Small worlds

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1 / 61 Pierre Senellart

17 March 2011

Collective Intelligence

Random networks and small worlds

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Small worlds

I proposed a more difficult problem: to find a chain of contacts linking myself with an anonymous riveter at the Ford Motor

Company — and I accomplished it in four steps. The worker knows his foreman, who knows Mr. Ford himself, who, in turn, is on good terms with the director general of the Hearst publishing empire. I had a close friend, Mr. Árpád Pásztor, who had recently struck up an acquaintance with the director of Hearst Publishing. It would take but one word to my friend to send a cable to the general director of Hearst asking him to contact Ford who could in turn contact the foreman, who could then contact the riveter, who could then assemble a new automobile for me, would I need one.

[...] Our friend was absolutely correct: nobody from the group needed more than five links in the chain to reach, just by using the method of acquaintance, any inhabitant of our Planet.

[Karinthy, 1929]

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17 March 2011

Six Degrees of Separation

Idea that two persons on Earth are separated bya chain of six individualswho know each other

Appears widely in popular culture:

It’s a small world!

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17 March 2011

Stanley Milgram’s Experiment [Travers and Milgram, 1969]

Stanley Milgram (1933-1984):social psychologist

Experiment:people are asked to send a message to some unknown person, by forwardingit to anacquaintancewho might be closer to this person

Results: only 29% of the messages arrived, with a mean number of acquaintances of5.2.

Validatessomehow the 6-degree theory!

Other more recent experiments [Dodds et al., 2003] confirm this order of magnitude.

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17 March 2011

Kevin Bacon’s Number

(David Shankbone, Wikimedia)

Kevin Bacon: Hollywood actor, played in numerous movies, mostly

secondary roles

Kevin Bacon’s number:

0 for Kevin Bacon himself

1 for actors who played in the same movie as Bacon

2 for actors who played in the same movie as someone with a number of 1

etc.

http://oracleofbacon.org/

Most actors have asmallBacon’s number!

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17 March 2011

Erd ˝ os number

(Kmhkmh, Wikimedia)

Paul Erd ˝os (1913-1996):

Mathematician and computer

scientist, worked across many fields, with may collaborators

Erd ˝os number:

0 for Paul Erd ˝os himself

1 for scientists who coauthored an article with Erd ˝os

2 for scientists who coauthored an article with someone with a number of 1

etc.

http://www.ams.org/mathscinet/

collaborationDistance.html Most scientists have asmallErd ˝os number!

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17 March 2011

Is there really apatternhere?

How can this be mathematicallymodeled?

Can weexplainwhat happens?

Anything else todiscoverin such networks?

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17 March 2011

Outline

Introduction

Basics of Graph Theory

Characteristics of Real-World Networks Models of Networks

Graph Mining Algorithms Conclusion

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17 March 2011

Graphs

1 2 3

4 5 6

Definition

Adirected graphis a pair(S,A)where:

Sis a finite set ofvertices(ornodes) Ais a subset ofS²defining theedges(or arcs)

1 2 3

4 5 6

Definition

Anundirected graphis a pair(S,A)where:

Sis a finite set ofvertices(ornodes)

Ais a set of (unordered) pairs of elements of Sdefining theedges(orarcs)

Remark

Graphis the mathematical term,networkis used to describe real-world graphs.

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17 March 2011

Paths and Connectedness

Definition

Apathis a sequence of verticesv₁. . .vn such thatv_k is connected by an edge tov_k₊₁for 1≤k ≤n−1.

Definition

Theunderlying undirected graphof a directed graphGis the graph obtained by adding all reverse edges.

Definition

An undirected graph isconnectedif for every two verticesu andv, there exists a path starting fromu and ending inv.

A directed graph isstrongly connectedif it is connected, and isweakly connectedif the underlying undirected graph is connected.

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17 March 2011

Connected Components

Definition

(S^′,A^′)is asubgraphof(S,A)ifS^′ ⊆SandA^′ is the restriction ofAto edges whose vertices are inS^′.

