Complex System Dynamics LTCC Course Feb-March 2012

1. Complex System DynamicsLTCC Course Feb-March 2012 Dr Hannah Fry hfry@math.ucl.ac.uk

2. Degree Distribution Uniform Lattice Cumulative Degree Distribution kmax = 4

3. Degree Distribution Random Network Cumulative Degree Distribution kmax = 15, p(k) = exp(-k^2)

4. Degree Distribution Scale Free Network Cumulative Degree Distribution kmax = 247, p(k) = Ck^-α

5. Scale Free Networks M.J. Newman, Networks: an introduction. 2010

6. Random Network Generators

7. Scale Free Network Generators A. Barabasi E. Bonabeau, Scale Free Networks, Scientific American (2003)

8. Scale Free Network Generators • Growth: Start with nodes • At each time step add a new node with edges • Preferential attachment: Connect free edges to existing nodes with probability

9. Scale Free Network Generators m,m0 = 1 m,m0 = 3 m,m0 = 5 m,m0 = 7 Barabasi, Albert, Jeong, Mean field theory for scale free random networks (1999)

10. Dynamics on Networks

11. Forrest Fires

12. Epidemic Modelling SIR

13. Epidemics and path length Clustering Coefficient: Average Degree: Average Path Length: Clustering Coefficient: Average Degree: Average Path Length:

14. Uniform Lattice

15. Random Network

16. Scale Free Network

17. TB in Wales Mark Temple, Finding both the needle and the haystack (2011)

18. Epidemics • “An epidemic is defined as outbreaks that affect a non zero fraction of the population in the limit of a large system size.” • λEffective spreading rate • ρ0 Average number of secondary infections caused by a single individual • R0 Reproduction rate

19. Epidemic Modelling SIS

20. Epidemic Modelling SIS

21. Surviving probability for viruses in the wild Pastor-SatorrasVespignani, Epidemic Spreading in Scale Free Networks (2000)

22. New Research: • Dynamics of Networks

23. Network Resilience A. Barabasi E. Bonabeau, Scale Free Networks, Scientific American (2003)

24. Network Resilience A. Barabasi E. Bonabeau, Scale Free Networks, Scientific American (2003)

25. Inter-dependent Networks Buldyrev, Parshani, Paul, Stanley,Havlin, Catastrophic Cascade of Failures in interdependent networks. Nature(2010)

26. Evolving sexual contact networks Liljeros, Edling, Amaral, Stanley, Aberg, The web of sexual contacts, Nature 2001 Holme, Saramaki, Temporal Networks (2011)

27. London Riots

28. The Physics of Transistors

29. Case Study III: Location Theory The Hotelling Model Von Thunen’s rings The retail model and analysis

31. The Hotelling Model

32. The Hotelling Model

33. Von Thunen’s Rings

34. 1. People like big shopping centers 2. People don’t like travelling far

35. The retail model: top down Money spent by residents of area i The floor space of center j Money spent in center j by residents of i The distance between i and j The benefit of shopping in j ( ) Constraints

36. The probability of a given flow configuration: Taking logs: Using Stirling's approximation:

37. Lagrangian multiplier: The entropy analogy:

38. Lagrangian multiplier: The entropy analogy: Some time later..

39. The retail model Relative attractiveness of shopping center j Total spending from residential area i Most likely set of flows between i and j

40. The origin constrained spatial interaction model Benefit associated with destination j Impedance associated with destination j Origin flows - known Flows between i and j

41. The origin constrained spatial interaction model Total attractiveness of destination j Origin flows - known

42. The origin constrained spatial interaction model Relative attractiveness of destination j Origin flows - known Flows between i and j