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Understanding spatial structure from network data : theoretical considerations and applications

Comprendre les structures spatiales à partir de données de réseaux : considérations théoriques et applications
Giovanni A. Rabino et Sylvie Occelli


Au cours de la dernière décennie, l'analyse de données de flux a été un des enjeux pratiques de nombreuses études de régionalisation, d'utilisation du sol et de transports. Les méthodologies existantes soulignent les difficultés qu'il y a à affronter la complexité croissante des structures spatiales, en raison de la diversité des rôles que les noeuds et les liens peuvent jouer dans la structure globale des interactions. En appliquant des considérations issues de la théorie des graphes, on pense qu'une investigation sur les réseaux peut révéler un éventail de "typologies" spatiales. La première section offre des arguments pour une meilleure compréhension des données de flux. On y montre qu'une conceptualisation des flux, comme un résultat des interactions entre les lieux définit une architecture générale selon laquelle un ensemble de formes d'interactions (ou de structures spatiales) peut être identifié. Deux applications empiriques sont décrites dans les sections suivantes. La troisième section présente un élargissement de l'approche dominante des flux, appliquée aux migrations pendulaires pour les régions du Piémont dans les trois dernières décennies. Au-delà de l'identification de la structure hiérarchique, la prise en compte des flux dominants et complémentaires permet de définir une typologies des relations et d'approfondir la question des changements dans l'organisation de la structure régionale de 1971 à 1991. Dans la quatrième section, une analyse de la mobilité résidentielle pour l'aire métropolitaine turinoise est menée. En observant le sous-ensemble des liens qui ont une stricte complémentarité, quelques structures en "circuit" peuvent être mise en évidence et interprétées comme des liaisons entre les aires de marchés résidentielles.

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Notes de l’auteur

Paper presented at the 28th International Geographical Congress, The Hague, August 4-10, 1996

Texte intégral


1In the past decade, analysis of flow data has been of practical concern in most studies dealing with regionalization and transportation land-use problems. Existing methodologies however show difficulties to cope with the increasing complexity of spatial structures because of the 'variety' of roles which nodes and links are likely to play within the overall interaction pattern. These difficulties have also been caused by some ambiguities and misunderstandings which peeped in the recent debate in the field of Regional Science about the juxtaposition of the concepts of system and network. The former being somehow associated with a hierarchical representation of areas; the latter with a grid of equipotential nodes, representing the functional relationships of the system.

2In the present study we argue that, besides being sterile, this juxtaposition, at least at a conceptual level, does not exist. In particular we show that an extension of some existing methodologies based on graph-theoretic considerations can provide useful insights for dealing empirically with both the hierarchical and network relationships of a spatial structure.

Theoretical considerations

Networks and systems in spatial analysis

3Both network and system are well established concepts in spatial analysis (see, Haggett,Cliff and Frey, 1977), where they have been used extensively in an interlinked or interchangeable way which has sometimes caused difficulties and misunderstandings.

4As far as the system concept is concerned, its diffusion dates back to the sixties in the fields of Quantitative Geography and new-born Regional Sciences. Its later consolidation was influenced to a varying extent by the different meanings which were given to the concept itself, relating to :

  • disclosing to the traditional spatial analysis the general underpinnings of the 'scientific approach' and its corner-stone analytical steps of measurement, modelling and experimentation;

  • establishing rigorous definitions of the spatial entities (the system as a collection of interlinked elements) on the basis of the System General Theory. A nomotetic aspect - aimed at recognising the laws of system organisation- and a taxonomic one - aimed at a rigorous definition of the system's elements - were at the heart of the approach. By focusing on the structural properties of spatial systems this approach had undoubtedly the merit to shed some lights into the complexity of system organisation (i.e. the 'urban dynamics', described by Forrester, 1969);

  • introducing in the geographical approach , on the epistemological ground, the 'reductionism' and 'rational-comprehensiveness', which actually negatively affected the current use of the system approach.

5The network concept, or, at least, the awareness of the importance of its many facets in the various fields of urban analysis, is more recent (see, for example, Batten, 1995, Batten, Casti and Thord, eds., 1995). One of the most relevant advancement concerns the definition of 'a new spatial pattern' of human settlements (Dupuy, 1985, Dematteis, 1990):

  • it overcomes the traditional spatial patterns, i.e. those represented by the hierarchical networks (equilibrium spatial systems, based on the complementarity relationships between market areas and nested ranked centres) and multi-centre networks ( dynamic spatial systems, consisting of nodes with different functions which interact according to the circular cumulative mechanisms ruling the agglomeration and polarisation processes);

  • it recognises the formation of equipotential networks, where locations are rather indifferent and urban functions are distributed casually as a result of contingent events in the location processes.

