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Random graphs: a probabilistic point of view Laurent M´ enard Modal’X, Universit´ e Paris Ouest Stats in Paris, November 2013
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Page 1: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Random graphs:a probabilistic point of view

Laurent MenardModal’X, Universite Paris Ouest

Stats in Paris, November 2013

Page 2: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

”Real world” networks

Collaboration graph of mathematicians[The Erdos number project, 2004]

Page 3: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

”Real world” networks

The internet topology in 1999[The internet mapping project]

Page 4: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

”Real world” networks

[Tellez 2013]

Page 5: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Outline

What are we looking for ?

Most common properties of ”real world networks”.

Page 6: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Outline

What are we looking for ?

Different models of random graphs and their properties

Most common properties of ”real world networks”.

Erdos-Renyi random graphs, configuration model,preferential attachment graphs, . . .

Page 7: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Outline

What are we looking for ?

Different models of random graphs and their properties

Convergence of random graphs

Most common properties of ”real world networks”.

Erdos-Renyi random graphs, configuration model,preferential attachment graphs, . . .

Local weak convergence and other notions

Page 8: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Outline

What are we looking for ?

Different models of random graphs and their properties

Convergence of random graphs

Statistical mechanics on random graphs

Most common properties of ”real world networks”.

Erdos-Renyi random graphs, configuration model,preferential attachment graphs, . . .

Local weak convergence and other notions

Contagion models, systemic risk, first passage percolation, . . .

Page 9: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling networks: Graph theory

Page 10: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling networks: Graph theory

A (simple, undirected) graphG = (V,E) consists of

• a set of vertices V = {1, . . . , n}• a set of edgesE ⊂ {{i, j} : i, j ∈ V and i 6= j}

Page 11: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling networks: Graph theory

A (simple, undirected) graphG = (V,E) consists of

• a set of vertices V = {1, . . . , n}• a set of edgesE ⊂ {{i, j} : i, j ∈ V and i 6= j}

Complete graph with• 6 vertices• 15 edges

Page 12: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling networks: Graph theory

A (simple, undirected) graphG = (V,E) consists of

• a set of vertices V = {1, . . . , n}• a set of edgesE ⊂ {{i, j} : i, j ∈ V and i 6= j}

Complete graph with• 6 vertices• 15 edges

Tree with• 11 vertices• 10 edges

Page 13: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling networks: Graph theory

A (simple, undirected) graphG = (V,E) consists of

• a set of vertices V = {1, . . . , n}• a set of edgesE ⊂ {{i, j} : i, j ∈ V and i 6= j}

Complete graph with• 6 vertices• 15 edges

Tree with• 11 vertices• 10 edges

Graph with• 21 vertices• 21 edges

Page 14: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Graph theory: some vocabulary

• Path from vertex i to vertex j:sequence of edges connecting i to j

i

j

Page 15: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Graph theory: some vocabulary

• Path from vertex i to vertex j:sequence of edges connecting i to j

• Length of a path:number of edges in the path

i

j

6

Page 16: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Graph theory: some vocabulary

• Path from vertex i to vertex j:sequence of edges connecting i to j

• Length of a path:number of edges in the path

• Geodesic path from i to j:shortest path from i to j (not necessarily unique)

i

j

Page 17: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Graph theory: some vocabulary

• Path from vertex i to vertex j:sequence of edges connecting i to j

• Length of a path:number of edges in the path

• Geodesic path from i to j:shortest path from i to j (not necessarily unique)

• Distance between i and j:dG(i, j)= length of a geodesic path from i to j.

i

j2

Page 18: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Graph theory: some vocabulary

• Path from vertex i to vertex j:sequence of edges connecting i to j

• Length of a path:number of edges in the path

• Geodesic path from i to j:shortest path from i to j (not necessarily unique)

• Distance between i and j:dG(i, j)= length of a geodesic path from i to j.

i

j

• Degree of a node i:di= number of edges i belongs to

di = 4

Page 19: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Graph theory: some vocabulary

• Path from vertex i to vertex j:sequence of edges connecting i to j

• Length of a path:number of edges in the path

• Geodesic path from i to j:shortest path from i to j (not necessarily unique)

• Distance between i and j:dG(i, j)= length of a geodesic path from i to j.

• Degree of a node i:di= number of edges i belongs to

• Connected component of a graph G:maximal connected subgraph

6 connectedcomponents

Page 20: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Graph theory: some vocabulary

• Path from vertex i to vertex j:sequence of edges connecting i to j

• Length of a path:number of edges in the path

• Geodesic path from i to j:shortest path from i to j (not necessarily unique)

• Distance between i and j:dG(i, j)= length of a geodesic path from i to j.

• Degree of a node i:di= number of edges i belongs to

• Connected component of a graph G:maximal connected subgraph

6 connectedcomponents

• Diameter of a connected component:largest distance between two vertices of the component

1 0 4

2

Page 21: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling large ”real world” networks: random graphs

• Size of the network: n vertices (n deterministic and large)• Network: random graph Gn (a random variable taking values in

the set of all graphs with n vertices)

Page 22: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling large ”real world” networks: random graphs

• Size of the network: n vertices (n deterministic and large)• Network: random graph Gn (a random variable taking values in

the set of all graphs with n vertices)

What properties do we want for Gn?

Page 23: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling large ”real world” networks: random graphs

• Size of the network: n vertices (n deterministic and large)• Network: random graph Gn (a random variable taking values in

the set of all graphs with n vertices)

What properties do we want for Gn?

SparseVertex degrees are very small

compared to the size of the network

Page 24: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling large ”real world” networks: random graphs

Small worldDistances are very small

compared to the size of the network

• Size of the network: n vertices (n deterministic and large)• Network: random graph Gn (a random variable taking values in

the set of all graphs with n vertices)

What properties do we want for Gn?

SparseVertex degrees are very small

compared to the size of the network

Page 25: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling large ”real world” networks: random graphs

Small world

Scale free

Distances are very smallcompared to the size of the network

Vertices with very high degreeare not uncommon

• Size of the network: n vertices (n deterministic and large)• Network: random graph Gn (a random variable taking values in

the set of all graphs with n vertices)

What properties do we want for Gn?

SparseVertex degrees are very small

compared to the size of the network

Page 26: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Modeling large ”real world” networks: random graphs

Small world

Scale free

Clustering(transitivity)

Distances are very smallcompared to the size of the network

Vertices with very high degreeare not uncommon

The friends of my friendsare more likely to be my friends

• Size of the network: n vertices (n deterministic and large)• Network: random graph Gn (a random variable taking values in

the set of all graphs with n vertices)

What properties do we want for Gn?

