Vol. 133 (2018) ACTA PHYSICA POLONICA A No. 6

Proc. 13th Econophysics Colloquium (EC) and 9th Symposium of Physics in Economy and Social Sciences (FENS), 2017

Application of the Four Colour Theorem to Identify Spatial

Regional Poles and Turnpikes of Economic Growth

A. Jakimowicza,* and D. Rzeczkowskib

aInstitute of Economics, Polish Academy of Sciences, Palace of Culture and Science,

Pl. De�lad 1, PL-00901 Warsaw, PolandbDepartment of Quantitative Methods, Faculty of Economic Sciences, University of Warmia and Mazury in Olsztyn,

M. Oczapowskiego 4, PL-10719 Olsztyn, Poland

This paper presents a method for identifying regional poles and the turnpikes of growth based on the followingfoundations: four colour theorem, Wikinomics business model in the form of platforms for participation, evaluationof the functionality of websites run by public administration municipal o�ces, and dual graph reduction. Theprovince of Warmia and Mazury, which is the subject of the study, is one of the poorest provinces in Poland in termsof economic development. It is therefore natural that the growth of this region requires external enterprise sources.This role can be best performed primarily by websites run by municipal o�ces, which initiate business activity intheir corresponding areas, and consequently, can be regarded as Wikinomics platforms of participation. Using the k-means clustering method, these websites were divided into four separate quality classes. These classes were assignedfour various colours, which were subsequently used for preparing the map of the province. Each municipality wasmarked with a colour corresponding to the quality class of the website run by the state administration unit operatingin a given area. The system of colours resulting from the four colour theorem and a corresponding dual graphserve as a frame of reference with regard to each empirical colour distribution and to another, related, dual graph.Thus, the four colour theorem describes the largest diversity of regional growth poles. The measure of the economicgrowth of the region is a degree of reduction of the dual graph corresponding to the empirical colour distribution,which identi�es actual growth poles and determines the turnpikes of growth. The ultimate development objective,although not always achievable, is a reduction of the dual graph to a single vertex, when all municipal o�ces inthe province have websites of the highest quality.

DOI: 10.12693/APhysPolA.133.1362

PACS/topics: 89.65.Gh, 02.10.Ox

1. Introduction

The relations between technology and economic devel-opment, emphasized by almost all schools of macroeco-nomics, have been known for a long time. This even ledto de�ning the term of �peak technology�, denoting thetechnology dominating in a given period, with teleinfor-matics ful�lling this role at present. Although until re-cently it was investigated as the factor behind growth andcompetitiveness mainly in the private sector, it has nowfound multiple applications in the public sector. Com-prehensive aspects of those changes are described by anew school in economics � Wikinomics.Teleinformatics in public administration sector con-

tributes to the creation of appropriate conditions inwhich two basic groups of business entities, i.e. enter-prises and households, operate. This takes e�ect primar-ily by improving the relationship between this sector andits economic surrounding, by streamlining the process ofproviding already existing services and implementing newones and by triggering the development of local industryand trade. What is of great importance at this point isknowledge sharing, which takes place through local gov-ernment websites. Therefore, in Wikinomics terms, they

*corresponding author

are referred to as platforms of participation. Website de-velopment provides evidence that the economic growthpole is emerging in a given community or region, sincethose websites become the main channels of economic in-formation �ow.

According to the research hypothesis assumed in thepaper, the process describing the emergence of munici-pal growth poles and their merging within the provinceinto turnpikes of growth can be identi�ed with the useof the four colour theorem. Consequently, economic de-velopment can be described in graph theory terms andit consists in reduction of the dual graph correspond-ing to the map of the province of Warmia and Masuria,with quality classes of the local government websites plot-ted on the map. This will help to reveal not only themere fact of local economic development or regress, butalso its qualitative and quantitative characteristics. Asa result of applying the above described method, it wasdemonstrated that achievements of past generations pro-vide the basic source of the current economic growth inthe province under examination. This proves the needto introduce an urgent change in the state policy withregard to initiating economic growth and development inthis region.

