15.1 Pa Rank Size Distribution
251
is not an isolated phenomenon but has grown out of ancient settlement patterns.
Thus, as we shall show, it is possible, for example, through a mathematical analysis
of Walton’s data to estimate a population growth rate for New Zealand averaged
over the full period of its settlement. Even the size of a founder population can be
estimated directly from the pa rank size distribution curve.
Throughout New Zealand’s history, the distribution of settlement sizes incorporated information about the original founder settlement and about the history of
development from that settlement but the survival of that information was inhibited by the perishable nature of the materials used to build settlements. With the
construction of enduring pa this information became manifest, became as it were
frozen in earth and wood. In the case of New Zealand pa, this evidence once secured
is effectively permanent. Despite the drastic population losses in the Little Ice Age
and consequent later mismatches between pa capacity and population levels, the
original population capacity of pa, represented by area alone, preserves evidence
for earlier population levels and settlement patterns.
When pa are ranked in decreasing order by size, and area graphed against rank,
the plot exhibits upper-tail power law behaviour. Similarly when pa are ranked in
increasing order by size, and area graphed against rank, the lower tail exhibits a
power law.
A parallel phenomenon has been known for half a century with cities and their
populations. Indeed, the phenomenon is true from major cities to villages of only
30 inhabitants [2]. That a pa rank size distribution follows power laws is hardly
surprising. If the pa area occupied per person and the proportion of pa area occupied
by people both vary over modest ranges, then the distribution of pa sizes can be
expected to reflect that of the sizes of human communities with only differences of
scale. Community sizes may be described in “equivalent pa areas” and the pa size
distribution serves as a proxy to the distribution of the sizes of the communities of
their occupants.
After some early difficulties, fairly satisfactory explanations are now available
for one and two tail rank-size laws for community sizes, although these involve
some technical points of probability and statistics. See, for example, Gabaix (1999)
[3], Reed and Hughes (2002) [4] and Reed and Jorgensen (2004) [5].
In the geometric Brownian motion model of Reed and Hughes an initial community grows in size and also produces daughter communities, which behave similarly.
Under mild assumptions we have asymptotically a collection of communities: the
size of a constituent community has (for appropriate positive constants α, β) a
double Pareto density function of the form
f (x) =
A(x/X 0 ) β x −1 if x ≤ X 0 ,
A(X 0 /x) α x −1 if x > X 0
which exhibits power law behaviour in both tails. The imprint of history is carried
in particular by the parameter X 0 which is the size of the original community.
It is worth commenting at this point that because the distribution of pa rank
size occurs asymptotically, it has no direct time frame. This can only be obtained
251
is not an isolated phenomenon but has grown out of ancient settlement patterns.
Thus, as we shall show, it is possible, for example, through a mathematical analysis
of Walton’s data to estimate a population growth rate for New Zealand averaged
over the full period of its settlement. Even the size of a founder population can be
estimated directly from the pa rank size distribution curve.
Throughout New Zealand’s history, the distribution of settlement sizes incorporated information about the original founder settlement and about the history of
development from that settlement but the survival of that information was inhibited by the perishable nature of the materials used to build settlements. With the
construction of enduring pa this information became manifest, became as it were
frozen in earth and wood. In the case of New Zealand pa, this evidence once secured
is effectively permanent. Despite the drastic population losses in the Little Ice Age
and consequent later mismatches between pa capacity and population levels, the
original population capacity of pa, represented by area alone, preserves evidence
for earlier population levels and settlement patterns.
When pa are ranked in decreasing order by size, and area graphed against rank,
the plot exhibits upper-tail power law behaviour. Similarly when pa are ranked in
increasing order by size, and area graphed against rank, the lower tail exhibits a
power law.
A parallel phenomenon has been known for half a century with cities and their
populations. Indeed, the phenomenon is true from major cities to villages of only
30 inhabitants [2]. That a pa rank size distribution follows power laws is hardly
surprising. If the pa area occupied per person and the proportion of pa area occupied
by people both vary over modest ranges, then the distribution of pa sizes can be
expected to reflect that of the sizes of human communities with only differences of
scale. Community sizes may be described in “equivalent pa areas” and the pa size
distribution serves as a proxy to the distribution of the sizes of the communities of
their occupants.
After some early difficulties, fairly satisfactory explanations are now available
for one and two tail rank-size laws for community sizes, although these involve
some technical points of probability and statistics. See, for example, Gabaix (1999)
[3], Reed and Hughes (2002) [4] and Reed and Jorgensen (2004) [5].
In the geometric Brownian motion model of Reed and Hughes an initial community grows in size and also produces daughter communities, which behave similarly.
Under mild assumptions we have asymptotically a collection of communities: the
size of a constituent community has (for appropriate positive constants α, β) a
double Pareto density function of the form
f (x) =
A(x/X 0 ) β x −1 if x ≤ X 0 ,
A(X 0 /x) α x −1 if x > X 0
which exhibits power law behaviour in both tails. The imprint of history is carried
in particular by the parameter X 0 which is the size of the original community.
It is worth commenting at this point that because the distribution of pa rank
size occurs asymptotically, it has no direct time frame. This can only be obtained