REGIONAL STATISTICS, 2013, VOL. 3: 30–40 DOI: 10.15196/RS03102
GABRIELL MÁTÉa – ANDRÁS KOVÁCSb – ZOLTÁN NÉDAc
Hierarchical Settlement Networks
Abstract
A network representation is introduced for visualizing hierarchical region structures on
various spatial scales. The method is based on a spring-block model approach borrowed
from physics and it was previously used with success to detect regions in any
geographical space. We illustrate here our network construction method for the case of
Transylvania, USA and Hungary.
Keywords: settlement networks, spring-block model, USA, Transylvania, Hungary.
Introduction
It is commonly believed that the core of any complex system is a network (Barabási
2005). The underlying network structure carries useful information for understanding the
evolution and measurable characteristics of large systems. Networks have proven that
they are useful for studying biological, social and economic systems (Barabási 2012).
A network approach might be helpful also in visualizing and understanding the complex
hierarchical inter-relationship between settlements in a given geographic region. The
method for such an approach is straightforward, assuming that one is able to detect
hierarchical region-like structures and region centers on different spatial scales. The
hierarchical space-division at various scale, could be synthesized in a compact and visually
interpretable manner by drawing up such a tree-like network topology. Here, based on our
previously elaborated region detection algorithm (Máté–Néda–Benedek 2011), we present
a method for the construction of such a hierarchical connection network.
This method can be helpful in the spatial analysis of any kind of territorial data. The
complexity of socio-economic processes with spatial dimension implies that natural
space delimitation cannot be possible (Dusek 2004), the determination of territorial units
(regions) depends on the examined phenomenon and its geographical characteristics
(Haggett 2001). Researching the spatial flow of goods, services and information one can
border regions with different spatial extent in different hierarchical structures. Thus it has
got high importance in bordering to clarify the regional structures on micro, mezzo and
macro regional level, based on different socio-economic data.
a Institute for Theoretical Physics, Heidelberg University, Philosophenweg 16 D-69120 Heidelberg, Germany E-mail:
G.Mate@tphys.Uni-Heidelberg.DE
b Edutus College, Department of International Business, 1114 Budapest, Villányi út 11–13., Hungary, E-mail:
kovacs.andras @edutus.hu
c Department of Physics, Babes-Bolyai University, str. Kogalniceanu 1, R0-400084 Cluj-Napoca, , Romania, E-mail:
zneda@phys.ubbcluj.ro
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GABRIELL MÁTÉ – ANDRÁS KOVÁCS – ZOLTÁN NÉDA
Our method enables us to border regions on different spatial levels by using different
kinds of data. This flexibility may contribute to a more efficient regional research in
bordering and in the horizontal and vertical division of socio-economic and geographical
space (Nemes Nagy 2009).
This paper can be divided into three main parts. First we will introduce the
methodological background and its geographical relevance, in the second part we
introduce three examples where spatial data are analyzed and visualized (examples are
from the USA, Transylvania and Hungary). In the last part we would like to point out the
socio-economic relevancies of this method in spatial analysis and regional science.
Method
Preliminaries
Spring-block models have been used widely to model various phenomena related to
relaxation, avalanche-like dynamics or self-organized criticality (Járai-Szabó–Néda 2012,
Kovács–Néda 2007). The model considers a system of blocks that are interconnected by
springs in a lattice like topology. The blocks can slide on a surface and apart of the
elastic forces experienced from their neighbors a friction force is damping their free
movement on the surface. The friction forces are usually velocity dependent. In the
simplest version there is a dynamic friction force that is independent of the sliding
block’s velocity and a static friction force (acting when the block is in rest relative to the
surface), which has a larger value than the dynamic one. Originally, the model was
considered in one-dimension by Burridge and Knopoff in order to model the power-law
like distribution of the earth-quakes magnitude (Burridge–Knopoff 1967). The model
was then extended to two-dimension by Olami, Feder and Christensen (1992). One of the
most spectacular application of this model is it’s success in modeling fracture and
fragmentation patterns resulting from quasi-static drying of granular materials in contact
with a frictional substrate (Leung–Néda 2010). The success of reproducing this
relaxation dynamics in drying granular materials, motivated the use of the spring-block
model for hierarchical region detection. Regions can be imagined as resulting from a
fragmentation dynamics in a geographic space. Once the interconnection topology of the
settlements is defined and the strength of the interconnecting forces are revealed, a
simple relaxation dynamics will lead us to a hierarchical grouping. This hierarchical
region-detection method was used in (Máté–Néda–Benedek 2011). Here, for the sake of
an easier understanding of the network construction method, we now briefly review the
main elements in our spring-block approach for region detection. For a more detailed
description we suggest the reader to consult (Máté–Néda–Benedek 2011).