Identi�cation of spatial regional poles and turnpikes ofeconomic growth is a challenge which has not yet beenproperly addressed [1]. However, it may be tackled in a

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Application of the Four Colour Theorem to Identify Spatial Regional Poles. . . 1363

non-standard manner by applying the four colour theo-rem. The subject of study is the region of Warmia andMasuria which is one of the least developed voivodshipsof Poland in terms of economy. The method applied in-volves a Wikinomics business model which entails plat-forms for participation. The role of such platforms isplayed by websites of the administrative authorities ofmunicipalities. An analysis of the functionality of thosewebsites provided information on the economic growth ofthe municipalities. Subsequently, by using the k-meansclustering method, four classes of functionality of plat-forms for participation of the municipalities were de�ned.Each municipality was marked on the map of Warmian-Masurian voivodship in a colour that re�ects the func-tionality of its website. In this way, spatial regional polesand turnpikes of economic growth were identi�ed. Thisresulted in a dual graph presenting the economic growthof the voivodship. To analyse the graph, a frame of refer-ence de�ned by another dual graph which complies withthe four colour theorem and shows the entire lack of polesof growth, was used.

2. The four colour theorem

The four colour theorem reads as follows: regions onevery planar map can be coloured in only four colours insuch a manner that each two adjacent regions have di�er-ent colours [2�3]. Adjacent regions mean regions whichborder each other along a line, not in a particular pointor even in a �nite number of points. In addition, a regionmeans a cohesive area consisting of a single area becauseotherwise it would be easy to name an example of a �ve-region map, where each region would border the otherfour regions. In other words, the four colour theorem isabout normal maps that meet two requirements: they donot border more than three regions in one point and noregion fully encircles any other region.Every non-normal map can be paired with a normal

map that requires at least the same number of colours.Thus, since the four colour theorem is valid for normalmaps, it is valid for all maps. Every normal planar mapcomplies with the following equation:

4p2 + 3p3 + 2p4 + p5 − p7 − 2p8

−3p9 − · · · − (N − 6) pN = 12, (1)

where pn is the number of regions on the map whichhave exactly n neighbours, and N is the largest num-ber of neighbours of any region; n = 0 and n = 1 areexcluded since normal maps do not have enclaves or is-lands. Therefore, the equation has to begin with p2. Thisway, every pn is either positive or equals zero and it ispositive only if n < 6. This means that at least one num-ber out of p2, p3, p4, or p5 has to be positive as the sumon the left side of Eq. (1) should be positive to equalise apositive number on the right side. One of the regions hasto have two, three, four, or �ve neighbours. There doesnot exist a normal planar map where each region has sixor more neighbours.

The foregoing allows two concepts underlying thegraph theory to be de�ned: an unavoidable set and re-ducible con�gurations. A set is unavoidable if it containsa group of con�gurations that consist of a region withtwo neighbours, a region with three neighbours, a regionwith four neighbours and a region with �ve neighbours.Its name is derived from the fact that every normal maphas to include at least one of the four con�gurations re-ferred to above. A con�guration is reducible if it cannotappear on a minimal �ve-chromatic normal map, i.e. onthe smallest map that requires �ve colours. The discov-ery of an unavoidable set of reducible con�gurations issu�cient to prove the four colour theorem.The four colour theorem was formulated in 1852 by

Francis Guthrie, a mathematics student at the UniversityCollege of London. When colouring a map of England, henoticed that only four colours are needed to make everytwo adjacent counties have di�erent colours. Mathemati-cians found it very di�cult to prove the four colour theo-rem, despite its simplicity, and it took them 125 yearsto prove it. The proof of concept was announced in1976 by Appel and Haken who based the theorem on thestructure of an unavoidable set of 1,936 reducible con-�gurations (as it turned out later, ultimately there were1,482 con�gurations) [4]. The proof was achieved withcomputer calculations. In subsequent years, that proofwas improved several times, but the need to use a com-puter was not eliminated [5�9]. This means that veri�ca-tion of the proof requires quite a di�erent understandingthan the one applied earlier in mathematics. This un-derstanding cannot be veri�ed by the human mind. Itsparked a discussion about the nature of a mathemati-cal proof [10]. Certain philosophical repercussions of thefour colour theorem were also debated by indicating thatit is not an a priori truth, which would be required in tra-ditional mathematics, but an a posteriori truth [11, 12].Perhaps a traditional proof of the four colour theoremexists and will be discovered in the future.

