Monday, October 20, 2014 After Zipf: From City Size Distributions to Simulations

1 The Zipf Seminars at EMU-UM

Thursday, April 20, 2023

After Zipf: From City Size Distributions to Simulations

Michael Batty & Yichun XieUCL & EMU

[email protected], [email protected] http://www.casa.ucl.ac.uk/


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


Scaling in Urban Systems: A Brief Scaling in Urban Systems: A Brief HistoryHistory

1. Gravitational analogies - Ravenstein (1888) 1. Gravitational analogies - Ravenstein (1888) for migration, Carey (1850), French Physiocratsfor migration, Carey (1850), French Physiocrats2. The Emergence of Social Physics from the 2. The Emergence of Social Physics from the 1940s on - Regional Science in the 1950s1940s on - Regional Science in the 1950s3. The simplest scaling - Zipf’s Law - the rank 3. The simplest scaling - Zipf’s Law - the rank size rulesize rule4. Transportation Modeling4. Transportation Modeling5. Scaling in terms of fractals from the 1980s on5. Scaling in terms of fractals from the 1980s on


Zipf’s Law and the Distribution of Zipf’s Law and the Distribution of Populations in Systems of CitiesPopulations in Systems of Cities

1. Zipf’s Law - the rank size rule1. Zipf’s Law - the rank size rule2. The confusion over its formulation as a 2. The confusion over its formulation as a probability distributionprobability distribution3. The original emphasis on description3. The original emphasis on description4. Most examples took the largest events known 4. Most examples took the largest events known such as the top 100 world cities as defined from such as the top 100 world cities as defined from yearbooks etcyearbooks etc5. There is hardly any attention to what these 5. There is hardly any attention to what these events really meanevents really mean


A Typical Distribution for World A Typical Distribution for World Population in 1994Population in 1994

Here we have 150 countries and we Here we have 150 countries and we will show how difficult it can be will show how difficult it can be from this kind of data to from this kind of data to demonstrate scaling demonstrate scaling


0123456789

10

0 0.5 1 1.5 2 2.5

Log Population against Log RankLog Population against Log Rank

Log

Pop

ulat

ion

Log Rank

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

1012

0 1 2 3log rank

log

po

pu

lati

on

Residuals against Rank Orders

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2 2.5

log rank

Re

sid

ua

ls99.1157.10 rPr

36.015.4 rPr


ProblemsProblems

1. Scaling - many indeed most distributions are not 1. Scaling - many indeed most distributions are not power functionspower functions2. The events are not independent - in medieval 2. The events are not independent - in medieval times they may have been but for the last 200 years, times they may have been but for the last 200 years, cities have grown into each other, nations have cities have grown into each other, nations have become entirely urbanized, and now there are global become entirely urbanized, and now there are global cities - the tragedy in NY tells us this - where more cities - the tragedy in NY tells us this - where more than half of those killed were not US citizensthan half of those killed were not US citizens3. Should we expect scaling ? We know that cities 3. Should we expect scaling ? We know that cities depend on history as well as economic growthdepend on history as well as economic growth


Problems (continued)Problems (continued)

4. Why should we expect 4. Why should we expect nono characteristic length characteristic length scale - when the world is finite ? We should avoid scale - when the world is finite ? We should avoid the sin of ‘Asymptopia’.the sin of ‘Asymptopia’.5. As scaling is often said to be the signature of self-5. As scaling is often said to be the signature of self-organization, why should we expect disparate and organization, why should we expect disparate and distant places to self-organize ?distant places to self-organize ?6. The primate city effect is very dominant in 6. The primate city effect is very dominant in historically old countrieshistorically old countries7. BUT should we expect these differences to 7. BUT should we expect these differences to disappear as the world become global ?disappear as the world become global ?


Let’s first look at arbitrary events - Let’s first look at arbitrary events - An Example for the UK based on An Example for the UK based on Administrative Units, not on trying Administrative Units, not on trying to define cities as separate fields to define cities as separate fields

These are 458 admin units, somewhat less than These are 458 admin units, somewhat less than full cities in many cases and some containing full cities in many cases and some containing towns in county aggregates - we have data from towns in county aggregates - we have data from 1901 to 1991 so we can also look at the 1901 to 1991 so we can also look at the dynamics of change - traditional rank size dynamics of change - traditional rank size theory says very little about dynamicstheory says very little about dynamics


3

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1991

1901

Log of Rank

Log

of

Pop

ulat

ion


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 This is what we get when we fit the rank size relation Prelation Prr=P=P11 r r - - to the data. The parameter is to the data. The parameter is

hardly 1 but it is more than 0.36 which was the hardly 1 but it is more than 0.36 which was the value for world population in 1994value for world population in 1994


-4.5

-4

-3.5

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-1

0 0.5 1 1.5 2 2.5 3

19011901

19911991

Log of RankLog of Rank

1991 Population based 1991 Population based on 1901 Rankson 1901 Ranks

Log

of

Pop

ulat

ion

Sha

res

Log

of

Pop

ulat

ion

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


Explaining City Size Distributions Explaining City Size Distributions Using Multiplicative ProcessesUsing Multiplicative Processes

The last 10 years has seen many attempts to The last 10 years has seen many attempts to explain scaling distributions such as these using explain scaling distributions such as these using various simple but stochastic processes.various simple but stochastic processes.