Connected component: maximal connected subgraph

Strongly connected component: maximal strongly connected subgraph

Weakly connected component: maximal weakly connected subgraph

1 2

4 5

3 6

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17 March 2011

Connected Components

Definition

1 2

4 5

3 6

Strongly connected components

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17 March 2011

Connected Components

Definition

1 2

4 5

3 6

Weakly connected components

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17 March 2011

Vocabulary

Incident: an edge is said to beincidentto a vertex if it it hasthis vertex for endpoint

Degree (of a vertex): number of edgesincident toa vertex, in an undirected graph

Indegree (of a vertex): number of edgesarriving toa vertex, in a directed graph

Outdegree (of a vertex): number of edgesleaving froma vertex, in a directed graph

Cycle: Path whose start and end vertex is thesame Distance: Length of theshortest pathbetween two vertices

Sparse: a graph(S,A)is sparse if|A| ≪ |S|²

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17 March 2011

Bipartite Graphs

Definition

Abipartitegraph is an undirected graph(S,A)such thatS=S₁∪S₂ (withS₁∩S₂=∅), and no edge ofAis incident to two vertices inS₁or two vertices inS₂.

graphs.

1 2 3 4

5 6 7

1 2

3 4

5 6

7

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17 March 2011

Bipartite Graphs

Definition

Paths of length 2 in a bipartite graph define two regular undirected graphs.

1 2 3 4

5 6 7

1 2

3 4

5 6

7

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17 March 2011

Bipartite Graphs

Definition

Paths of length 2 in a bipartite graph define two regular undirected graphs.

1 2 3 4

5 6 7

1 2

3 4

5 6

7

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17 March 2011

Beware of Graph Drawings

1 2

3 4

1 2

3 4

1

2

3 4

Three times the same graph! No “best” graph

Not always possible to have aplanar graph

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17 March 2011

Beware of Graph Drawings

1 2

3 4

1 2

3 4

1

2

3 4

Three times the same graph!

No “best” graph

Not always possible to have aplanar graph

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17 March 2011

Matrix Representation of a Graph

A graphGcan be represented by itsadjacency matrixM:

1 2 3

4 5 6

⎛

⎜

⎝

0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0

⎞

⎟

⎠

Adjacency matrices ofundirectedgraphs aresymmetric.

M^T is the adjacency matrix obtained fromGbyreversingthe arrows.

Mⁿis the matrix of the graph of allpaths of lengthninG.

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17 March 2011

Concrete Representation of Graphs

In programming, graphs are usually represented as:

itsadjacency matrix(stored as a multidimensional array), for non-sparsegraphs

the list of alledgesincident to each node, forsparsegraphs

1 2 3

4 5 6

1 2 2 2,4,5 3 6 4 1,5 5 4 6

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17 March 2011

Outline

Introduction

Characteristics of Real-World Networks Social Networks

Natural Networks Artificial Networks Models of Networks Graph Mining Algorithms Conclusion

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17 March 2011

Characteristics of Interest

Sparsity. Is the network sparse (|A| ≪ |S|²)?

All networks considered here will be sparse.

Typical distance. What is themean distancebetween any pairs of vertices?

Local clustering. Ifais connected to bothbandc, is the probability thatb is connected toc significantly greater than the probability any two nodes are connected?

Degree distribution. What is the distribution of the degree of vertices?

k P

Poisson Power-law Gaussian

𝜆^k

k! k^−𝛾 e^−k²

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17 March 2011

Outline

Introduction

Natural Networks Artificial Networks

Models of Networks Graph Mining Algorithms Conclusion

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17 March 2011

Acquaintance Network

As in the experiment by Milgram

. . . or as given bysocial networking sitessuch as Facebook, LinkedIn. . .

Logarithmic typical distance Strong local clustering

Gaussian degree distribution [Amaral et al., 2000]

k P

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17 March 2011

Acquaintance Network

As in the experiment by Milgram

. . . or as given bysocial networking sitessuch as Facebook, LinkedIn. . .Network characteristics

Gaussian degree distribution [Amaral et al., 2000]

k P

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17 March 2011

Actors and Scientists Networks

BipartitegraphsActor-MovieandScientist-Publication Corresponding undirected graphs:

actorsappearing in the same movie scientists whocoauthoreda paper

Bacon’s/Erd ˝os number:distancein the graph to the corresponding vertex

Power-law degree distribution (2≤𝛾 ≤3), with a possible tail cutoff [Amaral et al., 2000]

k P

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17 March 2011

Actors and Scientists Networks

BipartitegraphsActor-MovieandScientist-Publication Corresponding undirected graphs:

actorsappearing in the same movie scientists whocoauthoreda paper

Bacon’s/Erd ˝os number:distancein the graph to the corresponding vertex

Network characteristics

Power-law degree distribution (2≤𝛾 ≤3), with a possible tail cutoff [Amaral et al., 2000]

k P

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17 March 2011

Sex Networks

[Amaral et al., 2000]