SparseVertex degrees are very small

compared to the size of the network

Page 27: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The small world effect

Page 28: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The small world effect

Distances in large ”real world” networksare (very) small compare to their size

• [Milgram 1967] 6 degrees of separationsdeliver a letter in the US via intermidiaries knownon a first name basis

Page 29: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The small world effect

Distances in large ”real world” networksare (very) small compare to their size

• [Milgram 1967]

• [Watts 2000]

6 degrees of separationsdeliver a letter in the US via intermidiaries knownon a first name basis

larger scale with emails, similar results

Page 30: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The small world effect

Distances in large ”real world” networksare (very) small compare to their size

• [Milgram 1967]

• [Watts 2000]

• [Backstrom, Boldi, Rosa, Ugander, and 2011]

6 degrees of separationsdeliver a letter in the US via intermidiaries knownon a first name basis

larger scale with emails, similar results

average distance in Facebook = 5,diameter = 58 (but roughly 20 inside a country)

Page 31: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The small world effect

Distances in large ”real world” networksare (very) small compare to their size

• [Milgram 1967]

• [Watts 2000]

• [Backstrom, Boldi, Rosa, Ugander, and 2011]

• [The Erdos number project]

6 degrees of separationsdeliver a letter in the US via intermidiaries knownon a first name basis

larger scale with emails, similar results

average distance in Facebook = 5,diameter = 58 (but roughly 20 inside a country)

average collaboration distance between twomathematicians = 7.64, diameter = 23

Page 32: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The small world of Facebook

721 million active users, 69 billion friendship links: average degree = 191

Distances in Facebook[Backstrom, Boldi, Rosa, Ugander and Vigna 2011]

Page 33: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The small world of Facebook

721 million active users, 69 billion friendship links: average degree = 191

Distances in Facebook in different subgraphs[Backstrom, Boldi, Rosa, Ugander and Vigna 2011]

Page 34: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The small world effect: mathematical modeling

Two interesting criteria:

Small Diameter:

max16i,j6n

dGn(i, j)� n

Small average distance

2

n(n− 1)

n∑i,j=1

dGn(i, j)� n

Page 35: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The small world effect: mathematical modeling

Two interesting criteria:

Small Diameter:

max16i,j6n

dGn(i, j)� n

Small average distance

2

n(n− 1)

n∑i,j=1

dGn(i, j)� n

Both these quantities will grow very slowly with n, often as slowly as log n.

For example:• log(721 000 000) ' 20 (Facebook)• log(log(721 000 000)) ' 3• log(10 000 000 000) ' 23

Page 36: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property

Page 37: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property

”Some nodes have a very large degreecompared to the average degree in the graph.”

Page 38: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property

”Some nodes have a very large degreecompared to the average degree in the graph.”

Degree sequence of a graph Gn with n vertices:

dn = (d1(n), . . . , dn(n))

Degree distribution of Gn: proportion Pdn of vertices with given degree

Pdn ({k}) =1

n

n∑i=1

1{di(n)=k}

Pdn =1

n

n∑i=1

δdi(n)

If Gn is a random graph, Pdn is a (random) probability distribution:it is the law of the degree of a uniformly chosen vertex

Page 39: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property

”Some nodes have a very large degreecompared to the average degree in the graph.”

Degree sequence of a graph Gn with n vertices:

dn = (d1(n), . . . , dn(n))

Degree distribution of Gn: proportion Pdn of vertices with given degree

Pdn ({k}) =1

n

n∑i=1

1{di(n)=k}

Pdn =1

n

n∑i=1

δdi(n)

If Gn is a random graph, Pdn is a (random) probability distribution:it is the law of the degree of a uniformly chosen vertex

Scale free property: Pdn ”asymptotically has a heavy tail”

Page 40: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property: Facebook

Cumulative degree distribution in Facebook[Ugander, Karrer, Backstrom and Marlow 2011]

Page 41: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property: log-log plots

Degrees in the worldwide air transportation network[Ducruet, Ietri and Rozenblat 2011]

Page 42: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property: log-log plots

Number of links pointing to webpages in the African Web[Boldi, Codenotti, Santini, Vigna 2002]

Page 43: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property: log-log plots

Degree distributions in real world networks[Clauset, Shalizi and Newman 2007]

Page 44: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property: mathematical modeling

Degree distribution Pdn of the random graph Gn”asymptotically has a heavy tail”.

Page 45: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property: mathematical modeling

Degree distribution Pdn of the random graph Gn”asymptotically has a heavy tail”.

Most common example of random variable X with a heavy tail:power law with exponent τ > 1

P (X > k) = cτk−τ+1

logP (X = k)

log k= −τ

Page 46: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property: mathematical modeling

Degree distribution Pdn of the random graph Gn”asymptotically has a heavy tail”.

Most common example of random variable X with a heavy tail:power law with exponent τ > 1

P (X > k) = cτk−τ+1

logP (X = k)

log k= −τ

some properties• no exponential moments• infinite mean if τ ∈ (1, 2]• infinite variance if τ ∈ (1, 3]• moments of order < τ − 1

Page 47: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property and sparsity: technical assumptions

Take a sequence G = (Gn)n≥1 of random graphs such that for every n:• Gn has n vertices• degree distribution of Gn is Pdn

Page 48: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property and sparsity: technical assumptions

Take a sequence G = (Gn)n≥1 of random graphs such that for every n:• Gn has n vertices• degree distribution of Gn is Pdn

Sparsity Pdn converges weakly to a probability measure Pwith P ({0}) < 1 as n→∞:

Page 49: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property and sparsity: technical assumptions

Take a sequence G = (Gn)n≥1 of random graphs such that for every n:• Gn has n vertices• degree distribution of Gn is Pdn

Sparsity Pdn converges weakly to a probability measure Pwith P ({0}) < 1 as n→∞:

for every k: Pdn ({k}) →n→∞

P ({k})

Page 50: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property and sparsity: technical assumptions

Take a sequence G = (Gn)n≥1 of random graphs such that for every n:• Gn has n vertices• degree distribution of Gn is Pdn

Sparsity Pdn converges weakly to a probability measure Pwith P ({0}) < 1 as n→∞:

Regularity assumptions Dn r.v. with law Pdn and D r.v. with law P

• First moment

• Second moment

E [Dn] →n→∞

E [D] <∞

E[D2n

]→

n→∞E[D2]<∞

for every k: Pdn ({k}) →n→∞

P ({k})

Page 51: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free property and sparsity: technical assumptions

Take a sequence G = (Gn)n≥1 of random graphs such that for every n:• Gn has n vertices• degree distribution of Gn is Pdn

Sparsity Pdn converges weakly to a probability measure Pwith P ({0}) < 1 as n→∞:

Regularity assumptions Dn r.v. with law Pdn and D r.v. with law P

• First moment

• Second moment

Scale free P has a heavy tail (for example, it is a Power Law)

E [Dn] →n→∞

E [D] <∞

E[D2n

]→

n→∞E[D2]<∞

for every k: Pdn ({k}) →n→∞

P ({k})

Page 52: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Page 53: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Measures the network’s transitivity: the friends of my friends are morelikely to be my friends

Page 54: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Measures the network’s transitivity: the friends of my friends are morelikely to be my friends

me

my friend

Page 55: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Measures the network’s transitivity: the friends of my friends are morelikely to be my friends

me

my friend

a friend of myfriend

Page 56: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Measures the network’s transitivity: the friends of my friends are morelikely to be my friends

me

my friend

a friend of myfriend

?

