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1 The Zipf Seminars at EMU-UM Thursday, February 27, 2003 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 Xie UCL EMU m.batty@ucl.ac.ukm.batty@ucl.ac.uk yxie@emich.eduyxie@emich.edu http://www.casa.ucl.ac.uk/http://www.casa.ucl.ac.uk/ http://www.ceita.emich.edu/http://www.ceita.emich.edu/

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2 The Zipf Seminars at EMU-UM What we will do in this talk 1. 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.

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3 The Zipf Seminars at EMU-UM 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. Sum up and leave for discussion future directions 8. Take soundings about a further seminar

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4 The Zipf Seminars at EMU-UM 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

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5 The Zipf Seminars at EMU-UM 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

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6 The Zipf Seminars at EMU-UM Here are some classic examples from last week First from Zipf (1949)

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7 The Zipf Seminars at EMU-UM Now from Tom (2003) – top 135 cities

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8 The Zipf Seminars at EMU-UM 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

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9 The Zipf Seminars at EMU-UM Let’s look at some cities, countries, & spatial partitions USA-3149 cities R-sq = 0.992 = -0.81 Mexico-36 cities R-sq = 0.927 = -1.27 World-216 countries R-sq = 0.708 = -2.26 UK-459 areas R-sq = 0.760 = -0.58

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10 The Zipf Seminars at EMU-UM 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)

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11 The Zipf Seminars at EMU-UM Log Population Log Rank

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12 The Zipf Seminars at EMU-UM Log Population Log Rank The King or Primate City Effect Scaling only over restricted orders of magnitude A different regime in the thin tail

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13 The Zipf Seminars at EMU-UM

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14 The Zipf Seminars at EMU-UM 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

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15 The Zipf Seminars at EMU-UM 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 ?

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16 The Zipf Seminars at EMU-UM 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

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17 The Zipf Seminars at EMU-UM Log Population Log Rank

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18 The Zipf Seminars at EMU-UM

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19 The Zipf Seminars at EMU-UM This is what we get when we fit the rank size relation P r =P 1 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

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20 The Zipf Seminars at EMU-UM A Digression – Many other systems show such rank size – here we will look at geography of scientific citation –the Highly Cited

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21 The Zipf Seminars at EMU-UM

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22 The Zipf Seminars at EMU-UM Rank-Size Distributions of Highly Cited Scientists red institution, black place, grey by country straightline fits by institution (red) by place/city (black) by country (grey)

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23 The Zipf Seminars at EMU-UM The Highly Cited By Place

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24 The Zipf Seminars at EMU-UM 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

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25 The Zipf Seminars at EMU-UM 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

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26 The Zipf Seminars at EMU-UM 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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27 The Zipf Seminars at EMU-UM t=1000 t=900 Log of Rank t=1000 Population based on t=900 Ranks Log of Population Shares

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28 The Zipf Seminars at EMU-UM 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

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29 The Zipf Seminars at EMU-UM 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

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30 The Zipf Seminars at EMU-UM 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

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31 The Zipf Seminars at EMU-UM 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.

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32 The Zipf Seminars at EMU-UM This is population of countries

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33 The Zipf Seminars at EMU-UM And the average rank distances

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34 The Zipf Seminars at EMU-UM This is the telecoms data

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35 The Zipf Seminars at EMU-UM And the average rank distances

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36 The Zipf Seminars at EMU-UM 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

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37 The Zipf Seminars at EMU-UM 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.

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38 The Zipf Seminars at EMU-UM This is based on some network data produced by Martin Dodge and Naru Shiode in CASA from their web crawlers

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39 The Zipf Seminars at EMU-UM Number of Web Pages and 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

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40 The Zipf Seminars at EMU-UM Number of Web Pages,Total Links, GDP and Total World Populations

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41 The Zipf Seminars at EMU-UM 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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42 The Zipf Seminars at EMU-UM Two regimes for the in- degrees and out-degrees

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43 The Zipf Seminars at EMU-UM 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

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44 The Zipf Seminars at EMU-UM Resources References Papers Web 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/

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