In this particular case (small and incomplete community): [Amaral et al., 2000]

Unconnected network,longtypical distance No local clustering (the graph is almost bipartite!) But for larger studies [Liljeros et al., 2001]:

Logarithmic typical distance

No strict local clustering because of predominance of heterosexuality, butsome kind of locality

Power-law degree distribution (𝛾≈2.5 for females, 𝛾≈2.3 for males)

k P

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17 March 2011

Sex Networks

[Amaral et al., 2000]

In this particular case (small and incomplete community): [Amaral et al., 2000]

Unconnectednetwork,long typical distance No local clustering (the graph is almost bipartite!) But for larger studies [Liljeros et al., 2001]:

Logarithmic typical distance

No strict local clustering because of predominance of heterosexuality, butsome kind of locality

Power-law degree distribution (𝛾≈2.5 for females, 𝛾≈2.3 for males)

k P

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17 March 2011

Sociological aspects

An individual has two kinds of social network:

his actual connections with individuals (acquaintances, relationships, etc.) (explicit network)

his virtual connections with similar individuals (implicit network) Sociologically speaking, four kinds of connections between individuals [Smith et al., 2007], depending on the kind ofsocial capital[Lin, 2001] considered:

Implicit link

Yes No

Explicit link Yes Actual bonding Actual bridging No Potential bonding Potential bridging

Bothbondingandbridgingnecessary to make one’s life successful

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17 March 2011

Outline

Introduction

Natural Networks Artificial Networks

Models of Networks Graph Mining Algorithms Conclusion

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17 March 2011

Neural Networks

(Dorling Kindersley, dkimages)

Logarithmic typical distance [Watts and Strogatz, 1998]

Strong local clustering

Power-law degree distribution

k P

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17 March 2011

Neural Networks

(Dorling Kindersley, dkimages)

Logarithmic typical distance [Watts and Strogatz, 1998]

Power-law degree distribution

k P

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17 March 2011

Metabolic Networks

(Laboratory of Computer Engineering, Technical University of Helsinki)

Power-law degree distribution (2≤𝛾 ≤2.4) [Jeong et al., 2000]

k P

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17 March 2011

Metabolic Networks

(Laboratory of Computer Engineering, Technical University of Helsinki)

Power-law degree distribution (2≤𝛾 ≤2.4) [Jeong et al., 2000]

k P

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17 March 2011

Outline

Introduction

Natural Networks Artificial Networks Models of Networks Graph Mining Algorithms Conclusion

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17 March 2011

The Internet: physical connections be- tween LANs

http://www.opte.org/

Power-law degree distribution (𝛾 ≈2.2) [Faloutsos et al., 1999]

k P

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The Internet: physical connections be- tween LANs

http://www.opte.org/

Power-law degree distribution (𝛾 ≈2.2) [Faloutsos et al., 1999]

k P

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17 March 2011

The Web: logical hyperlinks between Web pages

[Broder et al., 2000]

Network characteristics Directed graph

Power-law indegree and outdegree distribution (2≤𝛾 ≤3) [Broder et al., 2000]

k P

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17 March 2011

The Web: logical hyperlinks between Web pages

[Broder et al., 2000]

Power-law indegree and outdegree distribution (2≤𝛾 ≤3) [Broder et al., 2000]

k P

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Scientific Citations Network

Vertices: Scientific publications Edges: Citation links

No cycles! No strong connectivity.

Strong local clustering (on the underlying undirected graph)

Power-law indegree and outdegree distribution (2≤𝛾 ≤3)

k P

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Scientific Citations Network

Vertices: Scientific publications Edges: Citation links

No cycles! No strong connectivity.