3. Wikinomics and platforms for participation

Wikinomics is a new school of economics that focuseson non-standard business models established in the digi-tal economy era [13]. One of these models are platformsfor participation which involve products and technicalinfrastructure shared with peer communities to let themuse such products and infrastructure to de�ne new val-ues and trigger innovative undertakings. Platforms forparticipation are based on all of the four Wikinomicsprinciples: openness, peering, sharing and acting glob-ally. Such a business model allows companies to acton a wider stage � when compared to the traditionalmodel � where many various partners can operate theirown economic activities or improve existing solutions.This results in an increase in the productive capacity ofcompanies without additional �xed costs to be incurredby them. A website with an e-commerce system forwarehousing, purchasing, and distributing goods could

1364 A. Jakimowicz, D. Rzeczkowski

be an example of a platform for participation. Theyare very frequently used to establish on-the-�y partner-ships with software developers who add new value towhat has already been achieved. Companies sharingtheir software services and databases via the applica-tion programming interface (API) contribute to gener-ating �uid webs of business partnerships, dubbed devel-oper ecosystems. Establishment of open platforms forparticipation increases the speed, coverage and chancesof an innovation's success.One of the most essential types of platforms for par-

ticipation are web service mashups. They are web-sites that combine various ready-to-go services or ap-plications made available via other websites, thus re-sulting in a new facility of higher quality and function-ality. In other words, it is about remixing data andsoftware into recombinant creations. This phenomenonis increasingly popular since more and more companiesare sharing their APIs.Platforms for participation can focus on various prod-

ucts or services, virtually anything that runs software,and that makes them useful in the public sector, espe-cially in public administration. Presently, websites ofthe administrative authorities of municipalities are farbehind platforms for commerce. This means that thelargest sources of public information are, in fact, notused, though they could be the basis for many new publicservices. Therefore, society exploits the most importantfactors of regional economic growth to a rather limitedextent. There is an urgent need to develop platformsfor participation in public administration so that citi-zens and companies communicate freely with their gov-ernment and the rest of the networked world. Mashups inpublic administration are an opportunity to extract newvalue from public information and to use it in businessbetter. A good example of this is the integration of pub-lic information bulletins with other non-proprietary datasources. Websites of public administration authoritiesmay be platforms for grassroots action and may includeplatforms for public disclosure and platforms for neigh-bourhood knowledge.

4. Assessment

of the functionality of the websites

of the municipal administrative authorities

An assessment of the functionality of the websites ofmunicipal administrative authorities was carried out inaccordance with sixteen criteria represented by symbolsfrom A01 to A16 [14]. The binary method was appliedwhich entails giving a website a point if a given crite-rion is met or a zero when it is not met. Subsequently,points were summed up. Thus, the functionality indexof a website was a number from a closed interval [0,16].Using the k-means method, the websites were dividedinto homogeneous subsets. This led to natural separa-tion of four subsets re�ecting four Wikinomics resourcecategories of the websites in question: low-quality, av-

erage quality, high quality, and very high quality. Thestudy was carried out twice � in 2009 and 2012.Table I characterises four clusters representing the

Wikinomics categories of quality of the websites of themunicipal administrative authorities. It also featurespercentage absorption of the website functionality cri-teria in individual years and clusters. The results shownin Table I are presented spatially, as two maps of theWarmian-Masurian voivodship, in which each municipal-ity is given a colour corresponding to the category of thequality of its website in a given year.

5. Speci�cation of the frame

of reference in the study

The four colour theorem is � despite appearances � oflittle signi�cance for map makers and cartographers [3].In fact, science lacks pro�table applications of the the-orem, i.e. applications solving an important scienti�cproblem. To date, there have been only a few attemptsto apply the four colour theorem to various �elds of sci-ence and all of them were rather unsuccessful [15�20].This paper seeks to demonstrate that this theorem is ofgreat importance not only in the identi�cation of eco-nomic growth poles, but also in regional growth mea-surement.The four colour theorem was used to establish a refer-

