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Abstract

Self-organisation in territories leads to the emergence of patterns in urban systems that shape the interactions and dependencies between cities, resulting in a hierarchical organisation. Governance follows a hierarchical structure as well, breaking the territory into smaller units for its management. The possible mismatch between these two organisations may lead to a range of problems, ranging from inefficiencies to insufficient and uneven distribution of resources. This paper seeks to develop a methodology to explore and quantify the correspondence between the hierarchical organisation given by the structure of governance and that given by the structure of the urban systems being governed, where Chile is used as a case study. The urban hierarchical structure is defined according to the connectivity of the system given by the road network. This is extracted through a clustering algorithm defined as a percolation process on the Chilean street network, giving rise to urban clusters at different scales. These are then compared to the spatial scales of the politico-administrative system. This is achieved by using measures of pair-wise distance similarity on the dendrograms, such as the cophenetic distance, by looking at the different clustering membership using Jaccard similarity and by analysing the topological diversity defined as the structural entropy. The results show that the urban sub-national structures present high heterogeneity, while the administrative system is highly homogeneous, replicating the same structure of organisation across the national territory. Such contrasting organisational structures present administrative challenges that can give rise to poor decision-making processes and mismanagement, in addition to impairing the efficient functioning of the systems themselves. Our results can help address these challenges, informing how to rebalance such mismatches through planning and political strategies that consider the complex interdependencies of territories across scales.

Keywords

Hierarchical organisation percolation cophenetic distance Jaccard similarity entropy

Introduction

Politics are inherently geographical and scalar (Agnew, 1997). Space is an important element of governance because governmental activities are territorially demarcated and have well-defined scales of operation (Hudson, 2007). These scales are spatially defined, so the territory over which governance is exercised is delimited by clear geographical boundaries such as communes, provinces and regions. In some cases, these boundaries are based on traditional and historical identities, while in others, they are imposed for utilitarian purposes and have little relation to existing territorial dynamics (Terlouw, 2012). The spatial extension of different levels of governance is often defined by the central state to accommodate administrative functions. State institutions play a crucial role in creating, modifying or removing boundaries and defining scalar organisations. In centralised governments, decision-making processes tend to occur distant to regional and local spaces of governance, leading to the implementation of standardised policies that do not account for the local particularities. In contrast, polycentric structures of governance operate at multiple scales, allowing for policies that better match territorial diversity (Cumming et al., 2013; Lebel et al., 2006). Developing effective regional policy requires relevant regional scalar data and a thorough understanding of regional structures. This information is key for decision-makers to identify the unique needs and challenges of different regions, allowing them to tailor their policies and plans accordingly (Rozenblat, 2009).

Scales of management and the scales of processes

The scalar thinking permeates different layers of organisation: the political, the geographical, the economical, the sociological and the infrastructural. The ubiquitous use of the concept has raised concern among scholars who stress the tension between different notions and conflated conceptions of scale (Brenner, 2001; Jonas, 1994; MacKinnon, 2011; Marston, 2000; Marston et al., 2016; Moore, 2008; Smith, 1992). Jonas (1994) and Moore (2008) have underlined the distinction between ‘operational’ and ‘discursive’ approaches to the scalar thinking. The operational scale is considered the real scale of processes where the sociospatial interactions occur. Discursive scales on the other hand, are a frame or set of abstractions used to understand and study the real processes (Jonas, 1994; Lukas and Flitner, 2019; Moore, 2008). In this context, discursive scales are the outcome of imposed boundaries (Lukas and Flitner, 2019), which are used to identify phenomena within a system of interest by setting ‘research scales’, or to implement policies relevant to the phenomena through ‘management scales’. Discursive scales have long been a subject of controversy, as they can be influenced by embedded assumptions, political implications and practicality considerations (Rangan and Kull, 2009). Some scalar narratives have been widely accepted without adequate scientific support, leading to the production of knowledge that is inconsistent with real processes (Lukas and Flitner, 2019). Geographical data, for example, is often provided aggregated into spatial jurisdictions and not necessarily on the basis of ‘scientifically derived’ scales (Jonas, 1994). Administrative boundaries have been used for a long time to address urban issues either by governance, research, planning or design approaches. These pre-given limits, provide all-purpose and durable in time urban definitions that can be easily implemented. Despite how convenient the straightforward use of these definitions may be, they are not always capable of reflecting the extent of the urban processes in question. Along with this, the speed of urban growth, the fuzzy urban-rural transition and the multiple extent of economic, ecological, social and political processes and their different scales, are challenging this fixed and ‘one-size-fits-all’ approach (Rozenfeld et al., 2008). According to Moore (2008), the attempt to fit complex processes into reduced pre-given scales ends up overlooking the diversity of processes, ignoring the granularity and variability of sociospatial systems (Moore, 2008).

