PROPOLIS

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Friday, August 15, 2014 After Zipf From
City Size Distributions to Simulations Michael
Batty Yichun Xie UCL EMU m.batty_at_ucl.ac.uk,
yxie_at_emich.edu http//www.casa.ucl.ac.uk/
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1.What is the Notting Hill Carnival A Two day
Annual event based on a street parade and street
concerts in inner London which is a celebration
of West Indian ethnic culture. Started in 1964 as
The Notting Hill Festival attracting 150,000
people by 1974 It attracts up to 1 million
visitors and spreads over an are of about 3.5 sq
miles Here are some pictures
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Scaling in Urban Systems A Brief History 1.
Gravitational analogies - Ravenstein (1888) for
migration, Carey (1850), French Physiocrats 2.
The Emergence of Social Physics from the 1940s on
- Regional Science in the 1950s 3. The simplest
scaling - Zipfs Law - the rank size rule 4.
Transportation Modeling 5. Scaling in terms of
fractals from the 1980s on
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Zipfs Law and the Distribution of Populations in
Systems of Cities 1. Zipfs Law - the rank size
rule 2. The confusion over its formulation as a
probability distribution 3. The original emphasis
on description 4. Most examples took the largest
events known such as the top 100 world cities as
defined from yearbooks etc 5. There is hardly any
attention to what these events really mean
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A Typical Distribution for World Population in
1994 Here we have 150 countries and we will show
how difficult it can be from this kind of data to
demonstrate scaling
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Log Population against Log Rank
The King or Primate City Effect
Scaling only over restricted orders of magnitude
A different regime in the thin tail
Log Population
Log Rank
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Problems 1. Scaling - many indeed most
distributions are not power functions 2. The
events are not independent - in medieval times
they may have been but for the last 200 years,
cities have grown into each other, nations have
become entirely urbanized, and now there are
global cities - the tragedy in NY tells us this -
where more than half of those killed were not US
citizens 3. Should we expect scaling ? We know
that cities depend on history as well as economic
growth
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Problems (continued) 4. Why should we expect no
characteristic length scale - when the world is
finite ? We should avoid the sin of
Asymptopia. 5. As scaling is often said to be
the signature of self-organization, why should we
expect disparate and distant places to
self-organize ? 6. The primate city effect is
very dominant in historically old countries 7.
BUT should we expect these differences to
disappear as the world become global ?
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Lets first look at arbitrary events - An Example
for the UK based on Administrative Units, not on
trying to define cities as separate fields
These are 458 admin units, somewhat less than
full cities in many cases and some containing
towns in county aggregates - we have data from
1901 to 1991 so we can also look at the dynamics
of change - traditional rank size theory says
very little about dynamics
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This is what we get when we fit the rank size
relation PrP1 r - ? to the data. The parameter
is hardly 1 but it is more than 0.36 which was
the value for world population in 1994
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Here is an example of the shift in size and ranks
over the last 100 years in GB
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Explaining City Size Distributions Using
Multiplicative Processes The last 10 years has
seen many attempts to explain scaling
distributions such as these using various simple
but stochastic processes. In essence, the
easiest which gives rise to distributions such as
these is a model of proportionate effect or
growth which leads to the lognormal distribution
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The key idea is that the change in size of the
object in question is proportional to the size of
the object and randomly chosen, that is
This leads to the log of differences across time
being a function of the sum of random changes
This gives the model of proportionate effect
or
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Heres a simulation which shows that the
rank-size rule is generated this way with much
the same properties as the observed data for UK
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This is a good model to show the persistence of
settlements, it is consistent with what we know
about urban morphology in terms of fractal laws,
but it is not spatial. However there are other
processes which we should note which have been
explored. I will list these as follows - and
please note that my survey is by no means
exhaustive.
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Other Stochastic Processes which have been used
to explain scaling 1. The Simon model - birth
processes are introduced which is not something
which is done in generating the lognormal 2.
Multiplicative random growth with constraints on
the lowest size - size is not allowed to become
too small otherwise the event is removed 3. Work
on growth rates consistent with scaling relations
involving Levy distributions
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The Second Example - distributions where the
events are unambiguous or less ambiguous - the
distribution of links on WWW Here we take a look
at the distribution of indegrees and outdegrees
formed by links relating to web pages - a web
page is pretty unambiguous. There is a lot of
work on this produced during the last three
years, notably the Xerox Parc group the Notre
Dame group
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Number of Web Pages and Total Links - indegrees
and outdegrees These are taken from relevant
searches of AltaVista for 180 domains in 1999
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Links as indegrees and outdegrees compared to
the Total Links
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Number of Web Pages,Total Links, GDP and Total
World Populations
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These are based on the general formula where q is
the parameter of the distribution As a general
conclusion, it does not look as though the event
size issue has much to do with the scaling or
lack of it.We urgently need some work on spatial
systems with fixed event areas, thus shifting the
focus to densities not distributions
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The internet is a great example - the densest
nodes are in the places where all the information
is concentrated - in the world cities - in short,
distances and locations in cyberspace mirror real
space - biggest hubs are in Manhattan, City of
London and so on e.g.
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Two regimes for the indegrees and outdegrees
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Last Comments and Future Work Scaling can be
shown to be consistent with more micro-based ,
hence richer less parsimonious models 1.