Page 57: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Measures the network’s transitivity: the friends of my friends are morelikely to be my friends

me

my friend

a friend of myfriend

?

Criterion that comparesthe number of triangles to the numberof connected triplets of vertices

Page 58: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Measures the network’s transitivity: the friends of my friends are morelikely to be my friends

me

my friend

a friend of myfriend

?

Criterion that comparesthe number of triangles to the numberof connected triplets of vertices

Global clusteringof a graph G

CL(G) =3× E (nb of triangles)

E (nb of connected triplets)

Individual clusteringof vertex i

CLi(G) =E (nb of triangles containing i)

E (nb of connected triplets centered at i)

Average clusteringof G

CL(G) =1

n

n∑i=1

CLi(G)

Page 59: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

Page 60: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

Page 61: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

Page 62: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

Page 63: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3

Page 64: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2

Page 65: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2

Page 66: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2

Page 67: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2 +1

Page 68: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2 +1

Global clustering coefficient CL(G) =3× 2

10=

3

5

Page 69: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2 +1

Global clustering coefficient CL(G) =3× 2

10=

3

5

Individual Clustering coeffiscients CLi(G)

Page 70: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2 +1

Global clustering coefficient CL(G) =3× 2

10=

3

5

Individual Clustering coeffiscients CLi(G)

1

Page 71: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2 +1

Global clustering coefficient CL(G) =3× 2

10=

3

5

Individual Clustering coeffiscients CLi(G)

1 2/3

Page 72: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2 +1

Global clustering coefficient CL(G) =3× 2

10=

3

5

Individual Clustering coeffiscients CLi(G)

1 2/3 ???

Page 73: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2 +1

Global clustering coefficient CL(G) =3× 2

10=

3

5

Individual Clustering coeffiscients CLi(G)

1 2/3 0

Page 74: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2 +1

Global clustering coefficient CL(G) =3× 2

10=

3

5

Individual Clustering coeffiscients CLi(G)

1 2/3 0

1

Page 75: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2 +1

Global clustering coefficient CL(G) =3× 2

10=

3

5

Individual Clustering coeffiscients CLi(G)

1 2/3 0

12/3

Page 76: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Clustering

Clustering: an example

2 trianges

10 connected triplets:

3 +2 +2 +2 +1

Global clustering coefficient CL(G) =3× 2

10=

3

5

Individual Clustering coeffiscients CLi(G)

1 2/3 0

12/3

Average Clustering coefficient CL(G) =1

5

(1 +

2

3+ 1 +

2

3+ 0

)=

2

3

Page 77: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Different models of random graphs

Page 78: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Different models of random graphs

Erdos-Renyi random graph

Simplest interesting model

Configuration model

Static random graph with prescribed degree sequence

Preferential attachment

Dynamical model, attachment proportional to degree plus constant

Inhomogeneous random graphs

Generalisation of Erdos-Renyi random graphs,independent edges with inhomogeneous edge occupation probabilities

Page 79: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph

Page 80: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph

ER(n, p)• n vertices• independant edges• edge between i and j with probability p

Egalitarian model: every vertex has the same role

Origins in [Erdos and Renyi 1959]

Page 81: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph

ER(n, p)• n vertices• independant edges• edge between i and j with probability p

Egalitarian model: every vertex has the same role

di degree of the node i: binomial r.v. with parameters (n− 1, p)

• If np→∞, di diverges almost surely

• Sparse graph when p =c

n, c > 0

Origins in [Erdos and Renyi 1959]

Page 82: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph

ER(n, p)• n vertices• independant edges• edge between i and j with probability p

Egalitarian model: every vertex has the same role

di degree of the node i: binomial r.v. with parameters (n− 1, p)

• If np→∞, di diverges almost surely

• Sparse graph when p =c

n, c > 0

Poisson approximation:Pdn converges weakly to a Poisson r.v. with parameter c

ER(n, c/n) is not scale free

Pdn ({k}) →n→∞

Pc ({k}) =ck

k!e−c

Origins in [Erdos and Renyi 1959]

Page 83: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: simulations

Erdos-Renyi random graph with 200 vertices and c = 0.5

Page 84: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: simulations

Erdos-Renyi random graph with 200 vertices and c = 1

Page 85: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: simulations

Erdos-Renyi random graph with 200 vertices and c = 1.5

Page 86: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: simulations

Erdos-Renyi random graph with 200 vertices and c = 2

Page 87: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: simulations

Erdos-Renyi random graph with 200 vertices and c = 5

Page 88: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: simulations

Erdos-Renyi random graph with 200 vertices and c = 10

Page 89: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: some properties

Random Graph ER(n, c/n),with high probability (with probability tending to 1 as n→∞),

Page 90: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: some properties

• O (log n) if c < 1• O

(n2/3

)if c = 1

• O (n) if c > 1, other connected components of size O (log n):unique giant component

Random Graph ER(n, c/n),with high probability (with probability tending to 1 as n→∞),

Size of the largest connected component:

Page 91: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: some properties

• O (log n) if c < 1• O

(n2/3

)if c = 1

• O (n) if c > 1, other connected components of size O (log n):unique giant component

Random Graph ER(n, c/n),with high probability (with probability tending to 1 as n→∞),

Size of the largest connected component:

If c > 1, diameter of the giant component is O (log n):

small world

Page 92: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: some properties

• O (log n) if c < 1• O

(n2/3

)if c = 1

• O (n) if c > 1, other connected components of size O (log n):unique giant component

Random Graph ER(n, c/n),with high probability (with probability tending to 1 as n→∞),

Size of the largest connected component:

If c > 1, diameter of the giant component is O (log n):

small world

Proof: Local weak convergence and comparison to branching processes

Page 93: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

The Erdos-Renyi random graph: some properties

• O (log n) if c < 1• O

(n2/3

)if c = 1

• O (n) if c > 1, other connected components of size O (log n):unique giant component

Random Graph ER(n, c/n),with high probability (with probability tending to 1 as n→∞),

Size of the largest connected component:

If c > 1, diameter of the giant component is O (log n):

small world

Clustering coefficient:

CL (ER(n, c/n)) =3× E (nb of triangles)

E (nb of connected triplets)=

3(n3

) (cn

)33(n3

) (cn

)2 =c

n

no transitivity

Proof: Local weak convergence and comparison to branching processes

Page 94: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Inhomogeneous random graphs

Page 95: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Inhomogeneous random graphs

Introduced by [Chung-Lu 2002]

Generalised by [Bollobas, Janson and Riordan 2007]

Generalisation of Erdos-Renyi random graphs

Page 96: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Inhomogeneous random graphs

Introduced by [Chung-Lu 2002]

Generalised by [Bollobas, Janson and Riordan 2007]

Random graphs with given expected degrees:

• independent edges

• inhomogeneous connection probabilities

edge between i and j with probability pi,j =wiwj∑n

k=1 wk + wiwj

Generalisation of Erdos-Renyi random graphs

Page 97: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Inhomogeneous random graphs

Introduced by [Chung-Lu 2002]

Generalised by [Bollobas, Janson and Riordan 2007]

Random graphs with given expected degrees:

• independent edges

• inhomogeneous connection probabilities

edge between i and j with probability pi,j =wiwj∑n

k=1 wk + wiwj

Generalisation of Erdos-Renyi random graphs

wi is close to the expected degree of i

Page 98: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Inhomogeneous random graphs

Introduced by [Chung-Lu 2002]

Generalised by [Bollobas, Janson and Riordan 2007]

Random graphs with given expected degrees:

• independent edges

• inhomogeneous connection probabilities

edge between i and j with probability pi,j =wiwj∑n

k=1 wk + wiwj

Generalisation of Erdos-Renyi random graphs

wi is close to the expected degree of i

Proper choice of (wi)1≤i≤n:

• unique giant component• power law degree sequence scale free• diameter of order log n small world• still has low clustering

Page 99: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model

Page 100: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model

Invented by [Bollobas 1980]

Construct a random graph with a given degree sequence:

Page 101: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model

Invented by [Bollobas 1980]

Construct a random graph with a given degree sequence:

• number of vertices: n• sequence of degrees: dn = (d1(n), . . . , dn(n))

Page 102: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model

Invented by [Bollobas 1980]

Construct a random graph with a given degree sequence:

• number of vertices: n• sequence of degrees: dn = (d1(n), . . . , dn(n))

n will be (very) large

dn will often be a sequence of i.i.d. random variables with given law

Page 103: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model

Invented by [Bollobas 1980]

Construct a random graph with a given degree sequence:

• number of vertices: n• sequence of degrees: dn = (d1(n), . . . , dn(n))

n will be (very) large

dn will often be a sequence of i.i.d. random variables with given law

Recall the regularity assumptions:Dn r.v. with law Pdn and D r.v. with law P

• First moment

• Second moment

E [Dn] →n→∞

E [D]

E[D2n

]→

n→∞E[D2]

• weak convergence: Pdn converges weakly to P

Page 104: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model

Invented by [Bollobas 1980]

Construct a random graph with a given degree sequence:

• number of vertices: n• sequence of degrees: dn = (d1(n), . . . , dn(n))

n will be (very) large

dn will often be a sequence of i.i.d. random variables with given law

Scale free: degree distribution converging to a power law

Recall the regularity assumptions:Dn r.v. with law Pdn and D r.v. with law P

• First moment

• Second moment

E [Dn] →n→∞

E [D]

E[D2n

]→

n→∞E[D2]

• weak convergence: Pdn converges weakly to P

Page 105: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: construction

1. Assign di(n) half edges to vertex i2. Pair half edges to create edges

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Configuration model: construction

1. Assign di(n) half edges to vertex i2. Pair half edges to create edges

assume total degreen∑i=1

di(n) is even

Page 107: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: construction

1. Assign di(n) half edges to vertex i2. Pair half edges to create edges

Different methods:

• List all the graphs obtained by pairing the half edges• Pick one uniformly at random

assume total degreen∑i=1

di(n) is even

Page 108: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: construction

1. Assign di(n) half edges to vertex i2. Pair half edges to create edges

Different methods:

• Pick two half edges uniformly at random and connect them• Repeat with the remaining half edges• Stop when all half edges are connected

• List all the graphs obtained by pairing the half edges• Pick one uniformly at random

assume total degreen∑i=1

di(n) is even

Page 109: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: construction

1. Assign di(n) half edges to vertex i2. Pair half edges to create edges

Different methods:

• Pick two half edges uniformly at random and connect them• Repeat with the remaining half edges• Stop when all half edges are connected

• List all the graphs obtained by pairing the half edges• Pick one uniformly at random

Same result: denote resulting (multi)-graph by CM(dn)

assume total degreen∑i=1

di(n) is even

Page 110: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: multiple edges and self-loops

CM(dn) can have multiple edges and self-loops, but very few of them

Page 111: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: multiple edges and self-loops

CM(dn) can have multiple edges and self-loops, but very few of them

• First moment regularity assumption:

In CM(dn), erase self-loops and merge multiple edges:new graph CM−(dn)

The degree distribution of CM−(dn) still converges weakly to P

Page 112: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: multiple edges and self-loops

CM(dn) can have multiple edges and self-loops, but very few of them

• First moment regularity assumption:

In CM(dn), erase self-loops and merge multiple edges:new graph CM−(dn)

The degree distribution of CM−(dn) still converges weakly to P

• Second moment regularity assumption:

As n→∞, the probability that CM(dn) is simple converges to

e−ν2−

ν2

4 where ν =E [D(D − 1)]

E [D]

Page 113: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: simulations

Configuration Model with 500 verticesand degrees power law with exponent 1.1

Page 114: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: simulations

Configuration Model with 500 verticesand degrees power law with exponent 1.2

Page 115: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: simulations

Configuration Model with 500 verticesand degrees power law with exponent 1.5

Page 116: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: simulations

Configuration Model with 1000 verticesand degrees power law with exponent 2

Page 117: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: simulations

Configuration Model with 1000 verticesand degrees power law with exponent 3

Page 118: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: simulations

Configuration Model with 1000 verticesand degrees power law with exponent 4

Page 119: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: properties

Recall ν =E [D(D − 1)]

E [D]and assume first moment regularity condition holds

Page 120: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: properties

• Phase transition: unique giant component iff ν > 1[Molloy and Reed 1995]

Recall ν =E [D(D − 1)]

E [D]and assume first moment regularity condition holds

true if ν =∞,e.g. D has a power law distribution with τ ∈ (2, 3)

Page 121: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: properties

• Phase transition: unique giant component iff ν > 1[Molloy and Reed 1995]