Scalar mismatch

According to Cumming (2006), scalar mismatches take place when the scale of management and the scale of processes are aligned in such a way that inefficiencies occur either affecting the functioning of the systems being managed or the functioning of the institutions responsible for their management (Cumming et al., 2013). Governance in these cases is less sensitive to the variety of local structures, lacking the appropriate information and oversimplifying the knowledge about processes. Deficient monitoring frameworks, together with the lack of clear responsibilities necessary to achieve the required scale of management, results in the formulation of misleading policies (Cumming et al., 2006). Under these circumstances, self-maintenance and self-organisation are undermined, affecting not only daily operations, but leading to a potential decrease of the urban resilience capacity (Cumming et al., 2006; Folke et al., 2005). Redressing scalar mismatches is therefore a crucial requirement to redefine the relationship between governance, planning and sociospatial processes. Alternative geographies of governance more responsive to scalar processes as Bulkeley (2005) suggests, need to be articulated. Managing scalar mismatches requires that scale is not taken for granted. Therefore, the organisational structure of systems needs to be identified beyond pre-given administrative definitions (Bulkeley, 2005). In order ‘to govern, it is necessary to render visible the space over which government is to be exercised’ (Rose, 1999: p. 36). The diversity of the territory needs to be decoded, so that political and planning strategies are not blind to the scalar particularities of the urban organisation. This challenges the understanding of sub-national scales such as communes, provinces or regions, as self-enclosed homogeneous entities and coherent bounded territories (Brenner, 2001; Hudson, 2007). Rather, it puts forward its conceptualisation as relational entities made up of intricate systems of cities.

Decoding and comparing hierarchical structures

Urban systems are inherently relational (Pumain, 2006b); therefore, the identification of different intensities of connectivity patterns is a key criterion for their definition (Bretagnolle et al., 2006; Pumain, 2006a). As objects of study, systems of cities could be defined by the nature of relationships holding them together (Batty, 2018; Rozenblat, 2022). Sub-national scales are then, the result of constituent elements and the way in which they are progressively connected to all others (Batty and Longley, 1994). By ordering connections at successive scales according to their orders of magnitude, a hierarchical structure can be discerned. As Simon (1977) observed ‘everything is connected, but somethings are more connected than others’. And it is from this process of heterogeneous interactions that the hierarchical structure of the urban territory emerges from the most local scale to the largest.

Hierarchical structures can be represented through dendrograms, which are branching graphical representations (Fowlkes and Mallows, 1983). To compare different dendrograms, certain structural similarities and differences can be visualised and quantified. The first methods were developed in biology for phenetic studies to evaluate results obtained from numerical taxonomic analysis. Distances on the basis of similarity between dendrogram elements, are widely used methods ever since Sokal and Rohlf (1962) introduced the use of cophenetic distance. Although its use is still more prominent in disciplines such as biology and ecology, it has been extended to other fields to assess the efficiency of different clustering methods and the consistency between outputs (Fischer, 1980; Rohlf, 1982; Saracli et al., 2013).

While theoretical discussions in various fields, such as ecology, politics and urban planning, have contributed to the understanding of scalar mismatches, a significant gap still exists in effectively quantifying them. To the best of our knowledge, there are no analytical papers in the current literature that specifically address this topic in such depth, particularly in the context of multi-scalar analysis. Our primary focus is to advance the study of urban mismatches from an analytical perspective by employing a multi-scalar and quantitative approach. In this paper, we compare hierarchical structures or dendrograms to explore scalar mismatches. Specifically, we examine the dendrogram of management scales resulting from the hierarchical structure of the administrative system, and the dendrogram of operational scales, resulting from the percolation analysis given the connectivity of the urban layout. Using Chile as a case study, we compare both dendrograms via three methods. First, we conduct a pair-wise comparison of the hierarchical distance of each city to all other cities in the system using the cophenetic distance. This provides information on the variation in relationship intensity between systems of cities in both structures. Then we compare the clusters in terms of their membership using Jaccard similarity to quantify how closely the city systems are grouped together in accordance with administrative boundary definitions. Finally, we examine the topological structure of the dendrograms by measuring the diversity of the furcations using structural entropy. The results of these analyses reveal the centralised organisation of the national system, the heterogeneous intra-regional structures and the relationships between systems of cities that go beyond administrative definitions.

Case study

Chilean decentralisation, a challenge for regional governance

The current politico-administrative division of Chile dates back to 1974, when it was established during the early years of the military dictatorship. The regionalisation process carried out at that time was seen as a political control strategy managed by the central government to strengthen national security and exercise administration (Arenas et al., 2007). The division of the territory into thirteen ‘homogeneous and equivalent’ regions (CONARA, 1974), was more about distributing power across the country than about empowering self-governing regions (OECD, 2017). As a result, the spatial structure of the governance system in Chile was primarily based on utilitarian purposes, ignoring territorial identities and the intrinsic local organisation. The regional administrative structure is homogeneously repeated throughout the country and does not necessarily align with the scalar logic of urban processes. Each region is organised mono-centrally around a capital city, which serves as the hub of development and main source of services. This political organisation has reinforced intra-urban regional disparities. In response to increasing demands for local identity and representation, three regions were re-delimited to better match the logic of local scales (Tarapaca and Los Lagos in 2007, and Biobio in 2017), while the rest of the territorial jurisdictions remained largely unchanged. The current Chilean politico-administrative system is organised into a simple 4-tier structure consisting of communes (346), provinces (56), regions (16) and the national level. The structure is equivalent at any jurisdictional level, with non-intersecting memberships, and does not consider intermediate scales or temporary jurisdictions for specific policy purposes.

Chile’s regional level is the core of decentralisation reforms, and is considered a crucial link between the national and local levels. Despite various attempts by democratic governments to promote regional reforms, Chile remains one of the most centralised countries in the OECD (OECD, 2017). The urgent need for a governance perspective that achieves the definitive decentralisation of the country has positioned regional governance as a top priority. To this end, the paper will focus more closely on the multi-scalar organisation of the regions.

