Wednesday, November 19, 2014 After Zipf: From City Size Distributions to Simulations

1. Wednesday, November 19, 2014 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.ukyxie@emich.edu http://www.casa.ucl.ac.uk/http://www.ceita.emich.edu/

2. What we will do in this talk • 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.

3. 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.

4. 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

5. 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

6. Here are some classic examples from last week First from Zipf (1949)

7. Now from Tom (2003) – top 135 cities

8. 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

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

10. 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)

11. Log Population Log Rank

12. The King or Primate City Effect Scaling only over restricted orders of magnitude A different regime in the thin tail Log Population Log Rank

14. 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

15. 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 ?

16. 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

17. Log Population Log Rank

19. 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

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

22. red institution, black place, grey by country straightline fits by institution (red) by place/city (black) by country (grey) Rank-Size Distributions of Highly Cited Scientists by country (grey)

23. The Highly Cited By Place

24. 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

25. 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

26. 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

27. t=1000 t=900 Log of Population Shares t=1000 Population based on t=900 Ranks Log of Rank

28. 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

29. 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

30. 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

31. 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.

32. This is population of countries

33. And the average rank distances

34. This is the telecoms data

35. And the average rank distances

36. 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

37. 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 1998 at CASA.

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

39. Number of Web Pages and Total Links - indegrees and outdegrees These are taken from relevant searches of AltaVista for 180 domains in 1999

40. Links as indegrees and outdegrees compared to the Total Links

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

42. 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

43. Two regimes for the indegrees and outdegrees

44. 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

46. 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

47. 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

48. 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

49. 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