Recall ν =E [D(D − 1)]

E [D]and assume first moment regularity condition holds

true if ν =∞,e.g. D has a power law distribution with τ ∈ (2, 3)

• No transitivity:average clustering coeffiscient of CM(dn) is of order 1/n

Page 122: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: properties

• Phase transition: unique giant component iff ν > 1[Molloy and Reed 1995]

Recall ν =E [D(D − 1)]

E [D]and assume first moment regularity condition holds

true if ν =∞,e.g. D has a power law distribution with τ ∈ (2, 3)

• No transitivity:average clustering coeffiscient of CM(dn) is of order 1/n

• Small world: [van der Hofstadt et al. 2005+]Hn distance between a uniform pair of vertices of the giantcomponent of CM(dn)

Page 123: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: properties

• Phase transition: unique giant component iff ν > 1[Molloy and Reed 1995]

Recall ν =E [D(D − 1)]

E [D]and assume first moment regularity condition holds

true if ν =∞,e.g. D has a power law distribution with τ ∈ (2, 3)

• No transitivity:average clustering coeffiscient of CM(dn) is of order 1/n

• Small world: [van der Hofstadt et al. 2005+]Hn distance between a uniform pair of vertices of the giantcomponent of CM(dn)

if second moment condition holds, Hn is of order log n

if D has a power law distribution with τ ∈ (2, 3),Hn is of order log log n

Page 124: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Configuration model: properties

• Phase transition: unique giant component iff ν > 1[Molloy and Reed 1995]

Recall ν =E [D(D − 1)]

E [D]and assume first moment regularity condition holds

true if ν =∞,e.g. D has a power law distribution with τ ∈ (2, 3)

• No transitivity:average clustering coeffiscient of CM(dn) is of order 1/n

• Small world: [van der Hofstadt et al. 2005+]Hn distance between a uniform pair of vertices of the giantcomponent of CM(dn)

if second moment condition holds, Hn is of order log n

if D has a power law distribution with τ ∈ (2, 3),Hn is of order log log n

in both cases, same growth for the diameter

Page 125: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs

Page 126: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs

First appearance in [Albert and Barabasi 1999]

Generalised by [Bollobas, Riordan, Spencer and Tusnady 2001]

Page 127: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs

First appearance in [Albert and Barabasi 1999]

Generalised by [Bollobas, Riordan, Spencer and Tusnady 2001]

Dynamical model:

• new vertices are more likely to be connected to verticeswith high degree

• vertices are added to the graph one at a time

Page 128: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs

First appearance in [Albert and Barabasi 1999]

Generalised by [Bollobas, Riordan, Spencer and Tusnady 2001]

Dynamical model:

• new vertices are more likely to be connected to verticeswith high degree

Rich get richer model

• vertices are added to the graph one at a time

Page 129: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs

First appearance in [Albert and Barabasi 1999]

Generalised by [Bollobas, Riordan, Spencer and Tusnady 2001]

Dynamical model:

• new vertices are more likely to be connected to verticeswith high degree

Old get richer model

• vertices are added to the graph one at a time

Page 130: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: construction

At time n, existing graph PAn(m, δ) has n vertices and degreesequence D(n) = (D1(n), . . . , Dn(n))

Two parameters: m ∈ N and δ > −m

Page 131: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: construction

At time n, existing graph PAn(m, δ) has n vertices and degreesequence D(n) = (D1(n), . . . , Dn(n))

• Add a single vertex with m edges

Two parameters: m ∈ N and δ > −m

• Connect the new vertex to vertex i with probabilityproportional to Di(n) + δ

Construction of PAn+1(m, δ):

Page 132: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: construction

At time n, existing graph PAn(m, δ) has n vertices and degreesequence D(n) = (D1(n), . . . , Dn(n))

• Add a single vertex with m edges

Two parameters: m ∈ N and δ > −m

• Connect the new vertex to vertex i with probabilityproportional to Di(n) + δ

Construction of PAn+1(m, δ):

Connected graph

Page 133: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: construction

At time n, existing graph PAn(m, δ) has n vertices and degreesequence D(n) = (D1(n), . . . , Dn(n))

• Add a single vertex with m edges

Two parameters: m ∈ N and δ > −m

• Connect the new vertex to vertex i with probabilityproportional to Di(n) + δ

Scale free: power law degree sequence with exponent

τ = 3 +δ

m> 2

Construction of PAn+1(m, δ):

Connected graph

Page 134: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: simulations

Barabasi-Albert graph with 200 verticeseach new vertex comes with 1 edge

Page 135: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: simulations

Barabasi-Albert graph with 200 verticeseach new vertex comes with 2 edges

Page 136: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: simulations

Barabasi-Albert graph with 500 verticeseach new vertex comes with 2 edges

Page 137: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: simulations

Barabasi-Albert graph with 200 verticeseach new vertex comes with 3 edges

Page 138: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: other properties

• Barabasi-Albert graph: m ≥ 2 and δ = 0, yielding τ = 3[Bollobas and Riordan 2004]:

Hn and diameter both of orderlog n

log log n

Page 139: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: other properties

• Barabasi-Albert graph: m ≥ 2 and δ = 0, yielding τ = 3[Bollobas and Riordan 2004]:

• General case when m ≥ 2 and δ 6= 0[Dommers, van der Hofstad and Hooghiemstra 2012]:

Hn and diameter both of orderlog n

log log n

if τ > 3, Hn and diameter both of order log n

Page 140: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: other properties

• Barabasi-Albert graph: m ≥ 2 and δ = 0, yielding τ = 3[Bollobas and Riordan 2004]:

• General case when m ≥ 2 and δ 6= 0[Dommers, van der Hofstad and Hooghiemstra 2012]:

Hn and diameter both of orderlog n

log log n

if τ > 3, Hn and diameter both of order log n

if τ ∈ (2, 3), Hn and diameter both of order log log n

Page 141: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Preferential attachment graphs: other properties

• Barabasi-Albert graph: m ≥ 2 and δ = 0, yielding τ = 3[Bollobas and Riordan 2004]:

• General case when m ≥ 2 and δ 6= 0[Dommers, van der Hofstad and Hooghiemstra 2012]:

Hn and diameter both of orderlog n

log log n

if τ > 3, Hn and diameter both of order log n

if τ ∈ (2, 3), Hn and diameter both of order log log n

No rigorous result on clustering, but empirical studies n−3/4:no transitivity

Page 142: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free random graphs: universal behavior

Page 143: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free random graphs: universal behavior

• Small worlds: every model we met has the small world property

small world when degrees have finite varianceultra small world when the variance is infinite

Page 144: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free random graphs: universal behavior

• Small worlds: every model we met has the small world property

• Low clustering:average clustering always goes to 0 with the size of the graph

small world when degrees have finite varianceultra small world when the variance is infinite