ence frame for the study. To this end, a map of municipal-ities of the Warmian-Masurian voivodship was colouredin four colours, as per the theorem. The map is presentedin Fig. 1. It allowed a corresponding dual graph to becreated. The capital cities or towns of all the municipal-ities were marked on the map and then the capital citiesor towns of neighbouring municipalities were linked witheach other by roads crossing their borders. The result isa dual graph of the original map and it is also presentedin Fig. 1. Here, the capital cities or towns of the mu-nicipalities are the vertices of the graph and the roadsare its edges. The dual graph was drawn in such a waythat all edges are straight line sections. The edges of thegraph divide the plane into faces. The capital cities ortowns of the municipalities, i.e. the vertices, correspondon the original map to the faces of the dual graph. Aftereliminating instances of full encirclement of one munici-pality by another, the map of the voivodship is a normalmap. This means that exactly three edges converge ateach vertex of the dual graph, hence all the faces haveto be triangles. Such a dual graph is called a triangula-tion. A number of edges converging at a given vertex iscalled the degree of the vertex and is equal on the origi-nal map to the number of neighbours of the municipalityrepresented by that vertex.According to the terms and de�nitions used in dual

graphs, a con�guration is part of a triangulation consist-ing of a set of vertices and edges that connect them. Thecontour of a graph is a �nite path consisting of verticesand edges which begins and ends in the same vertex,which does not cross itself and which divides the graphinto two parts: internal and external. An external con-

Application of the Four Colour Theorem to Identify Spatial Regional Poles. . . 1365

tour is called the boundary ring of the dual con�gura-tion. On the original map of the voivodship in question(Fig. 1), there are thirty-nine external municipalities en-circling their centre. The municipality of Dubeninki wasomitted for obvious reasons. After the dual graph wascreated, they were replaced with the boundary ring of

the dual con�guration, i.e. with an external contour con-sisting of thirty-nine vertices and thirty-nine edges. Thedual con�guration shown in Fig. 1 is dubbed a thirty-nine-ring con�guration as its ring has thirty-nine vertices.It corresponds to a ring of thirty-nine municipalities thatencircle the original con�guration.

TABLE I

Four clusters representing the Wikinomics categories of quality of the websites of the municipal administrative authorities.

Years 2009 2012

ClustersLow

qualityAveragequality

Highquality

Veryhigh

quality

Lowquality

Averagequality

Highquality

Veryhigh

qualitypoints (min�max) 0 5 � 9 10 � 11 12 � 16 0 7 � 10 11 � 13 14 � 16

centroid 0 7.72 10.45 13.35 0 9 11.75 14.45number

of municipalities17 22 37 40 6 11 49 50

Encoded functionality criteria of the websites of the municipal administrative authorities and their absorption in indi-vidual years and clusters (per cent)

A01 0 100 100 100 0 100 100 100A02 0 100 100 100 0 73 100 100A03 0 0 3 10 0 0 2 76A04 0 55 92 100 0 27 98 100A05 0 77 100 100 0 100 100 100A06 0 32 57 98 0 55 96 98A07 0 68 92 100 0 82 94 100A08 0 5 38 93 0 9 24 94A09 0 14 27 73 0 18 59 100A10 0 14 54 95 0 100 100 100A11 0 73 92 100 0 55 78 100A12 0 77 97 100 0 100 100 100A13 0 73 95 100 0 100 100 100A14 0 86 97 100 0 100 98 100A15 0 0 0 48 0 0 20 46A16 0 0 5 20 0 0 4 34

6. Identi�cation of economic growth poles

as a method to reduce a dual graph

Regional poles of economic growth should be under-stood as drivers which drive innovation and contributeto the improvement of economic performance of compa-nies and their surrounding industries [21�22]. In termsof Wikinomics, a growth pole is present wherever ad-vanced ICT tools are used in business. It may be a web-site of the administrative authorities of a municipalitywhich serves as a platform for participation and inte-grates the economic forces of a given municipality. Web-sites of very high quality have the best chance to becomelocal growth poles. If there are municipalities in a givenvoivodship with such websites, it is very probable thatthere are growth poles. In addition, the spatial distribu-tion of such municipalities may indicate economic growthin space and pinpoint obstacles.Two frames of reference can be used to assess the eco-