Diffusion and growth models 2. Agent-based
competition models 3. Treating the system as a
growing network - this latter model is worth
finishing with as it is particularly relevant to
the WWW and is probably close to interaction
models of cities as in transportation
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Network Approaches to Scaling Here we take a
look at the distribution of indegrees and
outdegrees formed by links relating to web pages
- a web page is pretty unambiguous. There is a
lot of work on this produced during the last
three years, notably the Xerox Parc group the
Notre Dame group let me start with some notions
of about graphs
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On the left a random graph, whose distribution of
the numbers/density of links at each node is near
normal - this has a characteristic length - the
average On the left, what is much more typical -
a graph which is scaling - one whose distribution
is rank size, following a power law P(k) k -
2.5
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Not only does the topology of web pages follow
power laws so does the physical hardware - the
routers and wires This and the last diagram are
taken from the article by Barabasi called The
Physics of the Web printed in the July 2001
issue of Physics World
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Here is some work that Steve Coast in our group
at CASA is doing on detecting and measuring the
distribution of the hardware of the web and
visualizing it - all this is prior to measuring
its properties - i.e. is it scaling, is it a
small world and so on
Challenge is to map real space onto cyberspace
and that so far has not really been attempted in
these new ideas about how network systems
grow This is the cluster of routers, and hubs
and machines in UCL
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Some statistics from Steves work - which imply
scale free networks
Lots and lots of issues here - we need models of
how networks grow and form, how does the small
world effect mesh into scale free networks ? We
need to map cyberspace onto real space and back,
and this is no more than mapping social space
onto real space and back - its not new. I will
finish
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Some of the most interesting work is being done
in virtual space - in cyberspace not in real
space. Here is an example of such a network
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Some references - Martin Dodge and Rob Kitchins
new book Steve Coasts web site www.fractalus.co
m/steve/ Our web site www.casa.ucl.ac.uk and
drill down to get to Martins www.cybergeography
.org
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World-216 countries R-sq 0.708 b -2.26
USA-3149 cities R-sq 0.992 b -0.81
Mexico-36 cities R-sq 0.927 b -1.27
UK-459 areas R-sq 0.760 b -0.58
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Figure 1 Rank-Size Distributions of Highly Cited
Scientists
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where
is the number of cited scientists at rank
is the mean number of cited scientists, and
is the number of institutions, places, or
countries for each of the three respective
aggregations6
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Number of Web Pages and Total Links - indegrees
and outdegrees These are taken from relevant
searches of AltaVista for 180 domains in 1999
54
Network Approaches to Scaling Here we take a
look at the distribution of indegrees and
outdegrees formed by links relating to web pages
- a web page is pretty unambiguous. There is a
lot of work on this produced during the last
three years, notably the Xerox Parc group the
Notre Dame group let me start with some notions
of about graphs
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As an introductory example, I will repeat what I
say in the editorial I handed out on small
worlds. You can read this later The term small
worlds was first coined in psychology and
sociology in the 1960s by Stanley Milgram but
remained a talking point only, for 30 years
largely because there was 1. No technical
apparatus to measure connectivity in very large
graphs - where you have say more than 1 million
nodes 2. There was no real way in which one could
handle processes taking place on graphs 3. There
was not much thinking about how real graphs
structures evolved - through time
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All these points needed to be resolved before one
could get anywhere and they are slowly being
resolved. An example of a small world - a kind
of connectivity in graphs
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• Examples
• Evolution of transport systems in big cities
• What makes small spaces in cities attractive and
  livable in
• Spread of disease - foot and mouth for example
• How social systems hold together
• Academic communities, like us
• Nervous systems, how particles interact, WWW etc

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Some of the most interesting work is being done
in virtual space - in cyberspace not in real
space. Here is an example of such a network
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The world wide web is a small world as are most
systems that dont break apart under tension -
thing about cities that break apart - London
currently with the fact that no decent freeway
system was built in the automobile age and the
subway hasnt been fixed for 50 years. Global
cities are small worlds. However there is a much
more general theory of networks being devised
which examines regularity and processes in such
structures. Recently it looks as though most
stable networks are scale free - this means that
when you examine their structure, there is no
characteristic length scale - they are fractal -
moreover as they grow, they grow through positive
feedback - dense clusters get denser - the rich
get richer - again think of cities - in short
they do not grow randomly
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On the left a random graph, whose distribution of
the numbers/density of links at each node is near
normal - this has a characteristic length - the
average On the left, what is much more typical -
a graph which is scaling - one whose distribution
is rank size, following a power law P(k) k -
2.5
61
Not only does the topology of web pages follow
power laws so does the physical hardware - the
routers and wires This and the last diagram are
taken from the article by Barabasi called The
Physics of the Web printed in the July 2001
issue of Physics World
62
Here is some work that Steve Coast in our group
at CASA is doing on detecting and measuring the
hardware of the web and visualizing it - all this
is prior to measuring its properties - i.e. is it
scaling, is it a small world and so on
Challenge is to map real space onto cyberspace
and that so far has not really been attempted in
these new ideas about how network systems
grow This is the cluster of routers, and hubs and
machines in UCL
63
Some more fancy visualizations of these networks
64
Some statistics from Steves work - which imply
scale free networks
Lots and lots of issues here - we need models of
how networks grow and form, how does the small
world effect mesh into scale free networks ? We
need to map cyberspace onto real space and back,
and this is no more than mapping social space
onto real space and back - its not new I
will finish
65
Some references - Martin Dodge and Rob Kitchins
new book Steve Coasts web site www.fractalus.com
/steve/ Our web site www.casa.ucl.ac.uk and
drill down to get to Martins www.cybergeography.
org