Collective Dynamics of ‘Small World’ Networks

C+ Elegans: Ilhan Savut, Spencer Telford, Melody Lim

29/10/13

Networks of dynamical systems

Networks of dynamical systems

• Can be represented by points (vertices or nodes) connected with lines (edges)

Other Networks

Overview

• Different types of networkso ‘Collective Dynamics of Small World Networks’ Duncan Watts and

Steven Strogatz, Nature 393:440-442 (1998)

• Properties of networkso New type of network!

• Applications

A Regular Network

A Random Network

Fine-Tuning p Changes Network Properties• p - rewiring of regular network

• Properties depend heavily on p

Properties of Networks

• Path Length, L

• Clustering coefficient, C

Properties of Networks

• Path Length, Lo “Degrees of separation”

• Clustering coefficient, Co Chances that your friends are friends with each

other

Path Length for Networks

Long Short

Clustering for Networks

High Low

An Example of a Regular Network

No Random Networks Exist in Nature

L and C Depend on p

New Type of Network• 0 < p < 1 (not fully random, not fully regular)

Small-World Networks

• Short path length, highly clustered

‘Small-world’ Network in the Middle

Small World Networks Are Natural

• Formation of networks favors small world • Most networks built from small elements• Evolutionary and natural processes favor the

formation of small-world networks

Many Real World Systems are Small World Networks

• Collaboration graph of actors (six degrees of separation study)

• Neural network of C. elegans• Power grid of Western US

Small World Networks Have New PropertiesRobustness• Resistant to random changes• Targeted ‘attacks’

Small World Networks Model the Spread of Disease

ConclusionNew class of network model• Low path length• High clustering coefficient• Models many natural systems!