Page 145: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free random graphs: universal behavior

• Small worlds: every model we met has the small world property

• Low clustering:average clustering always goes to 0 with the size of the graph

small world when degrees have finite varianceultra small world when the variance is infinite

Same behaviour for many other models:random intersection graphs, inhomogeneous random graphs ...

universality

Page 146: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free random graphs: universal behavior

• Small worlds: every model we met has the small world property

• Low clustering:average clustering always goes to 0 with the size of the graph

Common property that explains both small world property and low clustering:

small world when degrees have finite varianceultra small world when the variance is infinite

Same behaviour for many other models:random intersection graphs, inhomogeneous random graphs ...

universality

we considered locally tree like graphs

Page 147: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Scale free random graphs: universal behavior

• Small worlds: every model we met has the small world property

• Low clustering:average clustering always goes to 0 with the size of the graph

Common property that explains both small world property and low clustering:

small world when degrees have finite varianceultra small world when the variance is infinite

Same behaviour for many other models:random intersection graphs, inhomogeneous random graphs ...

universality

we considered locally tree like graphs

Models not locally tree like are much harder to deal with!

Page 148: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence

Page 149: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence

Introduced by [Benjamini and Schramm 2001]

Nice survey on applications to combinatorial optimization[Aldous and Steele 2003]

Page 150: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence

Introduced by [Benjamini and Schramm 2001]

Nice survey on applications to combinatorial optimization[Aldous and Steele 2003]

• Take a sequence of (random, growing) graphs Gn• For every n, choose uniformly at random a vertex on in Gn

What does Gn look like seen from on ?

Page 151: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence

Introduced by [Benjamini and Schramm 2001]

Nice survey on applications to combinatorial optimization[Aldous and Steele 2003]

• Take a sequence of (random, growing) graphs Gn• For every n, choose uniformly at random a vertex on in Gn

What does Gn look like seen from on ?

Local convergence: looking at the whole graph is too strong:

Page 152: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence

Introduced by [Benjamini and Schramm 2001]

Nice survey on applications to combinatorial optimization[Aldous and Steele 2003]

• Take a sequence of (random, growing) graphs Gn• For every n, choose uniformly at random a vertex on in Gn

What does Gn look like seen from on ?

Local convergence: looking at the whole graph is too strong:

What does Gn look like inside a fixed radius R around on ?

Page 153: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence

Introduced by [Benjamini and Schramm 2001]

Nice survey on applications to combinatorial optimization[Aldous and Steele 2003]

• Take a sequence of (random, growing) graphs Gn• For every n, choose uniformly at random a vertex on in Gn

What does Gn look like seen from on ?

Local convergence: looking at the whole graph is too strong:

What does Gn look like inside a fixed radius R around on ?

Convergence:

it should look like a limiting rooted graph (G∞, o)inside a radius R around its root o

Page 154: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence

Introduced by [Benjamini and Schramm 2001]

Nice survey on applications to combinatorial optimization[Aldous and Steele 2003]

• Take a sequence of (random, growing) graphs Gn• For every n, choose uniformly at random a vertex on in Gn

What does Gn look like seen from on ?

Local convergence: looking at the whole graph is too strong:

What does Gn look like inside a fixed radius R around on ?

Convergence:

it should look like a limiting rooted graph (G∞, o)inside a radius R around its root o

Possible limits: locally finite graphs(graphs with infinitely many vertices, but each vertex has finite degree)

Page 155: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: formal definition

G? = {locally finite rooted graphs}

Page 156: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: formal definition

G? = {locally finite rooted graphs}Local topology on G?:

Local weak convergence on G?:weak convergence in law for the local topology

Page 157: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: formal definition

Take R ∈ N and (G, o) ∈ G?,define the subgraph of G inside a radius R around o:

BallG(o,R) =

{Vertices ⊂ {v ∈ V (G) : dG(o, v) 6 R+ 1}Edges = {{v, v′} ∈ E(G) : dG(o, v) 6 R}

G? = {locally finite rooted graphs}Local topology on G?:

Local weak convergence on G?:weak convergence in law for the local topology

Page 158: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: formal definition

Take R ∈ N and (G, o) ∈ G?,define the subgraph of G inside a radius R around o:

BallG(o,R) =

{Vertices ⊂ {v ∈ V (G) : dG(o, v) 6 R+ 1}Edges = {{v, v′} ∈ E(G) : dG(o, v) 6 R}

G? = {locally finite rooted graphs}Local topology on G?:

Take (G, o), (G′, o′) ∈ G?, define:

dG? ((G, o), (G′, o′)) = inf

{1

R+ 1: BallG(o,R) = BallG′(o

′, R)

}

Local weak convergence on G?:weak convergence in law for the local topology

Page 159: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: formal definition

Take R ∈ N and (G, o) ∈ G?,define the subgraph of G inside a radius R around o:

BallG(o,R) =

{Vertices ⊂ {v ∈ V (G) : dG(o, v) 6 R+ 1}Edges = {{v, v′} ∈ E(G) : dG(o, v) 6 R}

G? = {locally finite rooted graphs}Local topology on G?:

Take (G, o), (G′, o′) ∈ G?, define:

dG? ((G, o), (G′, o′)) = inf

{1

R+ 1: BallG(o,R) = BallG′(o

′, R)

}dG? is a distance and (G?, dG?) is a polish space

Local weak convergence on G?:weak convergence in law for the local topology

Page 160: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: simple examples

1 2 3 n

Page 161: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: simple examples

1 2 3 n−→ (Z, 0)

Page 162: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: simple examples

1 2 3 n−→ (Z, 0)

12

n

−→ (Z, 0)

Page 163: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: simple examples

1 2 3 n−→ (Z, 0)

12

n

−→ (Z, 0)

1 n

n

1

−→(Z2, 0

)

Page 164: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: simple examples

1 2 3 n−→ (Z, 0)

12

n

−→ (Z, 0)

1 n

n

1

−→(Z2, 0

)Z/nZ× Z/nZ −→

(Z2, 0

)

Page 165: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: simple examples

1 2 3 n−→ (Z, 0)

12

n

−→ (Z, 0)

1 n

n

1

−→(Z2, 0

)Z/nZ× Z/nZ −→

(Z2, 0

)

binary tree ofheight n

1

n −→

Canopytree

o

Page 166: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: simple examples

1 2 3 n−→ (Z, 0)

12

n

−→ (Z, 0)

1 n

n

1

−→(Z2, 0

)Z/nZ× Z/nZ −→

(Z2, 0

)

binary tree ofheight n

1

n −→

Canopytree

o

Uniform random treewith n vertices

−→ Skeleton tree

Page 167: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: simple examples

1 2 3 n−→ (Z, 0)

