1 the zipf seminars at emu-um sunday, april 06, 2014 after zipf: from city size distributions to...

1 The Zipf Seminars at EMU-UM

Monday, April 10, 2023

After Zipf: From City Size Distributions to Simulations

Or why we find it hard to build models of how cities talk to each other

Michael Batty & Yichun XieUCL [email protected] [email protected] http://www.casa.ucl.ac.uk/ http://www.ceita.emich.edu/


What we will do in this talk

1. Continue Tom and John’s discussion of Zipf’s Law in particular and scaling in urban systems in general from last week

2. Review very briefly what this area is about from last week

3. Review the key problems – power functions v. lognormal, fat tails, thin tails, primate cities

4. Note the basic stochastic models where cities do not talk to each other but do produce ‘good’ simulations. Illustrate such a simulation.


What we will do in this talk

5. Outline some more examples of Zipf’s Law in terms of data applications – countries, spatial partitions, telecoms systems, the geography of citations

6. Note how connectivity or interaction is entering the debate through social networks and the web

7.


Zipf’s Law …

Says that in a set of well-defined objects like words (or cities ?), the size of any object (is inversely proportional to its size; and in the strict Zipf case this inverse relation is

This is the strict form because the power is -1 which gives it somewhat mystical properties but a more general form is the inverse power form

1 Krr

KPr

KrPr


In one sense, this is obvious – in a competitive system where resources are scarce, it is intuitively obvious that there are less big things than small things

And when you have a system in which big things ‘grow’ from small things, this is even more obvious

But why should the slope be -1 and why should the form be inverse power

In fact as we shall see and as Tom intimated last week this is highly questionable


Here are some classic examples from last weekFirst from Zipf (1949)


Now from Tom (2003) – top 135 cities


As you can see the curve is not quite straight but slightly curved – this is significant but there are some obvious problems

• Most researchers have taken the top 100 or so cities– they have disregarded the bottom but what happens at the bottom is where it all begins – where growth starts – the short tail

• Cities are not well defined objects – they grow into each other

• 3. Cities do not keep their place in the rank order –but shift but the order stays stable – how ?

• 4 Primate cities are problematic at the top of the long tail


Let’s look at some cities, countries, & spatial partitions

USA-3149 citiesR-sq = 0.992 = -0.81

Mexico-36 citiesR-sq = 0.927 = -1.27

World-216 countriesR-sq = 0.708 = -2.26

UK-459 areasR-sq = 0.760 = -0.58


Basically what these relations show is that as soon as you define something a little bit different from cities, you get Zipf exponents which are nowhere near unity. In fact it would seem that for countries we have much greater inequality than cities which in turn is much greater than for exhaustive spatial divisions

Now to show how different this all is, then I will show yet another set of countries where there are now only 149 countries, not 216 – from another standard data set (MapInfo)


0123456789

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ulat

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The King or Primate City Effect

Scaling only over restricted orders of magnitude

A different regime in the thin tail


Log Population versus Log Rank

02468

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0 1 2 3log rank

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po

pu

lati

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Residuals against Rank Orders

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ls99.1157.10 rPr

36.015.4 rPr


Related Problems

• Scaling - many indeed most distributions are not power functions

• 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

• Should we expect scaling ? We know that cities depend on history as well as economic growth


• Confusion over Zipf exponents and their value• Why should we expect no characteristic length

scale - when the world is finite ? We should avoid the sin of ‘Asymptopia’.

• As scaling is often said to be the signature of self-organization, why should we expect disparate and distant places to self-organize ?

• The primate city effect is very dominant in historically old countries

• BUT should we expect these differences to disappear as the world becomes global ?