6This latter pattern provides new ways of spatial representations also on an ideological base.

7If the 'area pattern' - the spatial representation most currently used - conveys an idea of objectivity, determinism and stability (to be ruled by conventional urban planning), the 'network pattern' is not just an antithetical representation but provides a mean by which making a discourse (as a metaphor or a language), capable of expressing new ideas, revisiting old ones as well as merging contradictory positions. This latter characteristic along with the assumption of the relativism of the 'visions' of spatial realities, classifies the network representation among those post-modern philosophical positions (i.e. hermeneutic, deconstructionism, etc.) characterised by the so-called 'weak thinking' or 'anti-scientific thinking' (Rossi, 1989).

8In spite of the impossibility to reconcile the system and network concepts on the epistemological ground (with some exceptions, see for example the critical rationalism à la Popper in Rabino's approach, Rabino, 1995), the two concepts have a major point in common as far as their role in spatial analysis is considered. They both succeeded in extending the focus of analysis from the 'local' (the 'area' logical operator and spatial point based modelling) to the 'relational' (the networking and functional organisation of the system). This extension was carried out in the two different cultural domains, quantitative and human geography, using the appropriate language:

  • the logical and formal language, i.e. considering the spatial and functional interactions between the components of the system, the cumulative causal mechanisms, the discrete choice processes, etc.;

  • the definition of ideal-types in the spatial configurations of settlements , i.e.the hierarchical pattern of Christaller's model, the multi-centre pattern of the marshallian district, etc.

9However, the difference of languages caused some ambiguities in the reciprocal interpretations of the two concepts. For the network 's scientists, the system approach, dealing with the functional relationships, was limited to the consideration of hierarchical or multi-centre networks. For the system's scientists the ideal-type of the equipotential network appeared so poorly defined to have little explanatory capability - if any- in interpreting the spatial structures.

10The misunderstanding originated from an inter-change between the two main terms which were involved: the identification of the object of analysis and the relevant attributes of the object.

11In the system approach in fact, the organisational structure (i.e. the hierarchical or connectivity network) is a specific attribute of the object under study (which is the system). On the contrary, in the network approach, the hierarchical or connectivity network (i.e. the organisational structure) itself is the object under study and the system relationships are its specific attributes.

12Whenever we take into account this duality we will be able to overcome the above difficulties.

13In this respect we believe that our treatment of network and hierarchical relationships as complementary aspects of a spatial organisation can be useful for both a system and network approach.

New methodological elements for the analyses of spatial structures

14One main stream of approaches for analysing system relationships and related spatial patterns is based on graph methodologies. Notwithstanding their extensive use in many domains of urban and regional research, it is rather surprising that so far their applications have not stimulated a discussion of their complementary analytical potentialities.One exception in this respect is Kipnis (1985) who applied three graph methods in order to disclose some distinctive as well as similar features of infra-metropolitan residential mobility. The proposed methods are named after the concept which is at their basis:

15a. the 'dominant flow' concept, as pioneered by Nystuen and Dacey (1961), involves a prior rule to define the 'centrality' of a place within a system (see also Slater, 1976, IRES, 1986). The notion of a hierarchy in the spatial organisation of places is at the very heart of the concept. A strict isomorphism is posited between the hierarchy of places and the pattern of flows. A dominant place is the one from which no maximum flow is directed to a less important place. According to this basic rule the identification of the spatial structure is formally equivalent to extracting the 'tree' of the 'graph' associated with the matrix of flows between places;

16b. the 'significant flow' concept primarily focuses on the intensity and direction of flows between places. A significant flow is an inflow or outflow whose value is greater than a given threshold. The rank of places within the urban hierarchy is of secondary importance; what matters is the 'existence' of the mutual relations between places. A crucial point in this respect is the 'saliency' of the relationships and consequently the likely association of places. This concept is at the basis of most of the well-established regionalization methodologies which have been developed (see Brown and Holmes, 1971, Keane, 1978, Istat-Irpet, 1989);

17c. the 'priority flow' concept involves both places and their mutual relationships. Priority flows are those flows which are greater than the expected (gravitational) interaction values. The concept intrinsically refers to the encompassing concept of spatial interaction as a 'movement or communication over space that results from a decision process' (Fotheringam and O'Kelly 1989, p.). Spatial interaction results from both the functional complementarities existing between the located activities and the accessibility of the different places. Conversely, spatial interaction is essential to the maintenance of the functional complementarities. The concept is central to a main field of approaches which have deserved a right of their own in the urban and regional literature (Wilson, 1974, IRES, 1988, Pooler 1994).

