Networks FIAS Summer School 6th August 2008 [email protected] Complex Networks 1.
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Transcript of Networks FIAS Summer School 6th August 2008 [email protected] Complex Networks 1.
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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?).
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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,...
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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
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Structural Metrics:Degree distribution (connectivity)
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* Divided in ‘in-degree’ and ‘out-degree’ for directed systems* High-degree nodes → ‘hubs’
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Structural Metrics:Clustering coefficient
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* How likely is that the friend of your friend is also your friend?
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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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Small-world networks
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Small-world 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
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Node dynamics and self-organization
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Node dynamics and self-organization
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Node dynamics and self-organization
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Node dynamics and self-organization
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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
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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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