The spring-block model for region detection
Settlements are regarded as blocks with masses m i proportional with their population
(Wi ). The blocks are displaced on a two-dimensional plane, so that their relative
coordinates are determined by the original GPS coordintes of the settlements. On each
block “i” a friction force is acting on behalf of the surface. For the sake of simplicity we
HIERARCHICAL SETTLEMENT NETWORKS
3
consider here an over-damped motion of the blocks. The value of the force acting on
block “i” is defined in analogy with classical mechanics:
(1)
Fi f = m i gm
Here g is a kind of gravitational constant, taken as g=1 (defining the units for forces),
and m is the static friction coefficient.
The blocks are interconnected by springs. In order to determine the interconnection
topology we use a Delanuay triangulation (Okabe–Boots–Sugihara–Chiu 2000). The
Delanuay triangulation performed on the point-like coordinates of the settlements will
yield the first-order neighbors (Figure 1a). Springs are placed between these first-order
neighbors, and as a result of this, a two-dimensional spring-block lattice will form
(Figure 1b). Relaxation is performed on this spring-block lattice.
Construction of the spring-block lattice
Figure 1
a) Delauney traingulation for detecting the neighbors of the settlements. Black dots represent the position of the
settlements, red polygons are the Voronoi cells around them and the black lines illustrate the Delauney triangulation. We
connect by springs the points connected by these edges.
b) A two-dimensional spring-block system. Red blocks represent settlements of different sizes and the springs connect the
neighboring blocks detected through the Delanuey triangulation.
Springs are modeling the socio-economic connections between the neighboring
settlements. Settlements that are closer to each other should attract each other in a
stronger manner and settlements that have a correlated tendency of evolution for some of
their relevant socio-economic indicators should also attract each other in a strong
manner. Taking into account these obvious facts, the spring-force between settlement “i”
and “j” is quantified as:
r
Fijs = -
kij
d ij
r
e ij
(2)
In the above equation kij denotes the connectivity strength between settlements “i” and
r
“j”, dij is the distance between them and e ij is a unit vector pointing from on settlement
to the other. In contrast with many earlier spring-blocks models, here the spring-forces
are not linear. The tension in the spring is increasing while the distance between the
settlements gets smaller. Another difference relative to the classical spring-block models
is that the here the springs cannot break, and there are able to resist any tension.
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GABRIELL MÁTÉ – ANDRÁS KOVÁCS – ZOLTÁN NÉDA
The crucial point in defining our spring-block model is to quantify the kij connectivity
strength. In order to achieve this, we consider a relevant socio-economic parameter, qi (t),
which is available on settlement level for a relatively long time-period, t. This can be for
example the population of the settlement, the GDP per inhabitant or yearly average tax
data per inhabitant. Furthermore, we suppose that we do have access to this data with a
uniform time-sampling interval, taken here as unity. We can define in such manner for
each settlement the relative change of this quantity for a unit time:
r i( t ) =
qi ( t + 1 ) - qi ( t )
qi ( t )
(3)
The main hypothesis for the used spring-block approach is that we assume that for the
relevantly connected settlements the time-variation of the ri relative changes has to be
strongly correlated. This means that we can quantify the kij connection strengths as the
time-like correlations between the ri quantities. Using Pearson correlations, we assume
thus:
k ij =
Here,
t
ri (t)r j (t) - ri (t)
t
t
r j (t)
t
s[ri (t)]s[r j (t)]
(4)
stands for an average taken over time, and s is the standard deviation:
s[ri (t)] =
ri2 (t) t - ri (t)
2
t
(5)
Please note the absolute value on the right side of equation (4). This is introduced due
to the fact kij must be positive, and both a high correlation and anti-correlation indicates
a strong connection between settlements “i” and “j”.
The Dynamics
Once the main elements of the model, and their relation to the relevant socio-economic
and geographic data are clarified, we sketch the dynamics that leads to the hierarchical
territory division.