18A closer look at these methods shows that beside suggesting different ways to operationally treat spatial relationships, their methodological underpinnings convey different views of the nodes-flows organisation by which the spatial structure of a system is conceptualised.

19In the 'dominant flow' concept a spatial structure consists of a set of nodes, represented as entities which are characterised by given punctual attributes (i.e. population, level of inflows) and inter-linked by one main relationship, reflecting the intrinsic characteristic of the node. In empirical applications this main relationship is usually the maximum outflow originating from a node and directed to another one which is considered more important. The notions of domination and dependency immediately derive from the application of this rule and the resulting spatial structure is a hierarchy represented by a tree graph.

20In the 'significant flow' concept the spatial structure is seen as a set of interacting nodes through some relationships consisting of both outflows and inflows which are significant (and not only necessarily maximum) in the overall flow pattern. In conventional empirical approaches then, the identification of the nodes interacting more strictly allows to recognise a set of areas (i.e. regions), which are homogeneous according to given interaction levels. The number and extension of these areas are the main characteristics of the resulting spatial structure .

21In the 'priority flow concept' the notion of spatial structure is dealt with implicitly and very few analysis have been aimed at its empirical description. To some extent, the spatial structure is 'inferred' from an examination of the distribution of the residuals between the observed and the expected distributions of flows.

22In the following we will elaborate on these ideas and put forward some conceptual elements which could be useful for identifying a whole range of spatial relationships.

23We assume to have a flow matrix, representing some spatial interactions occurring in a system. This can be described generally as a network consisting of a set of nodes connected by relationships, which are oriented and valued.

24As already mentioned, current approaches usually analyse the flow pattern considering one node at a time and assessing its relationships with the other nodes of the network. The relationships are usually the outflows of the node but can also be the inflows or both. The node and its relationships (outflows or/and inflows) constitute the main elementary entity (the pole) of the spatial structure. This latter is then identified according to stated rules about the pole - i.e. which take into account the saliency of its flows ( the significance of outflows or/and inflows), their direction and the importance of the destination node.

25Here we move the focus of the analysis from the pole to the dyad : that is, we consider two nodes, A and B, their mutual relationships (the outflows and inflows) and the impact of these latter on each node. While in the case of the pole only two relationships at most- i.e. the significant outflows and the significant inflows- can exist, in the case of the dyad four relationships at most can occur (the significant outflows and the significant inflows for each node). This is more clearly shown in the diagrams of Figure 1.

Figure 1

26As we are interested in those relationships between nodes A and B, which are perceived as relevant for each node, then, for a node A, four situations can occur in terms of the flows:

  • significant relationships with B in terms of both the outflows and the inflows;

  • significant relationships with B in terms of the outflows ;

  • significant relationships with B in terms of the inflows ;

  • no significant relationships with B.

27If these are referred to the dyad, then a typology of network relationships can be identified, which are graphically summarised in the scheme of Figure 2. As shown, the typology is obtained by cross-tabulating the above flow situations for each node. Of course, this typology depicts the qualitative 'relationships' which could exist between the nodes and not the flows really connecting them.

Figure 2 A typology of network relationships for a dyad

28Two general remarks can be made:

  • first, a symmetry between the types of network relationships exists along the main diagonal, although these are not equivalent (as their significance depend either on A or B) ;

  • second, it can be seen that the typology is exhaustive relatively to the range of interaction possibilities between the nodes A and B. Considering the main diagonal and the lower half of the scheme, we have in fact:

  1. two extreme situations in which all the relationships hold - i.e. the inflows and outflows are significant for both A and B ( case 1) - or no interaction at all exists (case 10);

  2. two situations in which only one relationship holds. The inflows (case 7) or outflows (case 9) are significant only for one node ;

  3. four situations in which two relationships hold. These include the most varied cases: a situation in which both the inflows and outflows are significant for only one node (case 4); a situation in which the flows are both significant outflows for a node and significant inflows for the other (case 6) ; a situation in which, for both the nodes, the flows are significant inflows (case 5) or significant outflows (case 8) ;

  4. two situations in which three relationships hold. These can be considered as a combination of the simpler cases already discussed. In particular they result from a case 4 situation, plus case 7 or, alternatively, case 9).

