a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs...
Transcript of a probabilistic point of view Random graphs · Modeling large "real world" networks: random graphs...
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Random graphs:a probabilistic point of view
Laurent MenardModal’X, Universite Paris Ouest
Stats in Paris, November 2013
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”Real world” networks
Collaboration graph of mathematicians[The Erdos number project, 2004]
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”Real world” networks
The internet topology in 1999[The internet mapping project]
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”Real world” networks
[Tellez 2013]
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Outline
What are we looking for ?
Most common properties of ”real world networks”.
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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, . . .
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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
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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, . . .
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Modeling networks: Graph theory
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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}
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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
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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
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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
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Graph theory: some vocabulary
• Path from vertex i to vertex j:sequence of edges connecting i to j
i
j
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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
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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
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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
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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
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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
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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
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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)
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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?
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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
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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
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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
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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
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The small world effect
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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
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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
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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)
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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
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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]
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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]
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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
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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
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Scale free property
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Scale free property
”Some nodes have a very large degreecompared to the average degree in the graph.”
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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
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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”
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Scale free property: Facebook
Cumulative degree distribution in Facebook[Ugander, Karrer, Backstrom and Marlow 2011]
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Scale free property: log-log plots
Degrees in the worldwide air transportation network[Ducruet, Ietri and Rozenblat 2011]
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Scale free property: log-log plots
Number of links pointing to webpages in the African Web[Boldi, Codenotti, Santini, Vigna 2002]
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Scale free property: log-log plots
Degree distributions in real world networks[Clauset, Shalizi and Newman 2007]
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Scale free property: mathematical modeling
Degree distribution Pdn of the random graph Gn”asymptotically has a heavy tail”.
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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
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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
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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
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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→∞:
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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})
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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})
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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})
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Clustering
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Clustering
Measures the network’s transitivity: the friends of my friends are morelikely to be my friends
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Clustering
Measures the network’s transitivity: the friends of my friends are morelikely to be my friends
me
my friend
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Clustering
Measures the network’s transitivity: the friends of my friends are morelikely to be my friends
me
my friend
a friend of myfriend
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Clustering
Measures the network’s transitivity: the friends of my friends are morelikely to be my friends
me
my friend
a friend of myfriend
?
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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
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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)
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Clustering
Clustering: an example
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Clustering
Clustering: an example
2 trianges
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Clustering
Clustering: an example
2 trianges
10 connected triplets:
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Clustering
Clustering: an example
2 trianges
10 connected triplets:
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Clustering
Clustering: an example
2 trianges
10 connected triplets:
3
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Clustering
Clustering: an example
2 trianges
10 connected triplets:
3 +2
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Clustering
Clustering: an example
2 trianges
10 connected triplets:
3 +2 +2
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Clustering
Clustering: an example
2 trianges
10 connected triplets:
3 +2 +2 +2
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Clustering
Clustering: an example
2 trianges
10 connected triplets:
3 +2 +2 +2 +1
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Clustering
Clustering: an example
2 trianges
10 connected triplets:
3 +2 +2 +2 +1
Global clustering coefficient CL(G) =3× 2
10=
3
5
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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)
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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
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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
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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 ???