nomic growth of the region of Warmia and Masuria.The

�rst of them is the map of the voivodship in questionpresented in Fig. 1, coloured in accordance with the fourcolour theorem. The dual graph created on its basis cor-responds to an instance of an absolute lack of growthpoles. The second of them applies when all the munic-ipalities have websites of the highest quality. In such acase, a dual graph is reduced to one vertex, which meansthat the entire voivodship has become a large growthpole and has a dominant position in the national econ-omy. Naturally, those two frames of reference are quiteunlikely to exist in the real world. Thus, the reality mustbe in between. However, they point to the nature ofthe method of identi�cation of growth poles and turn-pikes. It consists in reducing dual graphs correspondingto real maps (when compared to a graph correspondingto a master map). Such a reduction is done by combiningmunicipalities into clusters that correspond to individualcategories of the quality of their websites.

Figure 2 shows a map of the municipalities of theWarmian-Masurian voivodship divided into the cate-

1366 A. Jakimowicz, D. Rzeczkowski

gories of the quality of the websites of the municipalpublic administrative authorities in 2009. The details arepresented in Table I. Its comparison to the master map(Fig. 1), where no growth poles are present, shows essen-tial information on regional economic growth. The dualgraph corresponding to the map in Fig. 2 has far fewervertices and edges than the dual graph created based onthe master map. The graph created based on the mapof 2009 shows a certain characteristic of the voivodship

in question, i.e. the dominance of historically conditionedinfrastructural growth poles in that voivodship. Historicalconditions result from the fact that a signi�cant part ofthe Warmian-Masurian voivodship was once part of EastPrussia and it was �nally regained by Poland in 1945 aspart of the Regained Territories. Currently, it bordersthe Kaliningrad Oblast of Russia in the north. Polandtook over these areas with their existing roads, railwaytracks and inland waterways.

Fig. 1. Map of the municipalities of the Warmian-Masurian voivodship coloured in accordance with the four colourtheorem and a corresponding dual graph. It is a frame of reference without growth poles.

The municipal websites of high and very high qual-ity are strongly connected with transportation infrastruc-ture erected before 1945. Western areas of the voivod-ship include a cluster of websites of very high qualitysituated along the north-south axis. In those municipal-ities, there is the Elbl¡g Canal built in the 19th cen-tury. Moreover, there are the docks in Elbl¡g oversee-ing the tra�c of cargo barges on the Vistula Lagoon be-tween Poland and Russia, as well as the roads betweenNidzica, Olsztynek, Ostróda, Elbl¡g, and Królewiec withtwo road border crossings: Gronowo�Mamonowo andGrzechotki�Mamonowo II. Finally, there is the railwayline between Malbork, Elbl¡g, Braniewo, Królewiec withthe rail border crossing Braniewo�Mamonowo. The sec-ond cluster of websites of the highest quality is locatedto the east, also along the north-south axis, which par-tially results from geographical conditions. Two roadsare situated there. The �rst of them (Road No. 63)

goes through Pisz, Orzysz, Gi»ycko, and W¦gorzewo,and before 1945 it led to today's Kryªowo on the Rus-sian side of the border (there is no border crossingthere now). The second of them (Road No. 65) goesthrough Eªk, Olecko, Goªdap, and ends with the Pol-ish and Russian road border crossing at Goªdap�Gusev.On the map shown in Fig. 2, there is also a zoneof municipalities located in the centre of the voivod-ship, along the south-west and north-east axis. Na-tional Road No. 16 (Iªawa�Ostróda�Olsztyn�Mr¡gowo�Orzysz�Eªk) and a railway line (Iªawa�Ostróda�Olsztyn�Czerwonka�Korsze) go through there, ending with a railborder crossing Skandawa�Zheleznodorozhny. Slightly tothe left from the centre of the map, there is a barelyvisible growth line Olsztynek�Olsztyn�Dobre Miasto�Lidzbark Warmi«ski�Bartoszyce connected with Na-tional Road No. 51 and ending with the road bordercrossing Bezledy�Bagrationovsk.