1. Initially the settlements are placed on a plane according to their GPS coordinates.
The mi mass of each block is fixed according to the population of the
corresponding settlement (mi =Wi ). Neighbors are defined with the Delanuay
triangulation and springs are inserted between them. The dij distances between the
neighboring settlements and their k ij connectivity strength is calculated and fixed
for the whole duration of the simulation.
2. The resultant elastic force acting on each block is computed: Fr s = ĺ Fr s . In the
i
j( i )
ij
above equation we denoted by j(i) the neighbours of block i.
3. The friction coefficient is fixed to a large value, so
r sthat nof resultant force acting
on the blocks exceeds the friction force: "i ® Fi < Fi
4. We decrease the value of m , until the first block is allowed to slide, i.e. the
magnitude of the resultant
force is bigger than the static friction force
r s tension
f
acting on the block ( Fi > Fi ).
HIERARCHICAL SETTLEMENT NETWORKS
5
5. The block “i” inr one simulation step will be moved by a distance h in the
direction of the Fis force. The h value is chosen to be much smaller than the
smallest initial distance between any two settlements. As a result of this slip, if
any two blocks come closer to each other than a predefined dminvalue, these blocks
are united in one cluster. This cluster will be a new block in the system, inheriting
the mass and links of both blocks and in the next simulation steps will represent
all the blocks that are glued together in it as result of this coalescence process.
6. After each move the distances among the moving blocks and it’s neighbors are
recalculated.
7. In the next simulation step the moving block and it’s neighboring blocks are
checked whether they satisfy the slipping condition. If yes they are moved by a
distance h in the direction of the resultant
This simulation step is repeated
r s forces.
f
until no block can move. i.e. "i ® Fi < Fi
8. The movementrof blocks will continue in the consecutive simulation steps, until
s
f
the condition Fi > Fi is satisfied for the new position.
9. When no more blocks can move, the m friction coefficient is lowered until the
first block satisfies the slipping condition again. The simulation proceeds from
here by repeating the steps from 5.
10. The dynamics ends when all blocks have collapsed in one.
The above defined model and dynamics has only two freely adjustable parameters.
One is the h distance of the slip in one simulation step and the other is the dmin distance
for the collapse of two blocks. If we take them small enough, (as described earlier), their
value will not influence the hierarchical collapsing dynamics and the method will
become free of adjustable parameters.
Hierarchical territory division
The above algorithm will hierarchically group the elements respecting their connection
strength with neighbors and their initial spatiality. Smaller blocks will slide towards large
ones, since the friction forces acting on the latter ones are larger. These large settlements
will than become centers attracting other blocks. It is reasonable to assume that the above
presented spring-block relaxation dynamics will reveal regions at different scales, as the
system hierarchically collapse in larger and larger clusters. Keeping track of all these
collapsing events, and by projecting backward in time the collapsing order, one can
define a hierarchical clustering of the settlements for the investigated geographical
region.
The above presented method and model was implemented in an interactive JAVA
application (Máté–Néda–Benedek 2009), which allows the user to follow visually the
whole clustering process up to the end when all blocks concur in one. The program
memorizes all the intermediary situations and can interactively visualize the settlement
partitions corresponding to them. One can freely play with this model by visiting the
website dedicated to it (Máté–Néda–Benedek 2009).
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GABRIELL MÁTÉ – ANDRÁS KOVÁCS – ZOLTÁN NÉDA
The hierarchical settlement connection network
The collapsing dynamics obtained in the spring-block model can be used to draw a
hierarchical network structure for the settlements. In order to construct this network our
hypothesis is that each cluster is dominated by the largest settlement inside it. This means
that when two clusters collapse we assign a link between the dominating settlements. In
order to visualize different levels of hierarchy and to obtain a visually interpretable
network structure we draw lines with different colors or gray-shades according to the
collapsing stage. One possibility is to use the whole color palette from blue to red
according to the simulation time (step) of the time moment for the collapse. Blue lines
will represent the initial stage of the collapse, and dark red lines the latest stages when
only few clusters are present in the dynamics. In such manner interpreting the resulted
weighted graphs has to be done by taking into account the color code illustrated on the
attached legend. Another possibility is to use only warm colors, and indicate the collapse
stage with a color shifting from red to yellow. This convention generates more userfriendly networks, but it offers less resolution for illustrating different stages of the
collapse. It is also possible to use gray-shades instead of colors. In such case the darker
levels will correspond to earlier collapse time and lighter lines will indicate connections
made on bigger clusters level. We will illustrate in our examples all these methods.