29Some numerical examples of these types of relationships are presented in Figure 3, for a network consisting of 4 nodes.

30One major implication of the proposed scheme is that notwithstanding the manifold relationships which can exist between any set of nodes their relevance ultimately depends on the impact they have on each of them. What is emphasised here is the importance of the individual system of node-flow relationships within the overall interaction structure.

31In so far as the complexity of network interactions ultimately depends on both the network relationships and their impacts on the nodes themselves, the scheme provides some indications about some possible ways to deal with it. Firstly it puts in evidence a continuity among the types of relationships rather than a diversity. Second it explicitly shows that the decrease in complexity which can be observed moving away from the top left part of the scheme can be determined by neglecting one point-of-view (considering only the types of relationships in the upper or lower part of the scheme) as well as by a reduction of the relevance of the relationships for a node.

Figure 3 : Some examples of structures defined on the basis of the types of relationships of Figure 2

32Notwithstanding the fact that the empirical relevance of this typology should be further tested, those types of relationships associated with the above mentioned concepts could be accommodated in the scheme. As far as the dominant-flow concept is concerned for example, the representative type of relationships is clearly that represented by case 9. In the next paragraph we will show that an extension of the approaches based on this concept allows to take into account other types of relationships identified in the typology.

An extension of the dominant-flow approach.

33In order to extend the current methodological approaches based on the dominant-flow concept two preliminary steps are necessary:

  • the identification of the spatial structure. As already mentioned, this is given by the hierarchical tree graph which can be obtained from an interaction flow matrix considering the maximum outflow from the nodes;

  • the definition of a significant area. This consists of the set of nodes, associated with the tree graph, which are subordinate to the nodes having a given hierarchical level. The choice of this latter - which is referred to as the cut level of analysis - separates the tree graph in two main parts: the lower which contains the significant areas and the upper which identifies that part of the hierarchical structure which falls outside the areas themselves. The significance of an area depends on both the dimension (i.e. total population) and number of the nodes which are included.

34For a given interaction flow matrix, we assume to have identified its hierarchical structure and specified the cut level at which to carry out the analysis of the associated areas. The following types of relationship can then be defined, see Figure 4:

Figure 4 : Network relationships in a hierarchical tree graph

  • over-hierarchical flows. These are the flows occurring in the upper part of the hierarchy - i.e. those which take place outside the areas-;

  • endo-hierarchical flows. These are the flows which go from the upper to the lower part of the graph and thus enter into the areas ;

  • eso-hierarchical flows. These are the flows which go from the lower to the upper part of the graph and thus leave the areas;

  • inter-hierarchical flows. These are the flows - in the lower part of the hierarchical tree - which connect the areas belonging to a given hierarchical level;

  • primary hierarchical flows. These are the (maximum) outflows of the directly subordinate nodes which are directed to a higher level node along the same branch of the hierarchical tree;

  • secondary hierarchical flows. These are the (maximum) outflows of the lower subordinate nodes which are directed to a higher level node along the same branch of the hierarchical tree;

  • primary anti hierarchical flows. These are the inflows of the directly subordinate nodes which originate from a higher level node along the same branch of the hierarchical tree;

  • secondary anti hierarchical flows. These are the inflows flows of the lower subordinate nodes which originate from a higher level node along the same branch of the hierarchical tree;

  • para hierarchical flows. These are the outflows of the subordinate nodes which are directed to a higher (or equal) level node along different branches of the hierarchical tree;

  • anti para hierarchical flows. These are the outflows of the subordinate nodes which are directed to a lower level node along different branches of the hierarchical tree.

35As clearly shown in Figure 4, these specifications of flows can be accommodated in the previously discussed typology (as cases 4 and 7 in Figure 2). In particular, primary hierarchical and primary anti hierarchical flows represent an evident example of case 4, which was indicated as a situation characterised by a uni-directional domination.