Methods

The methodology of the paper is divided into two main sections. The first section outlines how to extract the hierarchical organisation of the system through percolation analysis, and then how to represent it in terms of a dendrogram. The second section describes the various methods used to compare the different dendrograms and evaluate the potential mismatches between them. These are the cophenetic distance, the Jaccard similarity and the structural entropy.

Hierarchical organisation

We use percolation analysis to retrieve the scalar structure of Chile based on the connectivity of the street network. Socio-historical processes have left traces in the way local entities organise. As explained in Arcaute et al. (2016), these footprints are contained in the street network, representing the main proxy for communication and exchange between settlements. The street network plays a crucial role in the organisation of urban areas, serving as the setting for urban life and the location for many social and economic interactions (Marshall et al., 2018). Streets form the underlying network for the functioning of urban systems, enabling the reachability of services and facilities and the flow of people and goods (Strano et al., 2012). The efficiency of these interactions is closely tied to the topological properties of the street pattern given by its connectivity (Cavallaro et al., 2014). Junctions, or intersections, play a key role in ensuring connectivity and providing access to a range of alternative routes. The more junctions there are, the greater the permeability of the network, the denser the urban activities and the greater potential there is for urban interaction.

Percolation analysis

Percolation theory can be applied to identify substructures in a system describing phase transitions of connected clusters that characterise the potential flow of information within the structure (Gallos et al., 2011). It provides information on the global scale structure of connectivity of a system from the bottom-up. The process starts with disconnected elements that merge together based on a distance threshold parameter until the largest cluster covers the entire system. Percolation-like processes have been applied to different fields of research like functional brain networks (Gallos et al., 2011), the spread of diseases (Gallos et al., 2012; Newman et al., 2002) and indeed, they have been used to study the structure of urban systems (Behnisch et al., 2019; Fluschnik et al., 2016; Piovani et al., 2017; Rozenfeld et al., 2008, 2011). In Arcaute et al. (2016), percolation was used to uncover the urban hierarchical organisation of Britain through multiple percolation transitions based on the distance between street intersections.

This research employs percolation processes as a hierarchical clustering algorithm to unveil its hierarchical structure and identify the pertinent scales of analysis. This decision is based on the method’s demonstrated effectiveness in various fields, revealing scalar organisational patterns in networks while also providing computational efficiency. We follow the procedure used in Arcaute et al. (2016); Piovani et al. (2017). The street network is obtained from OpenStreetMap (OSM) as the primary source of our spatial data. The OSM data contains detailed information on the street network, including attributes such as speed limits. The network is represented by nodes (intersections) and links (street segments between intersections) that are embedded in space and weighted by the time required to travel by car along each link. Using time as the weighting attribute provides a more comprehensive representation of the movement between spatial nodes. Its use also allow us to include ferries’ routes (considering associated travel and wait times), which are the only means of transportation in the southern archipelagos. In this way, we are able to capture the main structure of connectivity throughout the territory (see Supplemental Material). The process consists in extracting the sub-graphs of connected links whose weight $w_{i j}$ is lower than a given threshold $τ$ , $w_{i j} \leq τ$ . The process starts from the minimum time threshold at which all links are disconnected. Increasing the time threshold, leads to a series of percolation transitions where connected links with $w_{i j} \leq τ$ form clusters. The clusters appearing at lower thresholds, that is, for smaller travel time, do not necessarily imply physical proximity, but rather a greater density of intersections, and hence a higher potential for urban interactions. Important scale transitions can be identified when discontinuities occur in the evolution of the largest cluster size across the different time thresholds (Arcaute et al., 2016). These discontinuities do not imply actual breaks in the structure, but rather changes in the intensity of the interactions (Holling, 2001; Simon, 1962). Thresholds where these discontinuities occur are selected as the main levels of the hierarchical structure. As the threshold increases, cities are captured first, followed by urban conurbations, systems of cities and regions, until the entire country is obtained at the end of the process.

Dendrogram visualisation

The hierarchical structure of the urban system is reconstructed by ordering the different clusters obtained through the above-mentioned thresholding procedure. This is done using a dendrogram that is organised vertically into different scales, each representing a selected threshold with the highest at the top and the lowest at the bottom. This is possible because clusters generated using a lower threshold $τ_{1}$ are contained within the clusters obtained using a higher threshold $τ_{2}$ . Horizontally, clusters have strong intra-cluster interactions and weaker inter-cluster linkages. At the same hierarchical level, clusters are not closely connected and instead communicate at higher scales through cross-scalar relationships. The dendrogram shows the largest cluster of the system (in this case, the entire country) at the top, branching down and decomposing into subsystems according to the strength of network connectivity, until reaching the terminal nodes at the bottom scale (cities and towns). On the other hand, the dendrogram representing the politico-administrative structure is constructed based on the nested territorial subdivision of communes, provinces and regions, which are taken as hierarchical scales. To facilitate readability, we will refer to the governance dendrogram as $φ_{g}$ , and to the dendrogram of the urban structure resulting from the percolation as $φ_{s}$ . Both dendrograms have the same number of terminal clusters and use the same colour coding to indicate regional membership, which allows for easier comparison. For each dendrogram, let $N = {n_{1}, n_{2}, n_{3}, \dots, n_{k}}$ be the terminal clusters capturing cities and towns and $C_{τ} = {c_{1}, c_{2}, c_{3}, \dots, c_{k}}$ all the clusters at a specific threshold $τ$ .