Data
In order to obtain the connectivity strength (kij) between the settlements we need a
uniformly interpreted long-term dataset for some relevant socio-economic parameter
(qi (t)). The most convenient data in such sense is the population Wi of the settlements.
One might assume that the variations in the population data will reflect both cultural and
economic aspects of the settlements, so it seems appropriate for a well-balanced and
relevant socio-economic indicator. It also has the advantage that in general long-term
census data is easily accessible for all geographic territories. We have gathered freely
accessible data on the Internet for Transylvania on settlement level (Varga 2007)
(11 census data between 1850–2002) and for USA on county level (University of
Virginia Library n.y) (last 5 census). Due to the courtesy of Professor József Nemes
Nagy, we obtained excellent data on settlement level for Hungary (11 census data from
1870 to 1970). For the case of Hungary Prof. Nemes Nagy provided us also the average
local income tax paid per inhabitant for 20 consecutive years (data between 1990–2009).
This data offered the possibility to use also different socio-economic indicators for
determining the kij connectivity strength.
In order to initially position the blocks (settlements) on the simulation plane and to
determine their neighbors, we also needed the GPS coordinates of the settlements. This
coordinates are usually available in an electronic format. For Transylvania we got them
from (Astroforum n.y) and in the case of USA the data for the county center is available
at (Comcast n.y.). In the case of Hungary the data was provided by Prof. Nemes Nagy.
Once all these data are electronically available, the spring-block model can be
initialized and it’s hierarchical collapse will reveal the relevant network structure.
HIERARCHICAL SETTLEMENT NETWORKS
7
Results
An interactive version of the program running both on Windows, Linux and MacOsX
operation systems can be downloaded from (Máté–Néda–Benedek 2009). Once the Java
Runtime Environment (JVE) is installed on the computer and correctly running, one can
test the program. In the online version one will be able to select the studied territory
(Transylvania, USA or Hungary), the type of data used for the connectivity (population,
tax) and the visualization method. It is possible to visualize the “regions” obtained on
different hierarchical levels by rewinding the collapsing scenario. An alternative to this is
to plot the hierarchical network structure that is intended to synthesize visually the region
structures obtained at various collapsing stages. Results obtained for Transylvania, USA
and Hungary are given in Figures 2, 3 and 4.
For Transylvania we draw the edges using a color code extended from blue to dark
red (Figure 2). Someone who is familiar with the map of Transylvania will immediately
spot out on this graph as local hubs (gathering the tree-like connection topology in their
neighborhoods) the main region capitals (Cluj-Napoca, Timisoara, Sibiu, Brasov,
Oradea, Targu Mures and Baia Mare ). In order to illustrate the topology of the network
structure on lower lever, we have magnified the Banat region, with the obvious center in
Timisoara. This blow up indicates the hierarchical space division and the lower level
geographical regions in Banat.
For USA we have used a gray-scale code to illustrate the hierarchical network
structure on the county level (Figure 3). Two regions (one in the neighborhood of
Memphis, Tenesse, and one in the neighborhood of Salt Lake City, Utah) are magnified
to show the fine structure.
For Hungary (Figure 4) we use yet another representation, by considering only warm
colors ranging from red to yellow. Here we compare the settlement network structures
obtained using two very different datasets. One network is based on long-term population
census (from 1870 to 1970) and the other one is obtained by using data for 20
consecutive years (from 1990 to 2009) of income tax per inhabitant. Although the
topology is statistically very similar the two magnified regions show that the obtained
network structures are quite different. This illustrates well that the hierarchical networks
structures constructed in such manner are rather sensitive to the used data and the
spanned time-period.
After the visualization of the results the socio-economic implications will be taken
into account. The most important advantage of the applied method lies in the followings:
– The relation between time processes and spatial distribution can be seen in a
single map (see methodology). The spatial analysis and visualization of timebased data (time series) is possible with this method. It is based on long-term
socio-economic data, and the method helps to discover the space-time relations by
different variables. It can serve a basis in bordering of regions, gravity zones of
city regions, etc., because the strengths and directions of linkages can be seen in a
single map. These maps show the effect of geographical distance on regional
structures.
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GABRIELL MÁTÉ – ANDRÁS KOVÁCS – ZOLTÁN NÉDA
Figure 2
The settlement connection network for Transylvania (Romania) obtained by using
long-term population census data.