36The following remarks should also be pointed out.

37First, the above classification is exhaustive in the sense that it allows to take into account all the flows of an interaction matrix (and not only those which are maximal). Of course it could be further articulated according to the hierarchical level which the flows refer to. In this respect it is worth mentioning that whenever the analysis of the tree graph is referred to the highest hierarchical level (1° level), then the over endo and eso hierarchical flows are null by definition.

38Second, beside to the areas, the types of flow can also be obtained for each node of the hierarchy. Therefore the distribution of flows by types can be calculated for any part of the tree graph by simply aggregating the results by node.

39Finally, coherently with the point-of-view adopted in the field of 'the performance indicators' (Clarke and Wilson, 1987, Bertuglia, Clarke and Wilson,eds., 1994), in the operational procedures the computation of this distribution for the nodes can be distinguished depending on wether the nodes are considered as origin or destination of flows.

An analysis of journey-to-work flows for Italy and the Piedmont region

General premises

40The application of the extended methodology based on the dominant-flow concept was carried out for two geographical areas: the whole Italian country and the Piedmont region. This latter is situated in the north-western part of Italy, and is one of the most populous regions.

41For both areas, we considered the matrices of journey-to-work flows. These were obtained from the individual census data and aggregated at the commune level- the commune being the smallest administrative unit, according to which commuters' analysis at the sub-regional level is usually carried out-. As the areas include a great number of communes ( 8000 and 1200 for Italy and Piedmont respectively), each flow matrix consists of a relatively large dataset which is rather cumbersome to manage. In order to overcome this difficulty a package has been developed to implement the methodology. This is written in CLIPPER and can run on any standard PC. Besides making it possible to 'navigate' in the flow matrix, the procedures allow to determine the urban hierarchy and the distribution of flows by classes. As already mentioned, this latter can be determined at each pre-specified level of the urban hierarchy as well as at the commune level. In this case the procedure gives two possibilities of computation according to wether the commune is considered as an origin or a destination of flows.

42As for the Piedmont region the matrices at three census epochs were available, at 1991, 1981 and 1971, it was also possible to analyse the regional spatial changes over a relatively long time period. It is worth mentioning that while the total level of commuting in 1991 in Piedmont is almost as high as that observed twenty years earlier, the spatial distribution of flows is very different. In spite of a raising mobility rate all over the years, the growth of commuting has been counter-balanced by a slight reduction in employment and negatively influenced by the ageeing process of population. The most relevant changes however resulted in the spatial pattern as within -commune flows underwent a sharp decrease while between - commune flows increased considerably.

43In the following we will firstly comment on the position of the Piedmont areas within the Italian urban hierarchy and subsequently focus on some major features of the regional case.

Piedmont in the Italian urban hierarchy

44The analysis refers to the areas which were at the top in the Italian urban hierarchy (these are the areas which are subordinate to the 1° level urban nodes. To be selected the areas should have at least 100.000 inhabitants in 1991 and more than 10 communes).

45In 1991, Italy had 49 first level areas but only 14 with more than 1 million inhabitants, see Figure 5.

Figure 5 : The 14 largest areas at the top of the Italian urban hierarchy in 1991

46These include the regional capital cities which were also appointed as chief metropolitan town by the 1992 Local Government Law. Among these 14 areas, only 4, - those of Naples, Bari, Palermo and Catania , which are also the smallest -, are in the South.

47Not unexpectedly, the Milan area, in the Lombardy region, is the most populous and largest. It has more than 1500 comunes and a population of over 8 millions. It also extends in the Piedmont region, including its more peripheral North-Eastern provinces.

48The second largest area belongs to the national capital, Rome, with a population of 5.7 millions but only 500 communes.The largest area in Piedmont, that of Turin, occupies the fourth position in the national urban ranking as far as population is concerned but it is second in terms of number of communes.

49To get a clearer description of the characteristics of the Turin area relatively to the types of flows identified by the extensions of the approach based on the dominant-flow concept, a comparison was made with the Milan area, the whole Italian country and some macro-areas. These latter were obtained by aggregating the 49 urban areas according to two distinct classificatory dimensions: population size and geographical position.

50The results are summarized in Table 1. Endo, eso and over hierarchical flows are null by definition as reference is made to the highest level in the urban hierarchy.