Let’s 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


Log Population

Log Rank


Year t Correlation R2 Intercept Kt tKtP 101* Slope t

1901 0.879 6.547 3526157.772 -0.8171911 0.880 6.579 3801260.554 -0.8101921 0.887 6.604 4025650.857 -0.8121931 0.892 6.607 4046932.207 -0.8021941 0.865 6.532 3410371.276 -0.7401951 0.869 6.482 3034245.953 -0.7001961 0.830 6.414 2595897.640 -0.6511971 0.815 6.322 2101166.738 -0.6011981 0.816 6.321 2095242.746 -0.6011991 0.791 6.272 1872348.019 -0.577

This is what we get when we fit the rank size relation Pr=P1 r - to the data. The parameter is hardly 1 but it is more than 1.99 which was the value for world population in 1994


A Digression –Many other systems show such rank size – here we will look at geography of scientific citation –the Highly Cited

Table 2: Top Ten Ranking of Highly Cited Scientists by Country

Rank Country

No. Highly

Cited

No of Places

Concentration: Scientists/Places

Highly Cited per

Million Population

1

US 815

90

9.06

3.16

2 UK 100 24 4.17 1.72

3 Germany 62 21 2.95 0.78

4 Canada 42 15 2.80 1.53

5 Japan 34 14 2.43 0.27

6 France 29 11 2.64 0.50

8 Switzerland 26 5 5.20 3.78

9 Sweden 17 2 8.50 1.96

10 Italy 17

10 1.7 0.29


Table 1: Top Twenty Ranking of Highly Cited Scientists by Institution

Rankings Research Institution

No of Highly Cited Scientists

Percent Highly Cited Scientists

1

Harvard

52

4.3

2 Stanford 36 2.9 3 U-Cal, San Diego 30 2.5 4 MIT 26 2.1 5 NIH National Cancer Institute 19 1.6 6 U-Cal, San Francisco

Cornell 17 1.4

8 U-Cal, Berkeley University College London UK

16 1.3

10 CalTech 15 1.2 11 NIH Allergy & Infectious Diseases 13 1.1 12 Johns Hopkins

University of Cambridge UK Washington, Seattle Washington, St Louis

12 1.0

16 U-Cal, Davis U-Texas Cancer Center

11 0.9

18 Michigan Northwestern Yale

10 0.8


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ln [r/M]

ln [

P(x

)/<

x>]

Rank-Size Distributions of Highly Cited Scientists

red institution, black place, grey by countrystraightline fits

by institution (red)

)2.80( )5.90(

0.938 ,429,/ln816.0555.0)(ln 2

RMMrxxP

by place/city (black)

)8.76( )3.94(

0.962 ,232,/ln049.1768.0)(ln 2

RMMrxxP

by country (grey)

)6.21( )232(

0.949 ,27,/ln997.1583.1)(ln 2

.

RMMrxxP

by country (grey)


The Highly Cited By Place


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 stochastic processes. Most do not take any account of the fact that cities compete – talk to each other.

In essence, the easiest is a model of proportionate effect or growth first used for economic systems by Gibrat in 1931 which leads to the lognormal distribution


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

itit

it

P

P

t

iiit PP0

0loglog

ititit PP 1


Year t Correlation R2 Intercept Kt tKtP 101* Slope t

1 1 0 1 0900 0.840 -1.077 0.083 -0.7771000 0.844 -0.995 0.101 -0.824

Here’s a simulation which shows that the lognormal is generated with much the same properties as the observed data for UK

Note how long it takes for the lognormal to emerge, note also the switches in rank – too many I think for this to be realistic


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tt=1000=1000

tt=900=900

Log of RankLog of Rank

tt=1000 Population based =1000 Population based on on tt=900 Ranks=900 Ranks

Log

of

Pop

ulat

ion

Sha

res

Log

of

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ulat

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Sha

res


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.

In fact to demonstrate how this model works let me run a short simulation based on independent events – cities on a 20 x 20 lattice using the Gibrat process – here it is


Other Stochastic Processes which have been used to explain scaling

1. The Simon model - birth processes are introduced

2. Multiplicative random growth with constraints on the lowest size - size is not allowed to become too small otherwise the event is removed: Solomon’s model; Sornette’s work

3. Work on growth rates consistent with scaling involving Levy distributions – Stanley’s work

4. Economic variants – Gabaix, Krugman, LSE group, Dutch group, Reed etc


Dynamics of Rank-Size: Applications

We will now look again at countries and population change and then at penetration of telecoms devices by country

We have country data from 1980 to 2000, and telecoms data over the same period – we are interested in the dynamics – we can measure changes using the so-called Havlin plot defined as


This is the average difference in ranks over N cities or countries with respect to two time periods j and k.