4000
3500
3000
2500
2000
1500
1000
500
The color of the lines indicates the simulation time when the collapse in the relaxation dynamics occurred. Colder colors
are for earlier stages of the dynamics and warmer colors are for the later stages, as it is indicated in the attached legend. The
Banat region, with the obvious center in Timisoara is magnified for giving a better visualization of the local nature of the
obtained settlement network.
– The level of spatial structure can be determined by the researcher during the
visualization process. During the modeling procedure the researcher has the
opportunity to determine the desired number of formulating regions. It means that
one can draft a multilevel regional structure by stopping the simulation in
different stages. In the Transylvanian example a solid vertical structure of micro,
mezzo and macro regions can be observed (See the different colors and patterns.).
– The method is suitable for comparing of many different kinds of data with socioeconomic relevance. For horizontal and vertical division of socio-economic space
strong spatial links and long-term processes are needed. This method allows us to
compare the spatial linkages and strength in case of different variables. In the
Hungarian example (where we had two variables to work with) we have the
opportunity to compare the similarities and differences in spatial links by using
population and income data. As one can see in Fig 4, there are considerable
differences in the case of the two variables. The maps on population and income
have got partly different topology. The map on the census data shows stronger ties
HIERARCHICAL SETTLEMENT NETWORKS
9
on micro-regional scale than the map on per capita income does. It may come
from the different time dimensions of data, from the diverse behavior of social
and economic processes, etc.
– The original settlement (or regional) structure of the researched area is depicted in
the visualized network. The applied method uses GPS-data in the visualization
process, so not only the strength of the link between the neighboring settlements
can be identified, but the real settlement network as well. In the maps of the USA
and Hungary differences in density and spatial distribution of settlement are easy
to identify. In case of Hungary the small village areas of Somogy, Baranya and
Cserehát can be distinguished from the region of Alföld.
Figure 3
The county connection network revealed for USA by using population census data
3000
2500
2000
1500
1000
500
Salt Lake county
(Salt Lake city, Utah)
Shelby county
(Memphis, Tenesse
A gray-scale code (see the attached legend) indicates the simulation step when the collapse occurred in the relaxation
dynamics. Two rectangular regions are magnified for exemplifying the local connectivity topology.
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GABRIELL MÁTÉ – ANDRÁS KOVÁCS – ZOLTÁN NÉDA
The settlement connection network for Hungary constructed by using
long-term population census data (top) and average income tax data for
20 consecutive years (bottom)
Figure 4
population
income
The color code indicates the simulation step when the collapse occurred in the relaxation dynamics. Lighter colors (in the
direction of yellow) indicate later stages, as it is illustrated on the attached legends. Two regions are magnified. Although the
topology looks statistically similar, the networks obtained with the different input data are very different.
Conclusions
The hierarchical settlement connection networks plotted in the present study offers an
intuitive and visually appealing presentation for the complex inter-relationship between
the settlements. Due to the used “link” definition, the network has a tree-like structure,
and no loops are present. Settlements are connected only through the region centers
revealed at different spatial scales. These centers act as hubs of different sizes. The
shortest distance (minimal number of links) between any two nodes (settlements) will
indicate how strongly these settlements are connected. The hierarchy levels are illustrated
by using different color codes. Results obtained for the case of Transylvania, USA and
Hungary were used to exemplify the method.
Our approach and the applied techniques may help the regional research in several
fields. In regional and spatial planning it can contribute to the planning, realization and
control phases for determining an optimal regional structure (in advance), to identify
regions with special characteristics (backward areas, development axes, city regions, etc.)
or to measure and visualize the changes in settlement networks. In business it can
HIERARCHICAL SETTLEMENT NETWORKS
11
contribute to the spatial analysis of market data like spatial relations in revenues, profit,
costs, etc., to identify spatial distribution and concentration of business partners
(customers, suppliers) or business network elements like market centers, industrial parks,
clusters; or to find hierarchies in business activities (competitors, partners).
We believe that such representations could help in designing socio-economic
structures that would serve a better and more optimal territorial organization and business
activity.
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Acknowledgment
This research was supported in the framework of TÁMOP 4.2.4. A/2-11-1-2012-0001 'National Excellence
Program – Elaborating and operating an inland student and researcher personal support system' project. The
project was subsidized by the European Union and the State of Hungary. We thank Prof. Jozsef Nemes Nagy for
the provided data.