51The table clearly shows that:

52a) excluding that of Milan, the Turin area has, at least in relative terms, the lowest level of contacts with the other urban areas (i.e. the inter-hierarchical relationships are weak);

Table 1 : Network flow structure in some macro-areas and in those of Milan and Turin in 1991

53b) para-hierarchical flows for the Turin area also tend to be weaker than those for Milan and the Northern macro-area, although they are higher than the Italian average;

54c) the hierarchical structure , as determined by both hierarchical and anti hierarchical flows, results appreciately strongest in the Turin area .

55This latter aspect, in particular, represents a distinctive feature of the Turin area. Besides indicating the centrality of the city of Turin, it also reflects the polarized structure of the Piedmont city system, characterized by few large urban centers and a lot of small communes.

The spatial structure of Piedmont and its changes in the 1971-91 period

56In spite of the substantial stability in its overall structure, two kinds of changes affected the urban hierarchy of Piedmont in the 1971-1991 period:

57a) a weakening of the dependences on the regional capital for the communes in the North-eastern part of the region, which as already mentioned shifted to the Milan area;

58b) a re-articulation of the sub-regional areas, also accompanied in the last decade by a consolidation of the major ones. This was influenced by the diffusion processes which have been occurring since mid-seventies and which resulted in a significant movement of population and jobs toward the periphery of the main cities and along their main outward communication routes.

59The above changes are clearly reflected in the sub-regional areas which were at the top of the regional hierarchy at 1971, 1981 and 1991, see Figure 6 .

Figures 6a,6b,6c : First level sub area in Piedmont in 1971,1991 and 1981

60While in 1981 there was a tendency toward the formation of autonomous sub-regional spatial systems, in 1991, the reverse is occurring and the regional spatial configuration is made up of only two areas, centered on the regional capital Turin and the provincial capital Novara (which is near to the regional boarder with Lombardy).

61The tendency toward a consolidation of the sub-regional areas is also clearly revealed examining the flow average value by link within the main flow-types in the 1971-91 period, see Figure 7, which indicates a significant increase in both the total hierarchical and anti-hierarchical flows in 1991 (these include both primary and secondary types ).

Figure 7 : Mean value of flows by main flow-types in Piedmont at 1971, 1981 and 1991

62As already said, besides the possibility of aggregating the results of the flow distribution by types for any regional sub-areas the operational procedures allow to distinguish the computation by origin or destination. In Figure 8 we show some results for a sub-area of Piedmont, the metropolitan province, considered as an origin, where the city of Turin and the metropolitan rings are also distinguished. As in the previous figure, total hierarchical and para hierarchical flows account for both primary and secondary types .

63These results clearly indicate that:

  • the city of Turin has no para-hierarchical flows. Its position at the top of the urban ranking in Piedmont is such that dominance and subordination are the only type of relationships activated by the regional capital (as shown in Figure 8 in fact, for the city considered as an origin, only the anti hierarchical flows exist);

  • as for the region, the hierarchical flows in the metropolitan province underwent a decline in the 1971-91 period, while the para hierarchical flows tended to increase; this is not unexpected as this tendency was determined, to a large extent, by the spatial diffusion processes which took place in the metropolitan area in the last decades;

  • some differences however emerge at the sub-metropolitan level. For the first ring, the hierarchical flows remain relatively higher than the para-hierarchical in the whole period, although they decrease. The contrary occurs for the outer rings, with the exception of the second which behaves like the first ring till 1981.

Figure 8 : Distribution of flows by main types for the city of Turin, metropolitan rings and Turin province in 1971, 1981 and 1991

64To get a more detailed insight of the relative importante of the total hierarchical and para-hierarchical flows at the sub-regional level, their incidence at the commune level was calculated at the three epochs, 1971, 1981 and 1991, and the results mapped in Figures 9 and 10.

65A comparison of these maps indicates that:

  • notwithstanding the relevance of the hierarchical flows in the Piedmont region, their intensity decreased significantly between 1971 and 1991. The highest incidence values were those most affected by the decline. If in 1971 for nearly 1/3 of the communes the hierarchical flows represented more than 70% of their total outflows, in 1991 only 8% of the communes had such a high incidence. The reduction was particularly accentuated in those communes surrounding the provincial capital cities and above all in the metropolitan area (see Figure 9);

  • the general increase in the importance of the para-hierarchical flows tended to be accompanied by a consolidation of their spatial pattern (see Figure 10). In fact, it mostly affected those communes situated in the outer rings of the main regional towns, and in those parts of the region which are 'on the boarders' of the main local areas .

