Regional Economics George Horváth Department of Environmental Economics

Rank and Size Rules (Rank-Size Models) We rank the settlements of the country by population. Then, considering the size of the capital city, the size of a city of rank N will be roughly 1 / N th the size of the capital. When we plot this relationship with actual city data, the first few instances will be out-of-line. Somewhere between rank and , the points start to follow the parabola. 1

2 Ranking of European cities (by population)

3 Ranking of European cities (by GDP)

4 City ranks within some European states

Probability and physical analogies in spatial distribution Lots of things follow a normal distribution in nature Height, expected life span, intelligence, etc. Normal distribution is very frequent in physics and life sciences 5

Counting cells This method investigates the attractive and repulsive forces that affect gravitation to a city. Let’s say we draw a grid 10 cells by 10 cells on the pavement… …and we wait for the rain to come. After the rain, we count how many cells contain a certain number of drops. 6

Normal distribution in the cells 7

Perfect concentration and deconcentration 8

Distribution among cells The number of raindrops in the cells will be normally distributed (roughly) Such distributions can be found in how the population is dispersed in space. Size and population of settlements will define their mutual gravitation. If gravitational attractions are dominant, we will get a “perfect concentration” of the raindrops. If they are average, the raindrops will be normally distributed. If raindrops are uniformly distributed (“perfect deconcentration”), gravitation is practically zero. 9

Networks To keep together a network of interdependent settlements, we will need to investigate the infrastructural networks that keep them together. There is a mutual dependence: spatial networks will depend on the infrastructure for keeping together, but infrastructure development also depends on the gravitation between the settlements. In some cases, the creation of new infrastructure shifted the rank order of settlements. When railway routes were first designed, they may have connected to less important settlements, which later assumed a more important role. 10

Lösch’s Networks Each settlement has six neighbours. As they are roughly the same size, they will form a honeycomb pattern. If we connect each settlement with its neighbours by road, six roads will extend from each settlement. The network that is thus formed is very dense, therefore very costly and impractical. Lösch has also determined how many settlements there should be for a given length of infrastructure. 11

Lösch’s Networks 12

d = the average distance between settlements n = the number of settlements A = the area of the country/region Q = the total length of the network (theoretical) 13 Lösch’s road network connecting settlements

d = 3 km (from equation) n = A = km 2 Q = ? 14 Working it out for Hungary

Christaller’s road network, graphically 15

What does this tell us? If we were to connect every settlement with all their neighbours, the road network would have to be kilometres long. If we only connected all inferior-level settlements with their immediate superior-level settlement, the road network would have to be kilometres long. In practice, the actual length of the Hungarian main road network is kilometres. The length of all roads (local, rural, main roads and motorways) is kilometres 16

Analysing spatial networks using graphs What are graphs? Graphs show pairs of objects connected by links. Graphs have vertices (sing.: vertex), which are connected by edges. There are basically two kinds of graphs: topological and non-topological. Non-topological graphs show information about how vertices and edges are situated in space (e.g. directions and distances). Topological graphs do not care about directions and distances, they just tell us the order of objects and how they are connected to one another. 17

What kind of graph is this? 18

Properties of Graphs Connectivity of a graph: the ratio of edges to vertices (B = e/v) Diameter: the number of edges on the shortest route from one end of the graph to the other Shape of the graph: the number of edges divided by the diameter of the graph (∏ = e/D) Number of cut points: the number of such vertices whose removal would cut the graph into two separate graphs. 19

20 Number of vertices:36 Number of edges:36 Shape of the graph:36 / 9 = 4 Number of cut points:16 Connectivity:1 Mainline railway network of Hungary

21 Number of vertices:24 Number of edges:29 Shape of the graph:24 / 8 = 3 Number of cut points:7 Connectivity:29 / 24 = 1,21 Mainline railway network of Romania

Non-topological graph analysis Maximal connectivity Maximal fixed (establishment) costs Lower operational costs (fuel, time, etc.) 22 A B C D E

Non-topological graph analysis Tree graph Minimal connectivity Minimal fixed (establishment) costs Higher operational costs (fuel, time, etc.) 23 A B C D E N1N1 N2N2

Tree graphs Tree graphs are such graphs that do not include double connections or redundancies, and the length of the road network is minimal. 1.To minimise the length of the road network, we just need to define some auxiliary nodes N i 2.The number of additional nodes may not be more than two less than the number of total vertices (∑v-2) 3.The optimal location for auxiliary nodes is where the three connecting roads arrive at 120° 24

The Law of Least Exerted Effort Snellius described this principle in 1621 Fermat generalised these findings in 1650 Light „bends” when it passes between materials of different densities. It will choose the path with the least effort (energy) in needs to exert. This law has been generalised for economics by George Zipf. According to Zipf, it can be statistically shown that humans want to reach their goals with the least possible effort they need to exert. 25

The Snellius-Descartes Law 26 α β n1n1 n2n2

The Snellius-Descartes Law in practice 27 sea mainland port destination

Fractals Many things in nature are made up of similar shapes, but different sizes. When we need to determine the length of a coastline, we get very different results depending on what we take into consideration and what we omit. Trees, plants, crystals and snowflakes all resemble fractals Fractals are also used to explain some aspects of settlement growth 28

Estimating the length of a coastline Length L is estimated by the number of S compass radii, with various radii, on the same coastline. 29

Estimations from original article by Peitgen, Compass radius (km)Estimated length of coastline (km) Estimating the length of Britain’s coastline

What fractals are and what they look like Fractals essentially consist of the same shapes repeated over and over. The shapes are added on to the sides of the original shape, repetitively. Eventually, these fill up the space 31

32 Computer simulation of Cardiff’s growth