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Collective Dynamics of ‘Small World’ Networks C+ Elegans: Ilhan Savut, Spencer Telford, Melody Lim 29/10/13
Networks of dynamical systems
Can be represented by points (vertices or nodes) connected with lines (edges)
Overview Different types of networks o ‘Collective Dynamics of Small World Networks’ Duncan Watts and Steven Strogatz, Nature 393:440-442 (1998) Properties of networks o 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, L o “Degrees of separation” Clustering coefficient, C o Chances that your friends are friends with each other
Path Length for Networks LongShort
Clustering for Networks HighLow
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 Properties Robustness Resistant to random changes Targeted ‘attacks’
Small World Networks Model the Spread of Disease
Conclusion New class of network model Low path length High clustering coefficient Models many natural systems!
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