Hierarchical comparison

To investigate the potential for scalar mismatch, we use three methods (Figure 1) to compare the dendrograms (represented by $φ$ ) in terms of their members and structure.

Figure 1.

Methods for hierarchical comparison in dendrogram toy models. (a) The hierarchical distance of clusters in dendrograms by computing the cophenetic distance $D$ . (b) The membership similarity between clusters by calculating the Jaccard similarity index $J$ . (c) The clustering diversity of dendrograms by measuring Shannon’s entropy $H_{τ}$ of the topological structure.

Cophenetic distance

The Cophenetic distance ( $D$ ) allows us to compare the intensity of the relationships between the elements of the dendrogram. It is defined as the level or threshold at which two terminal clusters $n_{i}$ and $n_{j}$ are joined together in the same cluster (Sokal and Rohlf, 1962). The larger the value connecting two clusters, the more distant they are from each other, and therefore the lower the potential for interaction between them. In Figure 1(a) we can see in $φ_{a}$ that terminal clusters $n_{1}$ and $n_{5}$ are first connected in the threshold $τ_{1}$ , hence $D (n_{1}, n_{5}) = 1$ . For the same pair in $φ_{b}$ , $D (n_{1}, n_{5}) = 3$ . This means that $n_{1}$ and $n_{5}$ are more strongly connected in the dendrogram $φ_{a}$ . By plotting the distances of every city to all other cities within both dendrograms we are able to track down the dissimilar tendencies in the structuring process of these systems.

Jaccard similarity

To compare the dendrograms, we use the Jaccard similarity index ( $J$ ) (Jaccard, 1912), defined as follows:

J (c_{a}, c_{b}) = \frac{c_{a} \cap c_{b}}{c_{a} \cup c_{b}}

where

c_{a}

is a cluster in

φ_{a}

and

c_{b}

is a cluster in

φ_{b}

, then

c_{a} \cap c_{b}

is the number of common terminal clusters

n_{i} \in N

to both clusters, and

c_{a} \cup c_{b}

is the total number of terminal clusters

n_{i} \in N

in the two clusters. This index calculates the proportion of elements that are common to two clusters, out of the total number of members in both. We compute

J

between all clusters from both dendrograms, based on their common terminal elements in

N

. This allows us to investigate how cities cluster together and whether they follow the same organisational logic in both systems. The higher the

J

value, the greater the similarity in the membership of the clusters. In Figure 1(b), we observe that the higher membership similarity between clusters in

φ_{a}

and in

φ_{b}

is given between

A \in φ_{a}

and

K \in φ_{b}

, with

J = 0.8

Structural entropy

This measure allows us to explore the agglomeration preferences between urban centres and to identify monocentric, polycentric or disperse clustering processes across scales. To compute it, we look at the topology of the dendrogram, which is defined as the branching pattern relationship among its elements (Phipps, 1971). We are interested in exploring the organisational heterogeneity of the topology given by the diversity of the furcation patterns in the tree. We achieve this, by computing at each threshold $τ$ the structural entropy $H_{τ}$ , which is a measure derived from Shannon’s entropy function (Shannon, 1948), that determines the uncertainty of a city to join an urban cluster at that scale. This is defined as follows:

H_{τ} = - \frac{1}{\log | C_{τ} |} \sum_{k \in τ} p_{k} \log p_{k}

where

p_{k}

is the probability defined as

p_{k} = \frac{| c_{k} |}{| N |}

| c_{k} |

is the cardinality (or size) of the cluster

c_{k}

given by the number of elements

n_{i} \in N

it has, and

| N |

is the total number of elements

n_{i} \in N

in the whole system.

| C_{τ} |

is the total number of clusters

c_{k}

τ

. The use of

\log | C_{τ} |

normalises the outputs at every threshold allowing the comparison across scales and between dendrograms. In Figure 1(c) we can see the value of

H_{τ}

for every dendrogram at every threshold. For example, at

τ = 2

the lowest value of

H_{τ}

is for

φ_{a}

and the highest for

φ_{c}

Results and discussion

In this section, we apply the methodology to Chile. This is an interesting case study, because it lies along an elongated territory, favouring longitudinal relations between cities, but possessing a contradicting centralised structure in terms of its administrative governance. The analysis highlights the mismatches between the imposed administrative structure and the natural organisation of the country emerging from the proxies for interactions.

Scales of organisation

Percolation outputs allow us to visualise the formation of different systems of cities and regions along the thresholding process (Figure 2(b)). Going from lower to higher time thresholds, that is from clusters with stronger to weaker ties, we observe that one of the first transitions generated at $τ = 14 s$ is in good correspondence with the urban extent of cities and towns. From now on the clusters arising at this threshold are considered the lowest scale of urban organisation and correspond to the terminal nodes $n_{i} \in N$ in the dendrogram. It is important to note that at this transition, the cluster sizes are evenly distributed along the national territory. As the threshold increases, we observe a consistent giant central cluster, which absorbs the neighbouring clusters in a radial manner. At $τ = 2.22 \min$ , we see that this corresponds to a central urban corridor containing the capital city Santiago. At $τ = 4.56 \min$ , this giant central cluster is now merged with almost all central urban areas, while other smaller clusters are still dispersed in the north and south without constituting agglomerations at regional scale. At the transition $τ = 40 \min$ , the northern area is finally merged into a region and appended to the rest of the country. At the end of the percolation process at $τ = 7 h$ , the southern and Patagonian clusters are connected to the system, forming one cluster that covers all the national territory.