Strong local clustering (on the underlying undirected graph)

Power-law indegree and outdegree distribution (2≤𝛾 ≤3)

k P

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17 March 2011

Transportation Networks

Network characteristics Long typical distance Strong local clustering Limited degree variations

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17 March 2011

Transportation Networks

Network characteristics Long typical distance Strong local clustering Limited degree variations

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17 March 2011

Outline

Introduction

Random Networks Small Worlds

Scale-Free Networks Graph Mining Algorithms Conclusion

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17 March 2011

Outline

Introduction

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17 March 2011

Random Networks [Solomonoff and Rapoport, 1951, Erd ˝ os and Rényi, 1960]

Construction

1. Start withnvertices and a probabilityp.

2. For each pair of vertices(u,v), insert an edge betweenuandv with probabilityp.

Sparseifp≪1

Logarithmictypical distance (inside the giant connected component)!

No local clustering.

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17 March 2011

Random Networks [Solomonoff and Rapoport, 1951, Erd ˝ os and Rényi, 1960]

Construction

1. Start withnvertices and a probabilityp.

2. For each pair of vertices(u,v), insert an edge betweenuandv with probabilityp.

Sparseifp≪1

Logarithmictypical distance (inside the giant connected component)!

No local clustering.

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17 March 2011

Degree distribution in random networks

P(k) = (︃n

k )︃

p^k(1−p)^n−k ∼ (pn)^ke^−pn k!

k P

Exercise Prove this.

Remark

One can construct random graphs with anarbitrary degree distribution (more complicated); stillno local clustering, obviously.

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17 March 2011

Kolmogorov’s 0-1 Law

Andrey Kolmogorov (1903-1987):

Brilliant mathematician, laid the foundations of modern probability theory, between other very varied and important works, with impact in mathematics, physics, and computer science.

Kolmogorov’s 0-1 Law: Given an infinite sequence of independent random variables, events that are probabilistically independent of any finite subset of the random variables have probabilityeither 0 or 1.

Application:The probability that there is a (single)giantconnected component goes to 1 when the size of the graph goes either to 0 or to+∞, depending onp(actually, the transition isp≥1/n).

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17 March 2011

Outline

Introduction

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17 March 2011

Small Worlds [Watts and Strogatz, 1998, Watts, 1999]

Construction

1. Start with aregular lattice(a grid).

2. With probabilityp,rerouteeach edge randomly.

[Watts and Strogatz, 1998]

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17 March 2011

Small Worlds [Watts and Strogatz, 1998, Watts, 1999]

Construction

1. Start with aregular lattice(a grid).

2. With probabilityp,rerouteeach edge randomly.

[Watts and Strogatz, 1998]

Sparse.

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17 March 2011

Characteristics of Small Worlds

Forp=0: lattice (stronglocal clustering)

Forp=1: random graph (smalltypical distance) Somewhere in between:

Smalltypical distance (thanks torerouting) Stronglocal clustering (thanks to theinitial lattice) Degree distribution resembling a Poisson.

k P

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17 March 2011

Measuring the local clustering

C_G = 3×(number of triangles inG) number of connected triples inG C_fg=1for a fully connected graph

C_rg=pfor a random graph

A graphGhasstrong local clusteringifC_G ≫C_rg(for the random graph with the same number of edges)

Exercise ProveC_rg=p.

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17 March 2011

Outline

Introduction

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17 March 2011

Preferential Attachment [Barabási and Al- bert, 1999]

Construction

1. Start with a small graph of sizem₀, letmbe a constant with m<m₀.

2. One after the other,n−m₀vertices are added to the graph, connecting them tomexisting vertices; the probability of connecting to a vertex isproportionalto its degree.

Sparse ifm andn are chosen appropriately. Small typical distance.

Power-law degree distribution (actually, with𝛾 =3, but variations allow arbitrary exponents).

k P

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17 March 2011

Preferential Attachment [Barabási and Al- bert, 1999]

Construction

1. Start with a small graph of sizem₀, letmbe a constant with m<m₀.

2. One after the other,n−m₀vertices are added to the graph, connecting them tomexisting vertices; the probability of connecting to a vertex isproportionalto its degree.

Sparse ifm andn are chosen appropriately.

Small typical distance.

Power-law degree distribution (actually, with𝛾 =3, but variations allow arbitrary exponents).

k P

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17 March 2011

Scale-Free Graphs

Graphs with the power-law degree distribution are calledscale-free graphs:

There is notypical scale, or typical order of magnitude for the degree of nodes.