So at each time we can plot a curve of differences away from that time in terms of every other time period.

This lets us identify big shifts in rank and thus unusual dynamics.

2/12

N

rrR i

ikij

jk


This is population of countries


And the average rank distances


This is the telecoms data


And the average rank distances


Some More Issues

Note the way systems grow in terms of the telecoms data

Note the fact that there is no connectivity at all in these systems

Let’s finish by looking at connectivity – how cities talk to each other – can we say anything at all about models that take such interactions into account – its another seminar but let us sketch some ideas


Networks and Scaling

These are distributions where the events are unambiguous or less ambiguous - the distribution of links in and out of nodes defining networks have been shown to be scaling by many people over the last four years, notably by Barabasi and his Notre Dame group and by Huberman and his Xerox Parc now HP Internet Ecologies group

Here we take a look at the distribution of in-degrees and out-degrees formed by links relating to web pages - a web page is pretty unambiguous, and s is a link unlike a city. This is some work that we did in 1999 at CASA.


This is based on some network data produced by Martin Dodge and Naru Shiode in CASA from their web crawlers


Number of Web Pagesand Total Links - indegrees and outdegrees

These are taken from relevant searches of AltaVista for 180 domains in 1999

Note the notion of a system which is immature – in terms of the lognormal form


Number of Web Pages,Total Links, GDP and Total World Populations


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

Distribution Intercept log K Slope -q Correlation r2 P’(1)/P(1)No. Web Pages 21.22 2.91 0.90 35.84Total Links 18.60 1.60 0.92 1.35Incoming Links 21.48 2.98 0.89 37.28Outgoing Links 17.83 1.46 0.91 1.03GDP 11.98 2.18 0.80 22.67Population 23.39 2.00 0.72 12.64


Two regimes for the in-degrees and out-degrees

tribution Slope –q1 forupper ranks

Correlation r2

for upper ranksSlope –q2 forlower ranks

Correlation r2

for lower ranksw2q2 / w1q1

. Web Pages 0.88 0.97 4.25 0.98 31.05al Links 0.86 0.97 2.07 0.91 15.47

Incoming Links 1.04 0.98 4.49 0.97 26.30tgoing Links 0.78 0.97 1.87 0.88 17.29P 1.22 0.99 3.25 0.80 5.65

pulation 1.01 0.91 2.80 0.73 1.31


The Key Issues: Where do we go from here?

Scaling can be shown to be consistent with more micro-based, hence richer, less parsimonious models; but there is a disjunction between work on spatial fractals such as in our 1994 book Fractal Cities (Academic Press)and the rank size rule – very hard to know how to build consistent models that work at the spatial level and give fractal relations which translate into city size distributions


Resources

ReferencesPapersWeb Resources

We will assemble a list and put these on a web site. I will out this power point on the China Data Center site like Tom and John’s from last week if I can penetrate the Chinese walls of EMU ! Take a look at our web site where at least the web paper can be downloaded from the publications section and some of the work on cyberspace is reported

http://www.casa.ucl.ac.uk/http://www.cybergeography.org/http://www.casa.ucl.ac.uk/citations/


Network Approaches to ScalingNetwork Approaches to Scaling

Here we take a look at the distribution of Here we take a look at the distribution of indegrees and outdegrees formed by links indegrees and outdegrees formed by links relating to web pages - a web page is pretty relating to web pages - a web page is pretty unambiguous. There is a lot of work on this unambiguous. There is a lot of work on this produced during the last three years, notably produced during the last three years, notably the Xerox Parc group & the Notre Dame groupthe Xerox Parc group & the Notre Dame group

let me start with some notions of about graphslet me start with some notions of about graphs