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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
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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
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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
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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
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Different models of random graphs
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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
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The Erdos-Renyi random graph
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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]
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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]
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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]
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The Erdos-Renyi random graph: simulations
Erdos-Renyi random graph with 200 vertices and c = 0.5
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The Erdos-Renyi random graph: simulations
Erdos-Renyi random graph with 200 vertices and c = 1
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The Erdos-Renyi random graph: simulations
Erdos-Renyi random graph with 200 vertices and c = 1.5
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The Erdos-Renyi random graph: simulations
Erdos-Renyi random graph with 200 vertices and c = 2
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The Erdos-Renyi random graph: simulations
Erdos-Renyi random graph with 200 vertices and c = 5
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The Erdos-Renyi random graph: simulations
Erdos-Renyi random graph with 200 vertices and c = 10
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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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/90.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/91.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/92.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/93.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/94.jpg)
Inhomogeneous random graphs
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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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/96.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/97.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/98.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/99.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/100.jpg)
Configuration model
Invented by [Bollobas 1980]
Construct a random graph with a given degree sequence:
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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))
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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
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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
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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
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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
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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
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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
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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
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Configuration model: multiple edges and self-loops
CM(dn) can have multiple edges and self-loops, but very few of them
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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
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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]
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Configuration model: simulations
Configuration Model with 500 verticesand degrees power law with exponent 1.1
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Configuration model: simulations
Configuration Model with 500 verticesand degrees power law with exponent 1.2
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Configuration model: simulations
Configuration Model with 500 verticesand degrees power law with exponent 1.5
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Configuration model: simulations
Configuration Model with 1000 verticesand degrees power law with exponent 2
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Configuration model: simulations
Configuration Model with 1000 verticesand degrees power law with exponent 3
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Configuration model: simulations
Configuration Model with 1000 verticesand degrees power law with exponent 4
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Configuration model: properties
Recall ν =E [D(D − 1)]
E [D]and assume first moment regularity condition holds
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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)
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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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/122.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/123.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/124.jpg)
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
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Preferential attachment graphs
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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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/127.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/128.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/129.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/130.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/131.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/132.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/133.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/134.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/135.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/136.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/137.jpg)
Preferential attachment graphs: simulations
Barabasi-Albert graph with 200 verticeseach new vertex comes with 3 edges
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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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/139.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/140.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/141.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/142.jpg)
Scale free random graphs: universal behavior
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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
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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
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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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/146.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/147.jpg)
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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/148.jpg)
Local weak convergence
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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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/150.jpg)
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 ?
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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:
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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 ?
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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
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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)
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Local weak convergence: formal definition
G? = {locally finite rooted graphs}
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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
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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
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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
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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
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Local weak convergence: simple examples
1 2 3 n
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Local weak convergence: simple examples
1 2 3 n−→ (Z, 0)
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Local weak convergence: simple examples
1 2 3 n−→ (Z, 0)
12
n
−→ (Z, 0)
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Local weak convergence: simple examples
1 2 3 n−→ (Z, 0)
12
n
−→ (Z, 0)
1 n
n
1
−→(Z2, 0
)
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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
)
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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
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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
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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
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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
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Local weak convergence: Erdos-Renyi random graphs
Graph ER(n, c/n),degree distribution converges to Poisson r.v. P(c) with parameter c
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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
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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
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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
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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
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Local weak convergence: Configuration model
Graph CM(dn),degree distribution converges to r.v. D with law P
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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
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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})
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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})
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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})
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Other notions of convergence for graphs
[Berger, Borgs, Chayes and Saberi 2013]and [Dereich and Morters 2013]:preferential attachment graphs are locally tree-like
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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
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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
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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](https://reader035.fdocuments.us/reader035/viewer/2022070904/5f7228d04f345e619a5a24fc/html5/thumbnails/183.jpg)
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
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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, . . .
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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
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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 ?
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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∞
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Robustness of networks
Take preferential attachment PAn(m, δ) and remove verticesindependently with probability p: random attack
[Bollobas and Riordan 2003]
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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]
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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”
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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”
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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
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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
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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
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Contagion models and cascades: an example
q = 1/4
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Contagion models and cascades: an example
q = 1/4
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Contagion models and cascades: an example
q = 1/4
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Contagion models and cascades: an example
q = 1/4
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Contagion models and cascades: an example
q = 1/4
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Contagion models and cascades: an example
q = 1/4
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Contagion models and cascades: an example
q = 1/4
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Contagion and cascades: configuration model
[Lelarge 2011]
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Contagion and cascades: configuration model
If di < q−1, one neighbour is enough to convert i: pivotal player
[Lelarge 2011]
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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
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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)
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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
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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
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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 ?
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Thank you for your attentionand have a very nice week!