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Lecture 9: Network models

Lecture 9: Network models. CS 765: Complex Networks. Slides are modified from Networks: Theory and Application by Lada Adamic. Four structural models. Regular networks Random networks Small-world networks Scale-free networks. Regular networks – fully connected.

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Lecture 9: Network models

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  1. Lecture 9: Network models CS 765: Complex Networks Slides are modified from Networks: Theory and Application by Lada Adamic

  2. Four structural models • Regular networks • Random networks • Small-world networks • Scale-free networks

  3. Regular networks – fully connected

  4. Regular networks – Lattice

  5. Regular networks – Lattice: ring world

  6. Poisson distribution modeling networks: random networks • Nodes connected at random • Number of edges incident on each node is Poisson distributed

  7. Random networks

  8. Erdos-Renyi random graphs • What happens to the size of the giant component as the density of the network increases? http://ccl.northwestern.edu/netlogo/models/GiantComponent

  9. Random Networks

  10. modeling networks: small worlds • Small worlds • a friend of a friend is also frequently a friend • but only six hops separate any two people in the world Arnold S. – thomashawk, Flickr; http://creativecommons.org/licenses/by-nc/2.0/deed.en

  11. Small-world networks

  12. Small world models • Duncan Watts and Steven Strogatz • a few random links in an otherwise structured graph make the network a small world: the average shortest path is short regular lattice: my friend’s friend is always my friend small world: mostly structured with a few random connections random graph: all connections random Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:440-442.

  13. Watts Strogatz Small World Model • As you rewire more and more of the links and random, what happens to the clustering coefficient and average shortest path relative to their values for the regular lattice? http://ccl.northwestern.edu/netlogo/models/SmallWorlds

  14. Scale-free networks

  15. Scale-free networks • Many real world networks contain hubs: highly connected nodes • Usually the distribution of edges is extremely skewed many nodes with few edges number of nodes with so many edges fat tail: a few nodes with a very large numberof edges number of edges no “typical” number of edges

  16. But is it really a power-law? • A power-law will appear as a straight line on a log-log plot: • A deviation from a straight line could indicate a different distribution: • exponential • lognormal log(# nodes) log(# edges)

  17. Scale-free networks http://ccl.northwestern.edu/netlogo/models/PreferentialAttachment

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