Self-Organised Criticality and Complication in the U.K. Urban Distribution

1. Self-Organised Criticality and Complication in the U.K. Urban Distribution Alasdair (“Sasha”) Anderson a.anderson1@lancaster.ac.uk

2. Criticality Notion of criticality comes from the Greek kritikos: κριτικος = “able to discern or judge” Its application in the context of chemistry denotes a “phase transition” from one state of matter to another.

3. Criticality in Chemistry

4. Critical Transitions in History Equivalent “phase transitions” in human history: • c.40,000 BP: “Great Leap Forward” • Sophisticated co-operation in hunting; networks of barter trade • c.4000BC:Neolithic Revolution • Agriculture and rural settlement • c.AD1750: Industrial Revolution • Secondary manufacturing and urbanisation

5. Critical Transitions in History

6. Self-Organisation Structure appears within the system independently of external force. It is determined by local interactions of multiple component agents or degrees of freedom. Goal of system is an attractor in phase space.

7. Self-Organisation in History The essence of history - including that of the urban distribution - is the interactive process of constant self-organisation between four categories of agency: • Individuals • Collectives (societies, families, nations, classes) • Environments (location, resources, climate) • Memes (units of replicated information, notably ideas)

8. Self-organised Criticality Per Bak et al. introduced the concept inSelf-organised Criticality: an Explanation of 1/f noise (1987). • Sandpile model. • Forest fires • Earthquakes • Cellular Automata (J.H. Conway) • Evolutionary Biology (S. J. Gould) • Size and Frequency of Wars (G.G. Brunk) • Urban Distribution (M. Batty, Y. Xie)

9. Lessons from the Sandpile Model 1. System begins in a equilibrium state (i.e. flat). 2. Crosses the threshold to non-equilibrium that resolves to a power law distribution. 3. Non-equilibrium has a numerical value, the “angle of repose” (32 to 34º), suggesting a point attractor. 4. Scale Invariant Behaviour - it applies both to sand dunes and egg timers. 5. Connectivity between every grain of sand and every other. 6. Maintained by avalanches (“punctuations”) of varying sizes, also conforming to a power law.

10. Equilibrium Phase Early phases of settlement equate most closely to the random distribution of the environment (similar to sand in its flat configuration). In this scenario, individual leaders could exercise a strong influence on the future evolution of the system, reflected in the naming of settlements. • Edinburgh (Din Eidyn - “Eidyn’s Hill-fort”) • Anglo-Saxon settlements with -ing suffix (“people of …”)

11. Power Law Distribution G.K. Zipf (1949) in “Human Behaviour and the Principle of Least Effort” proposed the rank-size principle from the frequency of words in a text. Thence, the Zipf Law has been applied to cities, on the basis of their distribution to a power law.

12. Line of Criticality: 1520

13. Line of Criticality: 1600

14. Line of Criticality: 1670

15. Line of Criticality: 1700

16. Line of Criticality: 1750

17. Line of Criticality: 1801

18. Line of Criticality: 1821

19. London, Southwark, and Lambeth (1747)

20. London, Southwark, and Lambeth (1802)

21. London, Southwark, and Lambeth (1830)

22. Line of Criticality: 1851

23. Line of Criticality: 1861

24. Line of Criticality: 1871

25. Line of Criticality: 1881

26. Line of Criticality: 1891

27. Line of Criticality: 1901

28. Line of Criticality: 1901

29. Line of Criticality: 1911

30. Line of Criticality: 1921

31. Line of Criticality: 1931

32. Line of Criticality: 1938

33. Line of Criticality: 1947

34. Line of Criticality: 1951

35. Line of Criticality: 1961

36. Line of Criticality: 1971

37. Line of Criticality: 1975

38. Line of Criticality: 1981

39. Line of Criticality: 1985

40. Line of Criticality: 1989

41. Line of Criticality: 1991

42. Line of Criticality: 1993

43. Are the Data Wrong? Data between 1901 and 1961 contain anachronism of Greater London post-1965 London Boroughs, rather than Metropolitan and Municipal Boroughs. Data after 1961 contain spurious administrative units (Kirklees, Wirral, West Norfolk). However, this is insufficient to account for the persistence of the line.

44. Is the Theory Wrong? Cut-off point is misleading, as it truncates the “fat tail” or “long tail”. Data without a cut-off show settlements below a point that apparently violates the “law”. The primate city is consistently too populous for the prediction (possibly for reasons which may be explained). However, the very number of settlements conforming to the line and its persistenceacross an extended period of time support the conclusion that a power law is involved.

45. Is Reality Wrong? Should London be depopulated to conform to power law? Town & Country Planning Act (1947); Green Belt; New Towns; Decentralisation policies of 1960s and 70s attempted this (rather as the Elizabethans had done). N. Georgescu-Roegen in Entropy and the Economic Process (1971) was proposing “radical de-urbanisation”. Should population be “imposed” on the smaller settlements?

46. Power Law Line of Best Fit

47. Panocephalicity and Catouricity

48. Panocephalicity of London

49. Panocephalicity of London

50. Catouricity According to the theory, the cut-off point excludes the “rural” settlements, the origins of the urban system. (M. Batty). However, the smallest settlements are not all characteristically rural. This downward morphology is evident to the very tip - scale invariant behaviour is an essential feature of the distribution.