12

n

−→ (Z, 0)

1 n

n

1

−→(Z2, 0

)Z/nZ× Z/nZ −→

(Z2, 0

)

binary tree ofheight n

1

n −→

Canopytree

o

Uniform random treewith n vertices

−→ Skeleton tree

o

Page 168: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: simple examples

1 2 3 n−→ (Z, 0)

12

n

−→ (Z, 0)

1 n

n

1

−→(Z2, 0

)Z/nZ× Z/nZ −→

(Z2, 0

)

binary tree ofheight n

1

n −→

Canopytree

o

Uniform random treewith n vertices

−→ Skeleton tree

o

independant critical Galton Watson trees

Page 169: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: Erdos-Renyi random graphs

Graph ER(n, c/n),degree distribution converges to Poisson r.v. P(c) with parameter c

Page 170: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: Erdos-Renyi random graphs

Graph ER(n, c/n),degree distribution converges to Poisson r.v. P(c) with parameter c

Local weak limit: Galton-Watson tree with reproduction law P(c)Proof: breadth-first search of a connected component

Page 171: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: Erdos-Renyi random graphs

Graph ER(n, c/n),degree distribution converges to Poisson r.v. P(c) with parameter c

Local weak limit: Galton-Watson tree with reproduction law P(c)Proof: breadth-first search of a connected component

Erdos-Renyi random graphs are locally tree-like

Page 172: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: Erdos-Renyi random graphs

Graph ER(n, c/n),degree distribution converges to Poisson r.v. P(c) with parameter c

Local weak limit: Galton-Watson tree with reproduction law P(c)Proof: breadth-first search of a connected component

Erdos-Renyi random graphs are locally tree-like

Two applications:

• Phase transition: the Galton-Watson tree survives iff c > 1

Page 173: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: Erdos-Renyi random graphs

Graph ER(n, c/n),degree distribution converges to Poisson r.v. P(c) with parameter c

Local weak limit: Galton-Watson tree with reproduction law P(c)Proof: breadth-first search of a connected component

Erdos-Renyi random graphs are locally tree-like

Two applications:

• Phase transition: the Galton-Watson tree survives iff c > 1

• Distances (very sketchy!): height of a supercriticalGalton-Watson tree conditioned to have n vertices of order log n

Page 174: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: Configuration model

Graph CM(dn),degree distribution converges to r.v. D with law P

Page 175: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: Configuration model

Graph CM(dn),degree distribution converges to r.v. D with law P

Local weak limit: Unimodular Galton-Watson treewith reproduction law P

Page 176: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: Configuration model

Graph CM(dn),degree distribution converges to r.v. D with law P

Local weak limit: Unimodular Galton-Watson treewith reproduction law P

• root has reproduction law P• subtrees issued from first generation vertices are

Galton-Watson trees with reproduction law P ,size-biaised version of P :

P ({k}) = (k + 1)P ({k + 1})∑k≥0 kP ({k})

Page 177: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: Configuration model

Graph CM(dn),degree distribution converges to r.v. D with law P

Local weak limit:

Proof: breadth-first search of a connected component

Configuration models are locally tree-like

Unimodular Galton-Watson treewith reproduction law P

• root has reproduction law P• subtrees issued from first generation vertices are

Galton-Watson trees with reproduction law P ,size-biaised version of P :

P ({k}) = (k + 1)P ({k + 1})∑k≥0 kP ({k})

Page 178: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Local weak convergence: Configuration model

Graph CM(dn),degree distribution converges to r.v. D with law P

Local weak limit:

Proof: breadth-first search of a connected component

Configuration models are locally tree-like

Unimodular Galton-Watson treewith reproduction law P

• root has reproduction law P• subtrees issued from first generation vertices are

Galton-Watson trees with reproduction law P ,size-biaised version of P :

Phase transition: the tree survives iffE[D(D − 1)]

E[D]> 1

P ({k}) = (k + 1)P ({k + 1})∑k≥0 kP ({k})

Page 179: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Other notions of convergence for graphs

[Berger, Borgs, Chayes and Saberi 2013]and [Dereich and Morters 2013]:preferential attachment graphs are locally tree-like

Page 180: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Other notions of convergence for graphs

[Berger, Borgs, Chayes and Saberi 2013]and [Dereich and Morters 2013]:preferential attachment graphs are locally tree-like

Global notions of convergence:

• Scaling limits: in sparse Gn, typical distances of order log n

1. consider Gn as the (discrete) metric space (V (Gn), dGn)2. rescale the distances by a factor log n3. does (V (Gn), (log n)

−1dGn) converge to a limitingcontinuous random metric space ?Gromov-Hausdorf topology

Page 181: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Other notions of convergence for graphs

[Berger, Borgs, Chayes and Saberi 2013]and [Dereich and Morters 2013]:preferential attachment graphs are locally tree-like

Global notions of convergence:

• Scaling limits: in sparse Gn, typical distances of order log n

• Graphons: [Borgs, Chayes, Lovasz, Sos and Vesztergombi 2008]

1. consider Gn as the (discrete) metric space (V (Gn), dGn)2. rescale the distances by a factor log n3. does (V (Gn), (log n)

−1dGn) converge to a limitingcontinuous random metric space ?Gromov-Hausdorf topology

1. represent graphs by functions [0, 1]2 → [0, 1]2. metric on these functions that keeps track of the frequency

of appearance of any finite graph H in Gn3. works for sequences of dense graphs

Page 182: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Statistical mechanics on random graphs

Study random models or random evolutions on random graphs:random walks, percolations, ising model, . . .

Page 183: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Statistical mechanics on random graphs

Study random models or random evolutions on random graphs:random walks, percolations, ising model, . . .

First passage percolation

Crossing an edge has a cost

Percolation

Robustness under attacks

Contagion model

Game-theoretic diffusion model

Systemic risk

Default cascades in interbank networks

Page 184: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

First passage percolation

Large random graph Gn = (Vn, En)

Put positive weights on edges (Ye)e∈En :

• length of the edges• cost or congestion across edges, . . .

Page 185: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

First passage percolation

Large random graph Gn = (Vn, En)

Put positive weights on edges (Ye)e∈En :

• length of the edges• cost or congestion across edges, . . .

Take a path π in Gn, total weight of π: W (π) =∑e∈π

Ye

Now take uniformly at random two vertices v, v′ ∈ Vn, define

• average smallest weight: Wn = infπ:v→v′

W (π)

average cost• Hop count: Hn = length of smallest length path

between v and v′

time delay

Page 186: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

First passage percolation

Large random graph Gn = (Vn, En)

Put positive weights on edges (Ye)e∈En :

• length of the edges• cost or congestion across edges, . . .

