Networks FIAS Summer School 6th August 2008 Complex Networks 1

Overview Introduction Three structural metrics Four structural models Structural case studies Node dynamics and self-organization Visualization Bibliography 2

Introduction What is a network? What is a complex network? Networks in the real world Elementary features Motivations 3

What is a network? ● A network is a set of items (vertices or nodes) with connections between them called 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...) while edges can be defined by the relationship (are two channels synchronous or not?). 4

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,... 5

What is a complex network? ● A complex network is a network with non-trivial topological features (features that do not occur in simple networks such as lattices or random graphs) LatticeRandom ● Natural complex systems often exhibit such topologies. degree dist. clustering assortativity comunity hierarchical struct. 6

Networks in the real world: examples of complex networks Social, information, technological, biological,... 7

Elementary features: node diversity and dynamics 8

Elementary features: edge diversity and dynamics 9

Elementary features: Network Evolution 10

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 11

Three structural metrics Average path length Degree distribution (connectivity) Clustering coefficient 12

Structural metrics: Average path length 13 * Measures how quickly info can flow through the network

Structural Metrics: Degree distribution (connectivity) 14 * Divided in ‘in-degree’ and ‘out-degree’ for directed systems * High-degree nodes → ‘hubs’

Structural Metrics: Clustering coefficient 15 * 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 16

Regular networks – fully connected 17

Regular networks – Lattice 18

Regular networks – Lattice: ring world 19

Random networks 20

Random Networks 21

Small-world networks 22

Small-world networks 23

Small-world networks 24

Small-world networks 25

Scale-free networks 26

Scale-free networks 27

Scale-free networks 28

Scale-free networks 29

Scale-free networks 30

Case studies Internet World Wide Web Actors & scientists 31

The Internet 32

The Internet 33

The Internet 34

The World Wide Web 35

World Wide Web 36

World Wide Web 37

Actors 38

Mathematicians & Computer Scientists 39

Node dynamics and self- organization Node dynamics Attractors in full & lattice networks Synchronization in full networks Waves in lattice networks Epidemics in complex networks 40

Node dynamics: individual node 41

Node dynamics: coupled nodes 42

Node dynamics and self-organization 43

Node dynamics and self-organization 44

Node dynamics and self-organization 45

Node dynamics and self-organization 46

Node dynamics and self-organization 47

Node dynamics and self-organization: Epidemics in complex networks 48

Node dynamics and self-organization: Epidemics in complex networks 49

Visualization & analysis 50 ● Program for large networks analysis : Pajek ● Free ● Windows (on Linux too but not so smooth) *Vertices 3 1 “Source” 2 “Sink” 3 “Destination” *Arcs *Edges

Based on… Eileen Kramer & Kai Willadsen 51

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: Scale-free networks  Strogatz, S. H. (2001) Exploring complex networks. Nature, 410(6825): Exploring complex networks  Wang, X. F. (2002) Complex networks: topology, dynamics and synchronization. International Journal of Bifurcation and Chaos, 12(5):  Newman M. E. J. (2003) The structure and function of complex networks. arXiv:cond-mat/ v1 52