of the k -core decomposition

G is represented by its vertex list, where each one has aneighbors list.

1 Make an ordered array of lists, where each list iscomposed by the vertices with the same degree :O(n), where n is the number of vertices.

2 Compute each k -core starting by the minimum degreekmin, until any vertex remains in the graph :O(e), where e is the number of edges.

In general, the complexity is O(n + e)⇒ it is very useful for large networks .

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionComplexity of the k -core decomposition

G is represented by its vertex list, where each one has aneighbors list.

1 Make an ordered array of lists, where each list iscomposed by the vertices with the same degree :O(n), where n is the number of vertices.

2 Compute each k -core starting by the minimum degreekmin, until any vertex remains in the graph :O(e), where e is the number of edges.

In general, the complexity is O(n + e)⇒ it is very useful for large networks .

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionComplexity of the k -core decomposition

G is represented by its vertex list, where each one has aneighbors list.

1 Make an ordered array of lists, where each list iscomposed by the vertices with the same degree :O(n), where n is the number of vertices.

2 Compute each k -core starting by the minimum degreekmin, until any vertex remains in the graph :O(e), where e is the number of edges.

In general, the complexity is O(n + e)⇒ it is very useful for large networks .

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

k -core decompositionComplexity of the k -core decomposition

G is represented by its vertex list, where each one has aneighbors list.

1 Make an ordered array of lists, where each list iscomposed by the vertices with the same degree :O(n), where n is the number of vertices.

2 Compute each k -core starting by the minimum degreekmin, until any vertex remains in the graph :O(e), where e is the number of edges.

In general, the complexity is O(n + e)⇒ it is very useful for large networks .

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Plan

1 k -core decomposition

2 A k -core analysis of AS and IR Internet graphs

3 Network fingerprints

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Applying k -core decompositionto Internet graphs

We computed the k -cores of AS and IR Internet maps,then, we computed for each k -core :

their size as function of k

the cumulative degree distribution

the average clustering as a function of the degree

the average neighbors degree as a funtion of the degree

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Applying k -core decompositionto Internet graphs

We computed the k -cores of AS and IR Internet maps,then, we computed for each k -core :

their size as function of k

the cumulative degree distribution

the average clustering as a function of the degree

the average neighbors degree as a funtion of the degree

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Applying k -core to Internet graphs

1 10 100k

101

102

103

104

105

|k-c

ore|

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

k -core size as function of k

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Applying k -core to Internet graphs

10-3

10-2

10-1

100

P> (

d)1-core4-core7-core10-core14-core

1-core4-core7-core10-core14-core

100

101

102

d/<d>

10-4

10-3

10-2

10-1

P> (

d)

1-core5-core10-core15-core20-core

100

101

102

103

d/<d>

1-core2-core3-core4-core5-core

RV 2005/04 CAIDA 2005/04

DIMES 2005/05 INET 3.0

Cumulative degree distribution

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Applying k -core to Internet graphs

10-1

100

d nn (d

) / <

d nn> 1-core

4-core7-core10-core14-core

1-core4-core7-core10-core14-core

100 101 102 103

d

10-1

100

101

d nn (d

) / <

d nn> 1-core

5-core10-core15-core20-core

101 102 103 104

d

1-core2-core3-core4-core5-core

RV 2005/04 CAIDA 2005/04

DIMES 2005/05 INET 3.0

Average clustering distribution as degree function

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Applying k -core to Internet graphs

10-2

10-1

100

cc (d

) / <

cc>

1-core4-core7-core10-core14-core

1-core4-core7-core10-core14-core

100 101 102 103

d10-2

10-1

cc (d

) / <

cc>

1-core5-core10-core15-core20-core

100 101 102 103 104

d

1-core2-core3-core4-core5-core

RV 2005/04 CAIDA 2005/04

DIMES 2005/05 INET 3.0

Average neighbors degree distribution as degree function

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Plan

1 k -core decomposition

2 A k -core analysis of AS and IR Internet graphs

3 Network fingerprints

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Network fingerprintAS_CAIDA [5] 04/2005, 23 symmetrical sources

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Network fingerprintIR_CAIDA [5] 2003, 23 symmetrical sources

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Network fingerprintComparation between IR and AS of CAIDA

Autonomous System graph

Router graph

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Summary

k -core decomposition

Due to low complexity, it can be applied to analysis andvisualization of large scale networks, e.g. Internet.

Self-similarity of k -cores on the distributions (cumulativedegree, average neighbors degree and clustering)

Visualization using k -cores yields fingerprints useful tovalidate models and to compare different maps.

Public visualization tool :http ://xavier.informatics.indiana.edu/lanet-vi/

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Summary

k -core decomposition

Due to low complexity, it can be applied to analysis andvisualization of large scale networks, e.g. Internet.

Self-similarity of k -cores on the distributions (cumulativedegree, average neighbors degree and clustering)

Visualization using k -cores yields fingerprints useful tovalidate models and to compare different maps.

Public visualization tool :http ://xavier.informatics.indiana.edu/lanet-vi/

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Summary

k -core decomposition

Due to low complexity, it can be applied to analysis andvisualization of large scale networks, e.g. Internet.

Self-similarity of k -cores on the distributions (cumulativedegree, average neighbors degree and clustering)

Visualization using k -cores yields fingerprints useful tovalidate models and to compare different maps.

Public visualization tool :http ://xavier.informatics.indiana.edu/lanet-vi/

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Summary

k -core decomposition

Due to low complexity, it can be applied to analysis andvisualization of large scale networks, e.g. Internet.

Self-similarity of k -cores on the distributions (cumulativedegree, average neighbors degree and clustering)

Visualization using k -cores yields fingerprints useful tovalidate models and to compare different maps.

Public visualization tool :http ://xavier.informatics.indiana.edu/lanet-vi/

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

Summary

k -core decomposition

Due to low complexity, it can be applied to analysis andvisualization of large scale networks, e.g. Internet.

Self-similarity of k -cores on the distributions (cumulativedegree, average neighbors degree and clustering)

Visualization using k -cores yields fingerprints useful tovalidate models and to compare different maps.

Public visualization tool :http ://xavier.informatics.indiana.edu/lanet-vi/

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores

k -core decompositionA k -core analysis of AS and IR Internet graphs

Network fingerprints

V. Batagelj and M. Zaversnik.

Generalized Cores.CoRR, cs.DS/0202039, 2002.

M. Baur, U. Brandes, M. Gaertler, and D. Wagner.

Drawing the AS Graph in 2.5 Dimensions.In "12th International Symposium on Graph Drawing, Springer-Verlag editor", pages 43–48, 2004.

J. I. Alvarez-Hamelin, L. Dall’Asta, A. Barrat, and A. Vespignani.

k -core decomposition : a tool for the visualization of large scale networks.arxiv.org, cs.NI/0504107, 2005.

LArge NETwork VIsualization tool.http ://xavier.informatics.indiana.edu/lanet-vi/ .

Router-Level Topology Measurements" "Cooperative Association for Internet Data Analysis.http ://www.caida.org/tools/measurement/skitter/ router_topology/ .

University of Oregon Route Views Project.http ://www.routeviews.org/ .

"Distributed Internet MEasurements and Simulations".http ://www.netdimes.org .

Jared Winick and Sugih Jamin.

Inet-3.0 : Internet topology generator.Technical Report UM-CSE-TR-456-02, Department of EECS, University of Michigan, 2002.

J.I.Alvarez-Hamelin :: ECCS’05 Analysis and visualization using k -cores