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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15/10/2011 Slide 3/37Multilevel Networks: From local to global structure of interactions

• 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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15/10/2011 Slide 7/37Multilevel Networks: From local to global structure of interactions

1. Introduction

15/10/2011 Slide 8/37Multilevel Networks: From local to global structure of interactions

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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15/10/2011 Slide 11/37Multilevel Networks: From local to global structure of interactions

1. Introduction: Multilevel networks

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1. Introduction: Multilevel networks

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15/10/2011 Slide 13/37Multilevel Networks: From local to global structure of interactions

1. Introduction: Multilevel networks

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1. Introduction: Multilevel networks

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15/10/2011 Slide 15/37Multilevel Networks: From local to global structure of interactions

( , )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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15/10/2011 Slide 17/37Multilevel Networks: From local to global structure of interactions

( , , )M X E S=

1. Introduction: Multilevel networks

15/10/2011 Slide 18/37Multilevel Networks: From local to global structure of interactions

( , , )M X E S=

1. Introduction: Multilevel networks

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15/10/2011 Slide 19/37Multilevel Networks: From local to global structure of interactions

2. Local scale properties: Clustering

15/10/2011 Slide 20/37Multilevel Networks: From local to global structure of interactions

2. Local scale properties: Clustering

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2. Local scale properties: Clustering

15/10/2011 Slide 22/37Multilevel Networks: From local to global structure of interactions

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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15/10/2011 Slide 25/37Multilevel Networks: From local to global structure of interactions

2. Local scale properties: Clustering

15/10/2011 Slide 26/37Multilevel Networks: From local to global structure of interactions

2. Local scale properties: Clustering

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15/10/2011 Slide 27/37Multilevel Networks: From local to global structure of interactions

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

15/10/2011 Slide 30/37Multilevel Networks: From local to global structure of interactions

( , )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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15/10/2011 Slide 37/37Multilevel Networks: From local to global structure of interactions

References