Application of the Four Colour Theorem to Identify Spatial Regional Poles. . . 1367

A comparison of the dual graphs in Figs. 1 and 2proves that exerting an impact by local growth poleson the surroundings entails reducing the number of ver-tices and edges of the graph corresponding to the ac-tual map. Edges should be interpreted as obstacles togrowth occurring among neighbouring regions. The rea-sons for such phenomena vary and each of them needsto be analysed individually. Figure 2 shows that thelargest numbers of obstacles are in the north and the

south of the voivodship.Figure 3 presents, as per Table I, the location of the

Wikinomics growth poles in 2012. The dual graph cor-responding to that map has far fewer edges than thedual graph of 2009, which means that new growth poleshave appeared and many obstacles have been eliminated.The remaining obstacles are present in the north of thevoivodship; therefore, investment activities and innova-tion should be intensi�ed there.

Fig. 2. Map of the municipalities of the Warmian-Masurian voivodship divided into categories of the quality of thewebsites of municipal public administrative authorities in 2009 and a corresponding dual graph.

What notably contributed to such rapid developmentwas the construction of a 2,292 km long �bre-optic net-work with adequate infrastructure which allowed resi-dents, companies, and state authorities in that voivod-ship to use broadband Internet. This investment wasrealised in 2008�2015 and its value amounted to PLN327,041,042.07 (EUR 78,250,715.91) [23, 24]. This typeof economic growth can be associated with the turnpiketheory in economics [25, 26]. The idea originates from anAmerican English word that means a highway. The prob-lem concerns setting an economy on the path of maximalbalanced growth, as de�ned by von Neumann, which cor-responds to the use of the peak technology to generate thehighest revenues. The solution to this dilemma is similarto the choice between the use of a turnpike and a networkof minor roads when one wants to arrive at one's destina-tion as quickly as possible. If the starting point and thedestination are at a great distance from each other, it is

always better to use a turnpike, even if one needs to incurcosts at intermediate stages. Such an understanding wasapplied to the voivodship in question since according toWikinomics ICT is currently a peak technology, thus thelines connecting regional growth poles have to be calledturnpikes.

7. Methodological implications

Identi�cation of regional growth poles poses a greatchallenge for authorities in every voivodship. Thisaim can be reached by various methods developed byeconomists, as comprehensively described by Girejko [1].The methodology he presented concerns sustainable re-gional development in Belarus, therefore it may seem use-ful for examining the Warmian-Masurian voivodship inview of certain similarities between those regions. How-ever, the majority of the proposed methods are of thequalitative type, and they often refer to expert evalua-

1368 A. Jakimowicz, D. Rzeczkowski

tions, which can make their objectivity a questionable is-sue. Di�culties also lie in the need to determine a reliableframe of reference to be used for evaluation of economicachievements of all communes making up a given voivod-

ship. Another frequently overlooked aspect includes spa-tial conditions for conducting business activities, result-ing from the geographical location. Such objects as lakes,forests, or rivers can be both growth factors and barriers.

Fig. 3. Map of the municipalities of the Warmian-Masurian voivodship divided into the categories of the quality ofthe websites of municipal public administrative authorities in 2012 and a corresponding dual graph.

The application of the four colour theorem providessigni�cant progress in studies concerning regional polesand turnpikes of economic growth, since it removes themajority of cognitive constraints of the classical methods.In comparison to the conventional approach applied so farin economy, the bene�ts of the methodology presented inthe paper are as follows:1. The four colour theorem establishes two natural

frames of reference for evaluation of each empirical distri-bution of poles and turnpikes of economic growth in thecommunes. The �rst of them consists in the possibilityof referring any empirical map of the examined voivod-ship to the map presenting colour distributions consistentwith the four colour theorem. This permits to evaluatethe reduction degree for the dual graph corresponding tothe empirical map, which is tantamount to eliminating aspeci�c number of growth barriers. Consequently, iden-ti�cation of growth poles is made possible. The secondframe of reference is formed when the dual graph of theempirical map is reduced to a single vertex and the en-tire voivodship becomes a growth pole for the nationaleconomy. The entire reality lies between those two hypo-thetical states.