An analysis of Turin intra-metropolitan residential flows

General premises

66The study area covers the lowland part of the province of Turin and includes a relatively large number of communes which together concentrate about half of the population and jobs of Piedmont. This is the commuting area which has been defined within the Metropolitan Transportation Master Plan and includes the three major 'rings' surrounding the city of Turin. The area is significantly larger than most of the areas which have been recently suggested to be representative of the Turin metropolitan system (but less extensive, for example, than the functional region defined in Hall and Hay, 1980).

67A 1982-87 household migration flow matrix, F(i,j), was constructed on the basis of individual address change notes, recorded by Population Register at the commune level (this latter being the smallest administrative unit as well as the elementary statistical unit at which many official data are collected)(Ires,1994, 1995).

68F(i,j) denotes the flows from the origin area i to the destination area j, with i=j=144. 126 of the 144 areas are metropolitan communes and 18 are outer sub-regional areas ( according to which the rest of the Piedmont region has been subdivided).

69The approach which has been used is a direct application of some of the concepts developed in 2.2 As the analysis was aimed at disclosing the pattern of residential flows having a reciprocal movements between the metropolitan communes, the type of relationships which is of interest is that in which flows have a 'complete circularity' (the case 1 of Figure 2).

70From a methodological point-of-view, the following aspects characterise the approach which has been applied:

  • both versus and direction of the residential flows are taken into account;

  • the relative impact of the residential flows on the destination and source areas are evaluated jointly;

  • no prior assumption about the rank of areas is made. Each area is considered as an eligible participant in the mobility pattern no matter its importance (i.e. its socioeconomic characteristics or its size).

71As previously discussed, a 'strict complementarity' between area i and j is established if and only if the residential flows are significant for i and j and the significance should hold for the inflows and outflows of both i and j.

72The main steps of the approach can be summarized as follows:

  1. for each area i, the significant departures, SF(i,j) and the significant entries, SF(j,i) are identified indicating, respectively, the relevant destinations and source areas of residential mobility:
    SF(i,j) > c * OF(i) (1a)
    SF(j,i) > c * IF (j) (1b)
    where OF(i) and IF(j) are the mean value of the total outflows and inflows of an area, respectively, and c is a threshold value for flows significance;

  2. each area i is then matched with the selected areas J, j=1,..,n, for which both the relations (1a) and (1b) with area i reciprocally hold .

73Whenever a strict complementarity between areas i and j is established, the distinction between destination (j) and source area (j) does not hold anymore and a 'circularity' in the residential mobility network is identified.

74Before describing the main results of the empirical investigation two further points should be mentioned:

  • first, the resulting pattern of the complementary relations obviously depends on the threshold value assumed for the significance of flows, c. In this application a value of 1.5 has been considered. Its use has a twofold effect, see Figure 11: i) it enhances the importance of low mobility areas as well as reducing that of high mobility areas and ii) it averages the number of inks associated with the residential flows of each area;

  • second, a nodality index for each area has been superimposed on the pattern of complementary flows in order to provide a measure of the relative importance of areas in the whole mobility pattern. As the nodality index expresses the number of links allowing direct inflows to an area it can be considered a measure of areas' centrality in the mobility pattern (its correlation with the number of households moving to each area is higher than with the resident households).

Some main results

75The resulting spatial pattern and nodality indices for the Turin metropolitan area are shown in Figure 12.

76Despite the expected core-periphery movements, three distinctive dimensions of the intra-metropolitan residential mobility are revealed.

77Foremost is the fact that the city of Turin exhibits selective complementary relations : only areas (communes) of the first ring - not necesseraly geographically adjacent to the city but with a high value of the nodality index - do mutually exchange migrants with the regional capital. In addition, some authonomous mobility sub-networks are identified, mainly in the western part of the metropolitan ring. They reflect the higher consolidation of the functional organization and built up areas of this part of the metropolitan area .

Figure 11 : Nodality index for the metropolitan communes (ranked by the mean value of flows)

Figure 11 : Nodality index for the metropolitan communes (ranked by the mean value of flows)

Figure 12 : Complementary relationships between metropolitan communes in the Turin Metropolitan Area ( based on the total residential flows in the 1982-1987 period)

78The second dimension of mobility revealed by Figure 12 is an evident directionality in the complementary flows pattern. This latter in fact results more dense along some directions which correspond to the major outward communication axes from the city of Turin.