Figure 2.

Chilean administrative and urban connectivity organisation. (a) Limits of the administrative jurisdictions, where the colours show the different regions, and the lines display the province limits. (b) Sample of clusters at relevant percolation thresholds, from the most local scale, corresponding to cities and towns, to more regional scales, until reaching the entire national cluster. To get a clearer sense of what these time thresholds represent in terms of distance, at $τ = 14 s$ , for example, clusters are formed by street intersections that are on average 14 s apart. This is approximately equivalent to 120 m (an average block length) at a driving speed of 30 km/h (typical on residential streets). Connectivity at the city-scale level is detected at this threshold. The following threshold of 2.22 min correspond to the connectivity of clusters at intra-regional scales. This could be equivalent to 2.8 km road segments length between intersections, when travelled at 100 km/h (typical speed on intra-regional highways). Colours represent the rank of the clusters’ size in terms of street network intersection count.

The process of percolation unveils an accentuated centralised tendency of cluster formation, with the pervasiveness of the capital city in the evolution of the clusters accompanied by a less prominent conformation of other sub-national structures. As evidenced by the different phase transitions, the system grows radially, from the centre where the capital is located, to the peripheral regions of the country. The predominance of the capital city in the spatial organisation of the national system is in good agreement with its administrative hegemony. Santiago not only accommodates almost half of the population, concentrates big part of the labour productivity and is responsible for 69% of the GDP growth of the country (OECD, 2017), but it also concentrates most of the administrative power.

Regional formation

Regions are made up of elements that are connected with varying levels of intensity and are distributed unevenly in space. While some regions are spatially cohesive as expected given their administrative boundaries, others are composed of isolated and more dispersed city systems. In this section, we compute the hierarchical distance between cities, to illustrate the diversity of intra-regional relationships.

First, we can identify different patterns of formation of city systems by looking at the dendrograms in Figure 3. The urban connectivity patterns in $φ_{s}$ differ greatly from the national jurisdictional structure in $φ_{g}$ , which does not show major configurational variation between regions. In the latter, the levels of hierarchy represent administrative scales (communes, provinces and regions), while in $φ_{s}$ , the levels represent scales of organisation based on intensities of physical connectivity. In Figure 3, for example, we can see that in $φ_{s}$ : 2, a group of cities from the same region are clustered together at early thresholds, indicating lower hierarchical distance between them. In $φ_{s}$ : 4, a group of cities from the same region clusters together at the very end of the percolation process, indicating higher hierarchical distances between them and a weak potential for connectivity with the rest of the system. In $φ_{s}$ : 3, we identify cities that later in the thresholding process are still isolated, failing to connect with other cities before being absorbed by the larger cluster.

Figure 3.

Dendrograms representing the administrative and the urban hierarchical organisation of Chile. (a) $φ_{s}$ corresponds to the dendrogram of urban clusters resulting from the percolation process on the street network. The line chart in (a) represents the size of the 10 larger clusters along the percolation process. Discontinuities given in specific time thresholds are demarcated and defined as the relevant scales of the hierarchical organisation, which are used then to construct the dendrogram. The largest cluster is displayed in red, with the nine other larger clusters appearing in grey. (b) $φ_{g}$ corresponds to the governance dendrogram extracted from the politico-administrative subdivision of the country. Both dendrograms start with the same terminal clusters $n_{i} \in N$ representing cities and towns, and end with the national level at the top. Colour code in both corresponds to the administrative regional membership.

These significant disparities in regional clustering patterns, can also be examined by looking at the percentage of clustered entities per region at each percolation threshold in $φ_{s}$ . In Figure 4(a), while most central regions are nearly fully clustered at $τ = 2.8 \min$ the most isolated systems reveal a different pattern, with only around 30% to 60% of the cities within each region clustering together. The sparseness of the street network in the most extreme regions of the country, both in the north and south, reflects the limited proximity and urban density in these areas. The continuity of urban space weakens in such regions, primarily due to the prevalence of rural landscapes and challenging geography, limiting inter-city connectivity and regional spatial cohesiveness.

Figure 4.

Regional clustering process and hierarchical distance D. In (a) the percentage of clustered urban entities along the percolation process per region. In (b) the mean regional cophenetic distances for the governance dendrogram $φ_{g} (D_{g})$ and the urban connectivity dendrogram $φ_{s} (D_{s})$ . Central regions (V, MR, VI, VII) clustered faster during the percolation process in (a) and have lower mean $D_{s}$ between regional elements in (b).

We measure the hierarchical distances between cities within each region using the cophenetic distance $D$ . In Figure 4(b), we compare the average regional cophenetic distance of the urban connectivity dendrogram $φ_{s} (D_{s})$ with the cophenetic distance from the governance dendrogram $φ_{g} (D_{g})$ . We see that the central regions have a lower $D_{s}$ average, while the extreme regions have higher values. In the case of $D_{g}$ , the results do not show a clear pattern. The distances between cities in this case depend on the number of cities per region and not much on the inner hierarchical structure.