2. The method is based on ICT, which is thecurrent peak technology, and combines many quali-tative and quantitative factors a�ecting regional de-velopment in one coherent system represented by adual graph. The edges of this graph can be natu-rally interpreted as regional growth barriers. This ex-cludes, to a certain extent, the subjectivity resultingfrom a qualitative expert evaluation.3. It takes into account spatial diversi�cation of condi-

tions for conducting business activity in individual com-munes. This can be perfectly exempli�ed by cooperationof the communes situated along the Elbl¡g Canal. Thisinland water route is a course of infrastructural growthpoles, taking advantage of natural factors in the form ofa series of lakes situated in various communes. As resultsfrom Fig. 2, each of those communes have websites of thehighest quality, which in the western part of the voivod-ship creates the turnpike of economic growth along thenorth-south axis.4. The methodology used in the paper is based on the

�ow of economic information, both in individual com-munes and among them, which allows us to treat the en-tire voivodship as one system and to diagnose its strength

Application of the Four Colour Theorem to Identify Spatial Regional Poles. . . 1369

and weaknesses.5. The applied methodology is justi�ed both theoret-

ically and practically. The impact of technology on eco-nomic development has been a central issue in the theoryof economics for long. However, in the latest years, a neweconomic school emerged, known as wikonomics, whichis based on empirical patterns of economic cooperation.The graph theory combines both issues and makes it pos-sible to introduce synergistic reasoning to economics.6. The method facilitates real time monitoring of the

economic growth in individual regions. The study can bereplicated at any moment. Poles and turnpikes of eco-nomic growth are dynamic phenomena, as some of themcan emerge and develop in economic spaces, while otherscan disappear. Comparison of maps in Fig. 2 and Fig. 3allows us to examine the dynamics of those changes.7. Dynamic studies make it possible to distinguish

communes being de facto growth initiators, from thosewho are based only on imitation. This fact is of crucialimportance for regional economic policy, as it helps toallocate resources and �nancial means in a more e�ectivemanner.8. The applied method is not the last word on the sub-

ject of the regional growth pole identi�cation and showsa potential for further improvement and development.Moving the discussion into the graph theory �eld pro-vides the possibility for regional economics to enter newareas. This is bene�cial for the development of interdis-ciplinary and transdisciplinary studies.

8. Conclusions

Basing the economic growth of the Warmian-Masurianvoivodship on the historically conditioned infrastructuralgrowth poles explains why it is one of the least-developedregions in Poland. Since 1945, few key investments, espe-cially those related to innovation, have been implementedin the region. Two of the rare exceptions are the above-mentioned �bre-optic network and the modernisation ofthe Elbl¡g Canal, completed in 2015, which cost PLN115 million (EUR 27,515,911.37) [27]. It has to be high-lighted that this structure has been serving its purposewithout any major repairs for over 150 years [28]. Themain source of wealth in those areas remains the trans-portation infrastructure erected by previous generationsin previous centuries. There are plans for investments,e.g. development of the docks in Elbl¡g due to diggingthrough the Vistula Spit, creating a medium-sized feederport connecting the Tri-Cities and Russian ports, andfurther development of the ICT infrastructure as part ofthe 2025 Digital Plan for Warmia and Masuria.A certain question related to the economic history

of this region, though signi�cant nowadays, comes tomind: Did Poland play any part in creating infrastruc-tural growth poles in the region of Warmia and Masuriaprior to 1945? It seems that the answer to this questionis positive. Between 1226 and 1466, the region was ruledby the Teutonic Order. It contributed to the economic

growth of those lands by replacing hand-mills with wind-mills. In 1466, as a result of the Second Peace of Thorn,the State of the Teutonic Order was divided into RoyalPrussia and Teutonic Prussia (Ducal Prussia since 1525).In fact, both parts of Prussia became part of Poland,though the latter under �efdom [29]. The golden pe-riod for windmills in the region of Warmia and Masuriawas between the 15th century and the 18th century (andlater) when those lands were o�cially (or uno�cially)part of the Polish economy. The location of windmills inthe region from the turn of the 19th and 20th centuries isclosely related to the present transportation infrastruc-ture [30]. Windmills laid the path for the main trade andcommunication routes, which were essentially the sameroutes along which the �bre-optic network was recentlyconstructed. It indicates that Poland substantially con-tributed to creating historically conditioned infrastruc-tural growth poles in the region of Warmia and Masuria.

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