Take a path π in Gn, total weight of π: W (π) =∑e∈π

Ye

Now take uniformly at random two vertices v, v′ ∈ Vn, define

• average smallest weight: Wn = infπ:v→v′

W (π)

average cost• Hop count: Hn = length of smallest length path

between v and v′

time delay

Are Wn and Hn similar to average distance ?

Page 187: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

First passage percolation on configuration model

Gn = configuration model with iid power law degrees with exponent τ > 2

Edge weights Ye are iid exponential r.v.

[Bhamidi, Hooghiemstra and van der Hofstad 2010]:

There exists α > 0 such that for τ 6= 3:

Hn − α log n√α log n

−→ N (0, 1)

There exists γ > 0 such that for τ > 3:

Wn − γ log n −→W∞

For τ ∈ (2, 3):Wn −→W∞

d

d

dFor τ ∈ (2, 3):

Wn −→W∞

Page 188: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Robustness of networks

Take preferential attachment PAn(m, δ) and remove verticesindependently with probability p: random attack

[Bollobas and Riordan 2003]

Page 189: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Robustness of networks

Take preferential attachment PAn(m, δ) and remove verticesindependently with probability p: random attack

pn vertices are removed in average

Does the resulting graph still have a giant component?

[Bollobas and Riordan 2003]

Page 190: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Robustness of networks

Take preferential attachment PAn(m, δ) and remove verticesindependently with probability p: random attack

pn vertices are removed in average

Does the resulting graph still have a giant component?

[Bollobas and Riordan 2003]

Yes for every p < 1

”Large random graphs are robust against random attacks”

Page 191: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Robustness of networks

Take preferential attachment PAn(m, δ) and remove verticesindependently with probability p: random attack

pn vertices are removed in average

Does the resulting graph still have a giant component?

[Bollobas and Riordan 2003]

Yes for every p < 1

”Large random graphs are robust against random attacks”

Now, for p ∈ (0, 1), remove the first pn edges: targeted attack

There exists 0 < pc < 1 such that:• if p < pc, there is a giant component• if p > pc, there is no giant component

”Large random graphs are vulnerable against targeted attacks”

Page 192: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion models and cascades

Game-theoretic model from [Morris 2000]graph G, parameter q ∈ (0, 1)

• each vertex chooses between 2 behaviours:

• At the begining, every vertex is

• Interaction payoff:• Interaction payoff:

If two neighbours are , they both receive payoff q

If two neighbours are , they both receive payoff 1− qIf two neighbours disagree, they both receive 0

Page 193: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion models and cascades

Game-theoretic model from [Morris 2000]graph G, parameter q ∈ (0, 1)

• each vertex chooses between 2 behaviours:

• Consider vertex i, degree di:

i adopts if Ni > qdi

i adopts if Ni ≤ qdi

• At the begining, every vertex is

• Interaction payoff:• Interaction payoff:

If two neighbours are , they both receive payoff q

If two neighbours are , they both receive payoff 1− qIf two neighbours disagree, they both receive 0

Page 194: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion models and cascades

Game-theoretic model from [Morris 2000]graph G, parameter q ∈ (0, 1)

• each vertex chooses between 2 behaviours:

• Consider vertex i, degree di:

i adopts if Ni > qdi

i adopts if Ni ≤ qdi

• At the begining, every vertex is

Can we convert a macroscopic fraction of the graph toby forcing few vertices to adopt ?

Cascade:

• Interaction payoff:• Interaction payoff:

If two neighbours are , they both receive payoff q

If two neighbours are , they both receive payoff 1− qIf two neighbours disagree, they both receive 0

Page 195: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion models and cascades: an example

q = 1/4

Page 196: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion models and cascades: an example

q = 1/4

Page 197: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion models and cascades: an example

q = 1/4

Page 198: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion models and cascades: an example

q = 1/4

Page 199: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion models and cascades: an example

q = 1/4

Page 200: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion models and cascades: an example

q = 1/4

Page 201: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion models and cascades: an example

q = 1/4

Page 202: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion and cascades: configuration model

[Lelarge 2011]

Page 203: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion and cascades: configuration model

If di < q−1, one neighbour is enough to convert i: pivotal player

[Lelarge 2011]

Page 204: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion and cascades: configuration model

If di < q−1, one neighbour is enough to convert i: pivotal player

[Lelarge 2011]

Graph CM(dn) with degrees converging in law to D andthird moment regularity assumption

• Pn(v) = {v ∈ CM(dn) : dv < q−1}: set of pivotal players

• C(v, q): final numbers of vertices when at first only v issize of the cascade induced by v

Page 205: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Contagion and cascades: configuration model

If di < q−1, one neighbour is enough to convert i: pivotal player

[Lelarge 2011]

Graph CM(dn) with degrees converging in law to D andthird moment regularity assumption

• Pn(v) = {v ∈ CM(dn) : dv < q−1}: set of pivotal players

• C(v, q): final numbers of vertices when at first only v issize of the cascade induced by v

Let qc = sup{q : E[D(D − 1)1{D<q−1}

]> E[D]}:

• If q < qc, for any v ∈ Pn(q), with high probabilityC(v, q) = O(n)

• If q > qc, for any v ∈ Pn(q), with high probabilityC(v, q) = o(n)

Page 206: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Systemic risk

[Cont, Moussa and Bastos e Santos 2010]: Brasilian interbank network

Model for interbank network: directed random graph

• each vertex i has a capital ci > 0• weight Ei,j > 0 on directed edge (i, j): exposure of i to j

• Vertex i defaults if ci <∑j

Ei,j

Page 207: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Systemic risk

[Cont, Moussa and Bastos e Santos 2010]: Brasilian interbank network

Model for interbank network: directed random graph

• each vertex i has a capital ci > 0• weight Ei,j > 0 on directed edge (i, j): exposure of i to j

• Vertex i defaults if ci <∑j

Ei,j

Systemic risk:

• the default of a single vertex triggers a cascade of defaults bycontagion

• eventualy simultaneous with a market shock: for every i,ci becomes ci − εi

Page 208: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Systemic risk

[Cont, Moussa and Bastos e Santos 2010]: Brasilian interbank network

Model for interbank network: directed random graph

• each vertex i has a capital ci > 0• weight Ei,j > 0 on directed edge (i, j): exposure of i to j

• Vertex i defaults if ci <∑j

Ei,j

Systemic risk:

• the default of a single vertex triggers a cascade of defaults bycontagion

• eventualy simultaneous with a market shock: for every i,ci becomes ci − εi

Indentify institution posing a systemic risk ?

Page 209: a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs Small world Scale free Clustering (transitivity) Distances are very small compared

Thank you for your attentionand have a very nice week!


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