In Figure 5, we plot the cophenetic distances between elements in the urban connectivity dendrogram $φ_{s} (D_{s})$ on the x-axis, and the distances in the governance dendrogram $φ_{g} (D_{g})$ on the y-axis. Each of the plots extracts the distances between cities per regions. The red points highlight distances up to the regional level, which correspond to distances between cities within the same region. The blue points show distances within the national level, corresponding to inter-regional relationships.

Figure 5.

Regional cophenetic distances (D) between dendrograms $φ_{g}$ and $φ_{s}$ . Each plot represents a single administrative region, displaying the results of the cophenetic distances from every city within the region to other cities in the same or different regions. Intra-regional distances are shown in red and inter-regional distances in blue. The y-axis corresponds to the cophenetic distance in dendrogram $φ_{g}$ $(D_{g})$ with the different governance levels, and the x-axis corresponds to the cophenetic distance in dendrogram $φ_{s} (D_{s})$ with the thresholds from the street network percolation. The analysis allows us to identify various regional typologies which are grouped in (a), (b), (c) and (d).

Based on the patterns of inter-regional relationships (in blue) and intra-regional relationships (in red) as provided by $(D_{s})$ and $(D_{g})$ we define different categories (a-d) depending on whether the distances between cities are, on average low, high or exhibit significant variability. For example, Figure 5(a) shows that in region II and III the lower $D_{s}$ are found between cities in the same communes, then between provinces and regions, following a natural order in both dendrograms. In these regions, the highest connectivity intensities are found between cities belonging to the same region (in red), and there is a much greater distance to the rest of the national system (in blue). In these cases, intra-regional distances (blue) are very similar and overlap. In VII and XIII, intra-regional and inter-regional $D_{s}$ are similar, regardless of the governance distances $D_{g}$ . This indicates that the regional system has a high level of connectivity with most of the cities in the system. In X and XIV, the $D_{s}$ distances between the elements in dendrogram $φ_{s}$ do not represent the same organisation as those in $φ_{g}$ . Some distances between cities within the same region are much higher than inter-regional distances. The intensity of connectivity in regions is not homogeneous, and in these cases some cities are less accessible to the connectivity of the regional system. Finally in XI and XII, inter- and intra-regional distances are both high, meaning low regional and national spatial connectivity. The results of our analysis show the diverse scalar patterns of regional connectivity. In some regions, the hierarchical distances between the cities are shorter (Figure 5(b)), which suggests a greater potential for spatial cohesion and alignment with the governance structure. This can facilitate communication and collaboration and make decision-making more efficient. However, in regions with higher hierarchical distances (Figure 5(a) and (d)), some cities may have less potential for regional interaction, and less access to resources and opportunities, which can hinder their development and lead to disparities within the region. Regions with uneven connectivity and fragmented city systems may struggle to develop a shared regional vision and goals, as is the case of the regions in Figure 5(c), making it difficult for decision-makers to address regional challenges and promote the development of the region as a whole.

Regional membership

The output of this section reveals that potential strong relationships between urban systems are not always determined by politico-administrative definitions and are rather driven by the dynamics of the urban systems themselves. The multiplicity of regional membership within clusters is measured using the Jaccard similarity $J$ , which provides insights into the mismatch between governance and the urban inner structural organisation.

By examining the dendrograms displayed in Figure 3, we can see that in $φ_{s} : 1$ , there is a cluster formed by cities from three different regions. In this case, cities from regions VI (light pink) and VII (pink) are first connected to cities in the Metropolitan Region (red) before connecting to other cities in their assigned regions. This is not the case in $φ_{g}$ , where all cities belonging to the same region are first clustered together. To better understand the patterns observed in the dendrograms, we quantify $J$ between all clusters from both dendrograms displayed in Figure 3. This allows us to identify in which regions systems of cities cluster in more or less accordance with their territorial delimitations. Figure 6 shows the value of the highest Jaccard similarity $J$ for each region and the threshold at which it is obtained.

Figure 6.

Regional Jaccard Similarity between all clusters from dendrograms $φ_{g}$ and $φ_{s}$ . In the x-axis the $J$ maximum values per region and in the y-axis the percolation threshold from $φ_{s}$ at which the maximum $J$ is obtained. $J$ is computed in (a) considering the number of cities and towns within clusters, and in (b) using as a weight the size of the clusters given their street intersections count.

The results in Figure 6 also show that most of the regional clusters have their highest $J$ values at similar thresholds. This suggests that between these thresholds, we can detect regional formations in the urban system. In Figure 6 $a$ , we use the number of terminal clusters $n_{i} \in N$ (without considering their size) to calculate the $J$ value. In Figure 6 $b$ , we use the number of intersections in the street network within the clusters as a weight. As a result, larger cities will have a greater weight than small towns. This change leads to greater differences in $J$ , providing more detailed information about potential mismatches.

As shown in Figure 6, the metropolitan region MR, where the country’s capital is located, has a high Jaccard similarity value at a very low threshold, indicating a good correspondence between the administrative organisation and the urban organisation resulting from the local interactions. On the other hand, regions XI and XII only reach high $J$ values at high thresholds. Cities in these regions are grouped following the governance regional membership, nevertheless, unlike the Metropolitan Region MR, these cities connect spatially very late in the percolation process. This can be explained by the fact that these regions are located in the extreme south of the country, where cities are spatially segregated due to the fragmented geography. In contrast to the previous examples, Region II has a very low Jaccard similarity value. This indicates that the cities belonging to this region do not tend to group into the same clusters. When examining Figure 3- $φ_{s}$ , we observe that the cities from Region II (light green colour), are not contained in the same mother cluster. This indicates a large mismatch between territorial governance and the urban system structure.