In essence, the easiest which gives rise to In essence, the easiest which gives rise to distributions such as these is a model of distributions such as these is a model of proportionate effect or growth which leads to proportionate effect or growth which leads to the lognormal distributionthe lognormal distribution


itit

itP

P

t

iiit PP0

0]log[]log[

t

iiit PP0

0

ititit PP 1

The key idea is that the change in size of the The key idea is that the change in size of the object in question is proportional to the size of object in question is proportional to the size of the object and randomly chosen, that isthe object and randomly chosen, that is

This leads to the log of differences across time This leads to the log of differences across time being a function of the sum of random changesbeing a function of the sum of random changes

This gives the model of proportionate effectThis gives the model of proportionate effect

oror

16 The Zipf Seminars at EMU-UM-4.5

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-3.5

-3

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-1

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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

Pop

ulat

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Sha

res

Here’s a simulation which shows that the rank-size rule is Here’s a simulation which shows that the rank-size rule is generated this way with much the same properties as the observed generated this way with much the same properties as the observed data for UKdata for UK


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

This is a good model to show the persistence of This is a good model to show the persistence of settlements, it is consistent with what we know settlements, it is consistent with what we know about urban morphology in terms of fractal about urban morphology in terms of fractal laws, but it is not spatial.laws, but it is not spatial.However there are other processes which we However there are other processes which we should note which have been explored. I will list should note which have been explored. I will list these as follows - and please note that my these as follows - and please note that my survey is by no means exhaustive.survey is by no means exhaustive.


Other Stochastic Processes which Other Stochastic Processes which have been used to explain scalinghave been used to explain scaling

1. The Simon model - birth processes are 1. The Simon model - birth processes are introduced which is not something which is introduced which is not something which is done in generating the lognormaldone in generating the lognormal2. Multiplicative random growth with 2. Multiplicative random growth with constraints on the lowest size - size is not constraints on the lowest size - size is not allowed to become too small otherwise the event allowed to become too small otherwise the event is removedis removed3. Work on growth rates consistent with scaling 3. Work on growth rates consistent with scaling relations involving Levy distributions relations involving Levy distributions


The Second Example - The Second Example - distributions where the events are distributions where the events are unambiguous or less ambiguous - unambiguous or less ambiguous - the distribution of links on WWWthe distribution of links on WWW

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


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


Links as indegrees Links as indegrees and outdegrees and outdegrees compared to the compared to the Total LinksTotal Links


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


These are based on the general formulaThese are based on the general formulawhere q is the parameter of the distributionwhere q is the parameter of the distributionAs a general conclusion, it does not look as though As a general conclusion, it does not look as though the event size issue has much to do with the scaling the event size issue has much to do with the scaling or lack of it.We urgently need some work on spatial or lack of it.We urgently need some work on spatial systems with fixed event areas, thus shifting the systems with fixed event areas, thus shifting the focus to densities not distributionsfocus 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

qr KrP


The internet is a great example - the densest nodes are The internet is a great example - the densest nodes are in the places where all the information is concentrated - in the places where all the information is concentrated - in the world cities - in short, distances and locations in in the world cities - in short, distances and locations in cyberspace mirror real space - biggest hubs are in cyberspace mirror real space - biggest hubs are in Manhattan, City of London and so on e.g.Manhattan, City of London and so on e.g.


Two regimes for Two regimes for the indegrees the indegrees and outdegreesand outdegrees

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


Last Comments and Future WorkLast Comments and Future Work

Scaling can be shown to be consistent with Scaling can be shown to be consistent with more micro-based , hence richer less more micro-based , hence richer less parsimonious modelsparsimonious models

1. Diffusion and growth models 1. Diffusion and growth models 2. Agent-based competition models2. Agent-based competition models3. Treating the system as a growing network - 3. Treating the system as a growing network - this latter model is worth finishing with as it is this latter model is worth finishing with as it is particularly relevant to the WWW and is particularly relevant to the WWW and is probably close to interaction models of cities as probably close to interaction models of cities as in transportationin transportation


Network Approaches to ScalingNetwork Approaches to Scaling


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


Here is some work that Steve Coast in our group at CASA is Here is some work that Steve Coast in our group at CASA is doing on detecting and measuring the distribution of the doing on detecting and measuring the distribution of the hardware of the web and visualizing it - all this is prior to 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 measuring its properties - i.e. is it scaling, is it a small world and so onand so on

Challenge is to map real Challenge is to map real space onto cyberspace space onto cyberspace and that so far has not and that so far has not really been attempted in really been attempted in these new ideas about these new ideas about how network systems how network systems growgrow

This is the cluster of This is the cluster of routers, and hubs and routers, and hubs and machines in UCLmachines in UCL


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


Some of the most interesting work is being done in Some of the most interesting work is being done in virtual space - in cyberspace not in real space. Here is virtual space - in cyberspace not in real space. Here is an example of such a networkan example of such a network


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

Steve Coast’s web siteSteve Coast’s web site

www.fractalus.com/steve/www.fractalus.com/steve/

Our web siteOur web site

www.casa.ucl.ac.ukwww.casa.ucl.ac.uk

and drill down to get to and drill down to get to Martin’sMartin’s

www.cybergeography.orgwww.cybergeography.org



USA-3149 citiesR-sq = 0.992 b = -0.81

Mexico-36 citiesR-sq = 0.927 b = -1.27

World-216 countriesR-sq = 0.708 b = -2.26

UK-459 areasR-sq = 0.760 b = -0.58



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


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


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ln [r/M]

ln [

P(x

)/<

x>]

Figure 1: Rank-Size Distributions of Highly Cited Scientists


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



MrxxP /~)(

where )(xP is the number of cited scientists at rank r

x is the mean number of cited scientists, and M is the number of institutions, places, or countries for each of the three respective aggregations6





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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


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

Documents

Monday, October 20, 2014 After Zipf: From City Size Distributions to Simulations