Self-Organised Criticality and Complication in the U.K. Urban DistributionAlasdair (“Sasha”) Anderson

a.anderson1@lancaster.ac.uk

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.

Criticality in Chemistry

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

Critical Transitions in History

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.

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)

Self-organised Criticality

Per Bak et al. introduced the concept in Self-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)

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.

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 …”)

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.

Line of Criticality: 1520

Line of Criticality: 1600

Line of Criticality: 1670

Line of Criticality: 1700

Line of Criticality: 1750

Line of Criticality: 1801

Line of Criticality: 1821

London, Southwark, and Lambeth (1747)

London, Southwark, and Lambeth (1802)

London, Southwark, and Lambeth (1830)

Line of Criticality: 1851

Line of Criticality: 1861

Line of Criticality: 1871

Line of Criticality: 1881

Line of Criticality: 1891

Line of Criticality: 1901

Line of Criticality: 1901

Line of Criticality: 1911

Line of Criticality: 1921

Line of Criticality: 1931

Line of Criticality: 1938

Line of Criticality: 1947

Line of Criticality: 1951

Line of Criticality: 1961

Line of Criticality: 1971

Line of Criticality: 1975

Line of Criticality: 1981

Line of Criticality: 1985

Line of Criticality: 1989

Line of Criticality: 1991

Line of Criticality: 1993

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.

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 persistence across an extended period of time support the conclusion that a power law is involved.

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?

Power Law Line of Best Fit

Panocephalicity and Catouricity

Panocephalicity of London

Panocephalicity of London

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.

Catouricity in Small Settlements

Decentring and Punctuations

Fernand Braudel’s concept of “centring, decentring, and recentring” describes radical shifts of the world economic centre. These represent high-level landslide-like events in economic history:

• Venice (1380s-1500)

• Antwerp (1500-1550)

• Genoa (1550-1600)

• Amsterdam (1600-1780)

• London (1815-1900s)

• New York (1900s - )

Decentring and Punctuations

Shifts in rank and population size also correspond to the avalanches in the Per Bak model.

A manifestation of complication in human systems is that population transfers between urban centres (unlike sand) can either flow up or down the distribution

These shifts also equate to the “punctuated equilibria” proposed by Stephen Jay Gould in evolutionary biology.

Decentring and Punctuations

Decentring and Punctuations

Decentring and Punctuations

Self-Organised Criticality and Complication in the U.K. Urban DistributionAlasdair (“Sasha”) Anderson

a.anderson1@lancaster.ac.uk