The results indicate varying degrees of mismatch between the organisation of regional memberships. This information allows us to better understand how decisions made at one level of governance can align with the relationships of urban systems beyond jurisdictional boundaries. This can improve our understanding of the effects of policies on targeted urban systems and potential impacts on larger scales. When the scales of governance and the systems being governed match, it is more likely that measures can be effectively implemented. Otherwise, there may be a need for greater coordination between regional and lower levels of administration.

Regional structure

‘In the new regions a new homogeneous and equivalent institutionality will be established’

– National Commission for Administrative Reform (CONARA) 1974

From the national dendrogram we extracted a sub-dendrogram for each administrative region, keeping the same thresholds (Figure 7). In this case, the hierarchical comparison is made between all the regional dendrograms in order to visualise and quantify the heterogeneity of their topological organisation. This allows us to contrast the logic of intra-regional structures that emerge from the urban connectivity patterns with the monocentric administrative approach, that is replicated uniformly across regions in the country. Results show that although there are some regions that follow the mono-hierarchical organisation based on a regional capital, there are other poly-hierarchical as well as dispersed regional organisations.

Figure 7.

Regional hierarchical organisations and structural entropy H. In (a) the sub-dendrograms from the different regions. The first column identifies mono-hierarchical regional structures, in the second column the poly-hierarchical regions and in the last column the dispersed hierarchical organisations. The $H$ is computed for the thresholds with the maximum $J$ . In (b) the plots from three different regions in the relevant thresholds. In both (a) and (b), regional capital cities are indicated by an asterisk (*) for reference.

Figure 7 shows the hierarchical dendrograms constructed for every region, where clusters of cities and towns $n_{i} \in N$ belonging to the same regional membership sit at the first scale of organisation. In monocentric systems, as is the case of regions MR, VI and X, the percolation process evince a centre of growth that aligns with the cluster of the regional capital city, which tends to assimilate single or scarcely clustered cities as the region develops. In polycentric structures, such as V, VIII and VII, two or more principal urban clusters develop in parallel, while in dispersed structures, see III, I and II, most cities and towns remain in isolation without forming systems of cities between regional constituent elements. In addition to comparing the topological differences between the dendrograms in Figure 7, we can also quantify these differences by measuring the structural entropy $H_{τ}$ . This can provide a more objective way to classifying the structures. For each of the thresholds in each dendrogram we measure the diversity of clustering patterns.

The values in Figure 7 correspond to the $H_{τ}$ in thresholds at which each region obtained its highest value of $J$ . In the case, when for a given threshold, a cluster contains most of the terminal clusters $n_{i} \in N$ , the system will exhibit a skewed probability distribution, indicating that if a city or town is taken at random, it will most likely belong to that cluster. In information-theoretic terms, a lower entropy value $H_{τ}$ indicates reduced uncertainty in clustering patterns. On the other hand, if the probability distribution of the $n_{i} \in N$ across clusters in a threshold is similar, the uncertainty in the clustering pattern will be higher and therefore the entropy $H_{τ}$ will also be higher. When aggregation patterns are equiprobable, the entropy will be at its maximum.

For example, in Figure 7(a), first column, we can see that in the dendrogram of the Metropolitan Region (MR) the system evolves in relation to the cluster that is initially defined by Santiago, the regional capital, which is also the country’s capital city. As the clustering process progresses, clusters of cities and towns are attached to this main cluster. At different phase transitions, the dominance of the largest cluster and the absence of other significant clustering formations reflect the monocentric structure, which is consistent with the regional governance structure. The predominance of the cluster that contains the regional capital, reduces the uncertainty of the system, and therefore $H_{τ}$ is also low, with 0.25 in this case. In contrast, in regions like Valparaiso (V), in the second column, the clustering process evolves in the form of two independent intermediate systems before a regional system is established.

This polycentric structure is a result of the low connectivity between coastal and valley cities, which is hindered by the Coastal Mountain Chain that tends to isolate both sides. The regional capital also named Valparaiso, is part of the cluster formed in the coastal side. In this case, there is more uncertainty about how cities are grouped, as there is no clear dominant cluster. Instead, different systems of cities are forming. The entropy of poly-hierarchical dendrograms is higher than that of mono-hierarchical structures, with $H_{τ}$ = 0.47 in Valparaiso. Finally, dendrograms in the last column show a more disperse furcation structure, where small clusters of cities are scattered during the whole process and the regional capitals do not function as the centre of the clustering. An example is the Atacama Region (III), where the thresholding process does not show phase transitions defining clusters formed by systems of cities, and rather cities remain isolated until merging with the largest cluster at the latest levels. In this case, the lack of dominant clustering leads to a higher entropy value of $H_{τ}$ = 0.81.