P(𝛼k)

P(k) = (𝛼k)^−𝛾

k^−𝛾 =𝛼^−𝛾

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17 March 2011

Outline

Introduction

Graph Mining Algorithms Search

Discovery of communities Conclusion

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17 March 2011

Outline

Introduction

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17 March 2011

Google’s PageRank [Brin and Page, 1998]

Idea

Importantpages are pages pointed to byimportantpages.

{︃g_ij =0 if there is no link between pageiandj;

g_ij = _n¹

i otherwise, withn_i the number of outgoing links of pagei.

Definition (Tentative)

Probabilitythat the surfer following therandom walkinGhas arrived on pagei at some distant given point in the future.

pr(i) = (︂

k→+∞lim (G^T)^kv )︂

i

wherev is some initial column vector.

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Illustrating PageRank Computation

0.100 0.100

0.100

0.100 0.100

0.100

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Illustrating PageRank Computation

0.033 0.317

0.075

0.108

0.025 0.058

0.083

0.150

0.117

0.033

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Illustrating PageRank Computation

0.036 0.193

0.108

0.163

0.079 0.090

0.074

0.154

0.094

0.008

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Illustrating PageRank Computation

0.054 0.212

0.093

0.152

0.048 0.051

0.108

0.149

0.106

0.026

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Illustrating PageRank Computation

0.051 0.247

0.078

0.143

0.053 0.062

0.097

0.153

0.099

0.016

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Illustrating PageRank Computation

0.048 0.232

0.093

0.156

0.062 0.067

0.087

0.138

0.099

0.018

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Illustrating PageRank Computation

0.052 0.226

0.092

0.148

0.058 0.064

0.098

0.146

0.096

0.021

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Illustrating PageRank Computation

0.049 0.238

0.088

0.149

0.057 0.063

0.095

0.141

0.099

0.019

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Illustrating PageRank Computation

0.050 0.232

0.091

0.149

0.060 0.066

0.094

0.143

0.096

0.019

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Illustrating PageRank Computation

0.050 0.233

0.091

0.150

0.058 0.064

0.095

0.142

0.098

0.020

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Illustrating PageRank Computation

0.050 0.234

0.090

0.148

0.058 0.065

0.095

0.143

0.097

0.019

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Illustrating PageRank Computation

0.049 0.233

0.091

0.149

0.058 0.065

0.095

0.142

0.098

0.019

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Illustrating PageRank Computation

0.050 0.233

0.091

0.149

0.058 0.065

0.095

0.143

0.097

0.019

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17 March 2011

Illustrating PageRank Computation

0.050 0.234

0.091

0.149

0.058 0.065

0.095

0.142

0.097

0.019

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PageRank With Damping

May not always converge, or convergence may not be unique.

To fix this, the random surfer can at each steprandomly jumpto any page of the Web with some probabilityd (1−d:damping factor).

pr(i) = (︂

k→+∞lim ((1−d)G^T +dU)^kv )︂

i

whereU is the matrix with all _N¹ values withN the number of vertices.

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Iterative Computation of PageRank

1. ComputeG(often stored as its adjacency list). Make sure lines sum to 1.

2. Letu be the uniform vector of sum 1,v =u,w thezerovector.

3. Whilev isdifferent enoughfromw:

Setw =v.

Setv = (1−d)G^Tv+du.

Exercise

1 2

3 4

Run the first iteration of the PageRank computation.

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Using PageRank to Score Search Results

PageRank: globalscore, independent of the query

Can be used to raise the weight ofimportantpages, associated with some scoring function dependent of the query:

final(q,d) =score(q,d)×pr(d),

PageRank only useful indirectedgraphs! Proportional todegree otherwise

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17 March 2011

HITS [Kleinberg, 1999]

Idea

Two kinds of important pages: hubs and authorities. Hubs are pages that point to good authorities, whereas authorities are pages that are pointed to by good hubs.

G^′ adjacency matrix (with 0 and 1 values) of asubgraphof the Web.

We use the following iterative process (starting withaandhvectors of norm 1):

⎧

⎨

⎩

a:= _‖G′T¹h‖ G^′Th h:= _‖G¹′a‖ G^′a

Convergesunder some technical assumptions toauthorityandhub scores.

