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Collective Dynamics of ‘Small World’ Networks C+ Elegans: Ilhan Savut, Spencer Telford, Melody Lim 29/10/13

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Networks of dynamical systems

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

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Other Networks

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Overview Different types of networks o ‘Collective Dynamics of Small World Networks’ Duncan Watts and Steven Strogatz, Nature 393: (1998) Properties of networks o New type of network! Applications

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A Regular Network

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A Random Network

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Fine-Tuning p Changes Network Properties p - rewiring of regular network Properties depend heavily on p

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Properties of Networks Path Length, L Clustering coefficient, C

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Properties of Networks Path Length, L o “Degrees of separation” Clustering coefficient, C o Chances that your friends are friends with each other

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Path Length for Networks LongShort

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Clustering for Networks HighLow

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An Example of a Regular Network

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No Random Networks Exist in Nature

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L and C Depend on p

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

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Small-World Networks Short path length, highly clustered

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‘Small-world’ Network in the Middle

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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

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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

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Small World Networks Have New Properties Robustness Resistant to random changes Targeted ‘attacks’

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Small World Networks Model the Spread of Disease

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Conclusion New class of network model Low path length High clustering coefficient Models many natural systems!

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