79In this respect, two further aspects related to the metropolitan structure are exposed: a basic similarity with the commuting pattern and a continuity with the ' ring' and 'sector' movements which characterized Turin metropolitan growth in the past decades (Ires, 1983).

80The last dimension concerns the 'openess' of the metropolitan mobility 'field'. Although within-area residential mobility is relatively high (reaching 76% and 83% for out-migration and in-migration, respectively), some mutual relations between certain metropolitan communes and the outer areas show up clearly. Not unexpectly, most of these communes are the well-established sub-poles of the metropolitan urban structure.

81Notwithstanding the explorative nature of this approach, two general questions are raised which deserve further attention in future research.

82The first concerns the range of mobility patterns which could be obtained varying the significance threshold of flows. In this respect, it is argued that sensitivity analysis should be carried out along with an assessment of its implications relatively to the socioeconomic profile of metropolitan areas. The possibility that the patterns ad structure of flows are inherent components of the performance of spatial systems is an issue which has already been pointed out (Clarke and Wilson, 1987) and opens a new field of enquiry .

83The second question relates to the notion of flow complementarity and its identification on the empirical ground. In this case, for example, beside beaking down the flows by household types this should require a conceptual framework to rule out the expected kind of complementarities (i.e. relatively to the housing sub-markets). It is likely that the complementarities that we can recognize on the aggregate will not hold anymore if flows are articulated by socioeconomic types or categories. While this can be considered an intrinsic facet of the increasing complexity of today system organisation, the analysis of interaction flows can provide substantial contribution to its understanding.

Concluding remarks

84The aim of this paper was to suggest some directions for extending traditional approaches to the analysis of interaction data and related spatial structures. Building upon graph-theoretic considerations we showed that a number of stimulating insights could be gained for dealing with both the hierarchical and network relationships of a spatial structure.

85As far as the theoretical aspects are concerned we suggested that one main direction of advancement is represented by shifting the focus of the analysis from the 'node' to the 'dyad'. We argued that this would allow to deal with a richer and more articulate set of relationships. In particular, it would provide some new conceptual tools for identifying a whole range of relationships, beside those hierarchical, which nowadays have an increasing relevance in the spatial organisation of urban systems. These 'network relationships' however are not to be considered as a juxtaposition to hierarchical relationships but as a wider typology which encompasses the former as a specific case.

86A major aspect of novelty is represented by the fact that the relationships occurring between the nodes of the dyad are jointly evaluated by both of them. Of course further developments will be necessary in order to make the proposed approach fully operationalised, i.e. in order to recognise those structural properties characterising the different spatial patterns ( the mono-centric versus the multi-centre system).

87As far as the methodological aspects are concerned the extensions of the dominant -flow approach provide an empirically oriented tool for making a more complete diagnosis of a tree graph spatial structure. In this respect, as indicated by the applications which have been carried out in the Italian context so far, these methodological advancements have two major potentialities which are worth mentioning:

  • first, they can give a contribution to the implementation of a 'spatial evaluation' approach, along the arguments put forward for example by Masser(1983) for spatial accountability analysis;

  • and second, they provide an operational support by which empirical applications could be easily carried out thus favouring comparisons of the spatial structures of different urban systems.

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Titre Figure 11 : Nodality index for the metropolitan communes (ranked by the mean value of flows)
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Giovanni A. Rabino et Sylvie Occelli, « Understanding spatial structure from network data : theoretical considerations and applications », Cybergeo : European Journal of Geography [En ligne], Systèmes, Modélisation, Géostatistiques, document 29, mis en ligne le 26 juin 1997, consulté le 19 juillet 2019. URL : ; DOI : 10.4000/cybergeo.2199

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Giovanni A. Rabino

DISET - Dipartimento di Ingegneria dei Sistemi Edilizi e Territoriali, Polytechnic of Milan, Piazza Leonardo da Vinci , 32, 20133, Milan, Italy (tel.++39/2/23994102)

Sylvie Occelli

IRES - Istituto di Ricerche Economico Sociali del Piemonte, Via Bogino 21, 10123, Turin, Italy (tel.++39/11/88051)

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