On the left a random On the left a random graph, whose distribution graph, whose distribution of the numbers/density of of the numbers/density of links at each node is near links at each node is near normal - this has a normal - this has a characteristic length - the characteristic length - the averageaverage

On the left, what is much On the left, what is much more typical - a graph more typical - a graph which is scaling - one which is scaling - one whose distribution is rank whose distribution is rank size, following a power lawsize, following a power law

P(k) ~ kP(k) ~ k - 2.5 - 2.5


Not only does the topology of web Not only does the topology of web pages follow power lawspages follow power laws

so does the physical hardware - the so does the physical hardware - the routers and wiresrouters and wires

This and the last diagram are taken This and the last diagram are taken from the article by Barabasi called from the article by Barabasi called “The Physics of the Web” printed in “The Physics of the Web” printed in the July 2001 issue of the July 2001 issue of Physics Physics WorldWorld


Some statistics from Steve’s work - which imply scale free networks

Lots and lots of issues here - we need models of how Lots and lots of issues here - we need models of how networks grow and form, how does the small world effect networks grow and form, how does the small world effect mesh into scale free networks ? We need to map mesh into scale free networks ? We need to map cyberspace onto real space and back, and this is no more cyberspace onto real space and back, and this is no more than mapping social space onto real space and back - its than mapping social space onto real space and back - its not new.not new.

I will finishI will finish


Links as indegrees and outdegrees compared to the Total Links


Number of Web Number of Web PagesPagesand Total Links - and Total Links - indegrees and indegrees and outdegreesoutdegrees

These are taken These are taken from relevant from relevant searches of searches of AltaVista for 180 AltaVista for 180 domains in 1999domains in 1999


Network Approaches to ScalingNetwork Approaches to Scaling

Here we take a look at the distribution of Here we take a look at the distribution of indegrees and outdegrees formed by links indegrees and outdegrees formed by links relating to web pages - a web page is pretty relating to web pages - a web page is pretty unambiguous. There is a lot of work on this unambiguous. There is a lot of work on this produced during the last three years, notably produced during the last three years, notably the Xerox Parc group & the Notre Dame groupthe Xerox Parc group & the Notre Dame group

let me start with some notions of about graphslet me start with some notions of about graphs


As an introductory example, I will repeat what I say in the editorial I handed out on ‘small worlds’. You can read this laterThe 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 nodes2. There was no real way in which one could handle processes taking place on graphs3. There was not much thinking about how real graphs structures evolved - through time


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


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


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


The world wide web is a small world as are most systems that don’t 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 hasn’t 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


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 averageOn 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


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


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 growThis is the cluster of routers, and hubs and machines in UCL


Some more fancy visualizations of these networks


Some statistics from Steve’s 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


Some references - Martin Dodge and Rob Kitchin’s new book

Steve Coast’s web sitewww.fractalus.com/steve/

Our web sitewww.casa.ucl.ac.uk

and drill down to get to Martin’s www.cybergeography.org


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19011901

19911991

Log of RankLog of Rank

1991 Population based 1991 Population based on 1901 Rankson 1901 Ranks

Log

of

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Sha

res

Log

of

Pop

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Sha

res

Here is an example of the shift in size and ranks over Here is an example of the shift in size and ranks over the last 100 years in GBthe last 100 years in GB


Figure 1: Rank-Size Distributions of Highly Cited Scientists

red institution, black place, grey by country

straightline fits by institution (red)

)2.80( )5.90(

0.938 ,429,/ln816.0555.0)(ln 2

RMMrxxP

by place/city (black)

)8.76( )3.94(

0.962 ,232,/ln049.1768.0)(ln 2

RMMrxxP

by country (grey)

)6.21( )232(

0.949 ,27,/ln997.1583.1)(ln 2

.

RMMrxxP

1 the zipf seminars at emu-um sunday, april 06, 2014 after zipf: from city size distributions to...

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