Networks FIAS Summer School 6th August 2008 [email protected] Complex Networks 1.

Networks

FIAS Summer School

6th August 2008

[email protected]

Complex Networks

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Overview

Introduction Three structural metrics Four structural models Structural case studies Node dynamics and self-organization Visualization Bibliography

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Introduction

What is a network? What is a complex network? Networks in the real world Elementary features Motivations

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What is a network?

● A network is a set of items (vertices or nodes) with connections between themcalled edges. Mathematicians call them “graphs”.

● Need not to be physical connections: nodes can be any type of entities and edges any type of abstract relationships.

● Ex.:nodes can be the channels of any multirecording device (EEG, MEG, multielectrode arrays, etc...) whileedges can be defined by the relationship (are two channelssynchronous or not?).

RT

RF

ROLO

LT

LF

LP RP

RCLC

ZF

ZC

ZP

ZO

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What is a network?

● Edges can be undirected or directed (arcs).

● Graphs can allow (friendship networks) or disallow loops (citation networks), parallel edges, ...

● Different types of networks: different types of vertices or edges, weighted networks, digraphs, bipartite graphs,evolving networks,...

RT

RF

ROLO

LT

LF

LP RP

RCLC

ZF

ZC

ZP

ZO

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What is a complex network?

● A complex network is a network with non-trivial topological features (featuresthat do not occur in simple networks such as lattices or random graphs)

Lattice Random

● Natural complex systems often exhibit such topologies.

• degree dist.• clustering• assortativity• comunity• hierarchical struct.

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Networks in the real world: examples of complex networks

Social, information, technological, biological,...

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Elementary features:node diversity and dynamics

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Elementary features:edge diversity and dynamics

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Elementary features:Network Evolution

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Motivations

complex networks are the backbone of complex systems every complex system is a network of interaction among numerous

smaller elements some networks are geometric or regular in 2-D or 3-D space other contain “long-range” connections or are not spatial at all understanding a complex system = break down into parts + reassemble

network anatomy is important to characterize because structure affects function (and vice-versa)

ex: structure of social networks prevent spread of diseases control spread of information (marketing, fads, rumors, etc…)

ex: structure of power grid / Internet understand robustness and stability of power / data transmission

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Three structural metrics

Average path length Degree distribution (connectivity) Clustering coefficient

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Structural metrics: Average path length

13* Measures how quickly info can flow through the network

Structural Metrics:Degree distribution (connectivity)

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* Divided in ‘in-degree’ and ‘out-degree’ for directed systems* High-degree nodes → ‘hubs’

Structural Metrics:Clustering coefficient

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* How likely is that the friend of your friend is also your friend?

Four structural models

Regular networks Random networks Small-world networks Scale-free networks

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Regular networks –fully connected

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Regular networks –Lattice

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Regular networks –Lattice: ring world

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Random networks

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Random Networks

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Small-world networks

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Scale-free networks

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Scale-free networks

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Scale-free networks

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Scale-free networks

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Scale-free networks

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Case studies

Internet World Wide Web Actors & scientists

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The Internet

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The Internet

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The Internet

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The World Wide Web

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World Wide Web

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World Wide Web

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Actors

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Mathematicians &Computer Scientists

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Node dynamics and self-organization

Node dynamics Attractors in full & lattice networks Synchronization in full networks Waves in lattice networks Epidemics in complex networks

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Node dynamics: individual node

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Node dynamics:coupled nodes

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Node dynamics and self-organization:Epidemics in complex networks

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Node dynamics and self-organization:Epidemics in complex networks

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Visualization & analysis

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http://vlado.fmf.uni-lj.si/pub/networks/pajek/

● Program for large networks analysis : Pajek

● Free● Windows (on Linux too but not so smooth)

*Vertices 31 “Source”2 “Sink”3 “Destination”*Arcs*Edges1 2 12 3 1

Based on…

Eileen Kramer & Kai Willadsen

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Bibliography

Reviews Barabási, A.-L. (2002) Linked: The New Science of

Networks.Perseus Books. Barabási, A.-L. and Bonabeau, E. (2003)

Scale-free networks. Scientific American, 288: 60-69. Strogatz, S. H. (2001) Exploring complex networks.

Nature, 410(6825): 268-276. Wang, X. F. (2002) Complex networks: topology,

dynamics and synchronization. International Journal of Bifurcation and Chaos, 12(5): 885-916.

Newman M. E. J. (2003) The structure and function of complex networks. arXiv:cond-mat/0303516v1

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Networks FIAS Summer School 6th August 2008 [email protected] Complex Networks 1.

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