multilevelnetworks: fromlocal toglobal...
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15/10/2011 Slide 1/37Multilevel Networks: From local to global structure of interactions
Multilevel Networks: From local to global structure of interactions
Regino Criado (joint work with Miguel Romance)
Group G.A.M.E.Applied Mathematics Department (URJC)
15/10/2011 Slide 2/37Multilevel Networks: From local to global structure of interactions
1. Introduction
• Motivation
• Multilevel Networks
2. Local scale properties: Clustering and degreedistribution.
3. Global-scale properties: Geodesic structure• Paths and distances
• Analytical results
• An application to transportation networks
4. Summary
Outline
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• A (directed/undirected) complex network is a graph, that is, a pair of sets G =(V,E) , where
– V is finite (the set of nodes), and
– E is a subset of (unordered/ordered) pairs of V (edges or links),
1. Introduction
V =n
E =m
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1. Introduction
Transport networks:underground, train, airlines …
Nodes: stations, airports,…
Links: direct connections between them
But,
What about lines?
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1. Introduction
Yes: the Mesoscales
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1. Introduction
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1. Introduction
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1. Introduction
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15/10/2011 Slide 9/37Multilevel Networks: From local to global structure of interactions
Classical model (networks) :Inadequate
Social networks: Many individuals with diversesocial interactions and relationships between them.
We can only see the “projection”:
1. Introduction
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1. Introduction: Multilevel networks
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1. Introduction: Multilevel networks
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1. Introduction: Multilevel networks
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1. Introduction: Multilevel networks
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1. Introduction: Multilevel networks
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( , )X E
( , , )M X E S=
{ }1 2 3, ,S S S S=
1. Introduction: Multilevel networks
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( , , )M X E S=
1. Introduction: Multilevel networks
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( , , )M X E S=
1. Introduction: Multilevel networks
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( , , )M X E S=
1. Introduction: Multilevel networks
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2. Local scale properties: Clustering
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2. Local scale properties: Clustering
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2. Local scale properties: Clustering
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2. Local scale properties: Clustering
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2. Local scale properties: Clustering
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2. Local scale properties: Clustering
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2. Local scale properties: Clustering
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2. Local scale properties: Clustering
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3. Global scale properties: Geodesic structure
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3. Global scale properties: Geodesic structure
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3. Global scale properties: Geodesic structure
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( , )X E
a
b
cd
Auxiliary Graph:
a3
b3
c3
a2
b2
c2
c1d1
a1
d2
{ } { }2 3 1 2( , ) 4 ( , , , )Md b d d b b d d= =
1S
2S
3Sa
b
c
a
b
c d
ac
d
3. Global scale properties: Geodesic structure
M ( )G Mɶ
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3. Global scale properties: Geodesic structure
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3. Global scale properties: Geodesic structure
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3. Global scale properties: Geodesic structure
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3. Global scale properties: Geodesic structure
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3. Global scale properties: Geodesic structure
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4. Summary
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References