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Using HITS to Order Web Query Results

1. Retrieve the setDof Web pagesmatchinga keyword query.

2. Retrieve the setD^* of Web pages obtained fromDby addingall linked pages, as well as allpages linking topages ofD.

3. Build fromD^*the correspondingsubgraphG^′ of the Web graph.

4. Computeiterativelyhubs and authority scores.

5. Sort documents fromDbyauthority scores.

Less efficient than PageRank, becauselocalscores.

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Outline

Introduction

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Discovery of communities

Classical problem in social networks: identifyingcommunitiesof users (or of content) using thegraph structure

Two subproblems:

1. Given some initial vertex or vertex set, finding the corresponding community

2. Given the graph as a whole, finding a partition in communities

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Maximum Flow / Minimum Cut

/6 /2

/1 /5 /2

/3

sink source

/4

Tarjan, 1988] to separate aseedof users from the remaining of the graph

ComplexityO(n²m)(n: vertices,m: edges)

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Maximum Flow / Minimum Cut

/6 /2

/1 /5 /2

/3 source

4 0

3 2

1 /4 4

1 sink

Use of a maximum flow computation algorithm [Goldberg and Tarjan, 1988] to separate aseedof users from the remaining of the graph

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Maximum Flow / Minimum Cut

/6 /2

/1 /5 /2

/3

sink source

4 0

3 2

1 /4 4

1

Use of a maximum flow computation algorithm [Goldberg and Tarjan, 1988] to separate aseedof users from the remaining of the graph

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Markov Cluster Algorithm (MCL) [van Don- gen, 2000]

Graphclusteringalgorithm

Based as well on maximum flow simulation, in the whole graph Iteration of a matrix computation alternating:

Expansion(matrix multiplication, corresponding to flow propagation)

Inflation(non-linear operation to increase heterogeneity) Complexity: O(n³)for an exact computation,O(n)for an approximate one

[van Dongen, 2000]

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Markov Cluster Algorithm (MCL) [van Don- gen, 2000]

Graphclusteringalgorithm

Based as well on maximum flow simulation, in the whole graph Iteration of a matrix computation alternating:

Expansion(matrix multiplication, corresponding to flow propagation)

Inflation(non-linear operation to increase heterogeneity) Complexity: O(n³)for an exact computation,O(n)for an approximate one

[van Dongen, 2000]

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Deletion of the edges with the highest be- twenness [Newman and Girvan, 2004]

Top-downgraph clustering algorithm

Betwennessof an edge: number of minimal paths between two arbitrary vertices going through this edge

General principle:

1. Compute thebetweennessof each edge in the graph 2. Removethe edge with the highest betweenness

3. Redo the whole process, betweenness computation included Complexity: O(n³)for a sparse graph

[Newman and Girvan, 2004]

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Outline

Introduction

Graph Mining Algorithms Conclusion

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To remember

What you should remember

1. Most (but not all!) real-worldnetworks:

are sparse

have small typical distance have strong local clustering

2. In addition, a large class of them arescale-free

3. Three simplemodels of networks, modeling (and explaining?) some or all of these properties:

Random graphs Small worlds

Preferential attachment

4. A collection of algorithms formining graphs

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Applications of the Models

Epidemiology

Network fault detection

Efficient search in P2P networks . . .

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To go further

[Watts, 1999]: an easy-to-read book describing the small world problem and small-world models, with concrete applications

[Newman et al., 2006]: an in-detail survey of the most fundamental works on network theory, networks models, and experimentations on real-world networks

[Chakrabarti, 2003]: the particular example of the World Wide Web

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L. A. Amaral, A. Scala, M. Barthelemy, and H. E. Stanley. Classes of small-world networks. PNAS, 97(21):11149–11152, October 2000.

Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. Science, 286(5439):509–512, October 1999.

Sergey Brin and Lawrence Page. The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems, 30(1–7):107–117, 1998.

Andrei Broder, Ravi Kumar, Farzin Maghoul, Prabhakar Raghavan, Sridhar Rajagopalan, Raymie Stata, Andrew Tomkins, and Janet Wiener. Graph structure in the web. Computer Networks, 33(1-6):

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17 March 2011

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