The variety of regional hierarchical structures could suggest the need for a more diverse system of governance that can take into account local particularities. The current regional administrative approach centred around a well-defined regional development hub or capital city, may not always align with the spatial organisation of regions, leading to a mismatch that can have negative effects on planning. The outputs of the analysis show some regions with a mono-hierarchical organisation which aligns well with the administrative logic, while others do not. Regions with more diverse and dispersed hierarchical structures may benefit from a more flexible, multi-scalar governance approach that takes into account the varying capacities of different cities to collaborate and influence development. This could help promote positive spill over effects and support the overall growth of the region.

Conclusions

In this paper, we employ several quantitative methods, including cophenetic distance, Jaccard similarity and structural entropy, to compare dendrograms from different regions. This enables us to investigate the mismatches between the physical urban connectivity and the geographical administrative structures. We begin by examining the hierarchical distance between elements, calculating the cophenetic distance. When there are mismatches, we observe regions where cities are widely distant in terms of their relational intensity, which can hinder their ability to function as a unified entity. In such cases, governance faces greater challenges in establishing common development goals and promoting equitable growth. Next, we examine the membership of the dendrograms by computing the Jaccard similarity. When mismatches are present, we may find cases where the administrative boundaries of regions do not align with the clustering of cities. This can lead to cities from different regions being grouped together with higher levels of connectivity rather than forming strong regional clusters first. This is important for governance, in order to understand the impact of regional policies at different levels. Finally, we analyse the regional organisational structure by measuring the structural entropy of the dendrograms. In cases where mismatches exist, the clustering process may not align with the administrative logic based on a single centre of regional development, resulting in polycentric rather than monocentric regional structures. This presents a major challenge for governance, as it highlights the need to include different regional development strategies, including polycentric structures.

In this work, we focused on the scalar composition of regions in Chile, so that territorial disparities can be addressed. We found that current regional management approaches often overlook local dynamics and the heterogeneities within city systems. Our analysis provides valuable insights into the hierarchical structure of urban systems and emphasises the importance of considering the scalar composition of regions in decision-making. We found that scalar mismatches suggest that a ‘one-size-fits-all’ approach to regional governance may not be effective, and that decision-makers should take into account the specific organisation of each region in order to promote the development of the region as a whole.

It is worth noting that the definition of clusters is influenced by aggregation criteria and the nature of the network, particularly time and street networks in our case. Therefore, conducting hierarchical clustering analysis with different chosen criteria and networks could capture different organisational structures of the system. While the hierarchical clusters resulting through percolation on street networks may not capture the overall system patterns, they undoubtedly serve as valuable proxies to enhance our understanding of urban dynamics and the potential interactions between urban settlements. In future work, we plan to expand our analysis to include functional networks, bringing together structure and flows. Exploring alternative measures for examining and comparing dendrogram structures, along with different hierarchical clustering approaches used in this paper, could offer valuable paths for future research.

Overall, this work contributes to advancing our understanding of the complexity of the spatial regional morphology from a scalar approach. The different methods for comparing dendrograms are crucial for a comprehensive understanding of scalar mismatch, providing a framework for studying the interplay between the administrative organisation and the structure that emerges from the spatial connectivity of the urban system.

Supplemental Material

Supplemental Material - The scalar mismatch of regional governance: A comparative analysis of hierarchical structures

Supplemental Material for The scalar mismatch of regional governance: A comparative analysis of hierarchical structures by Valentina Marin, Carlos Molinero and Elsa Arcaute in Environment and Planning B: Urban Analytics and City Science

Footnotes

Acknowledgements

VM thanks The National Research and Development Agency (ANID) for the financial support provided as a PhD scholarship Becas Chile.

Author contributions

VM developed the theoretical framework, designed the experiments and conducted the analysis. CM and EA contributed to the theoretical framework. VM wrote the paper, and all authors contributed to the structure and fine tuning of the manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported by The National Research and Development Agency of Chile (ANID).

Data Availability Statement

The street network data is sourced from OpenStreetMap (OSM) obtained from Geofabrik ().

Supplemental Material

Supplemental material for this article is available online.

Valentina Marin holds a PhD in Urban Complexity from the Centre for Advanced Spatial Analysis at University College London where she is a Research Assistant. With an architecture background and a Master of Science degree in Spatial Design: Architecture and Cities from The Bartlett School of Architecture at UCL, she specialises in using evidence-based methods to study cities and urban systems. Her areas of interest include modelling and analysing multi-scalar urban processes and interactions, with a focus on gaining a deeper understanding of urban hierarchies and resilience through quantitative analysis.

Carlos Molinero holds a Degree and Master in architecture from the Superior Technical School of Architecture of Madrid, Universidad Politécnica de Madrid and an M.Res. and Ph.D. in Computer Science from the School of Computer Science of the Universidad Complutense de Madrid. He was part of the Complexity Science Hub in Vienna as Senior Scientist, where he continued to expand his research in the field of urban science. Currently, he is a Honorary Research Fellow at the Centre for Advanced Spatial Analysis at the University College of London, where he worked as Research Associate. His research interests lay in the understanding of cities as evolving systems and their fractal nature.

Elsa Arcaute is a Professor of Complexity Science at the Centre for Advanced Spatial Analysis, University College London, and an Honorary Professor at Hong Kong University. She holds a masters and a PhD in theoretical physics from the University of Cambridge. Before joining UCL, she investigated processes of self-organisation in ant-colonies and social systems at the Complexity and Networks group at Imperial College. For the past 10 years, her research has focused on cities and urban systems in general, such as the emergence of hierarchies, the characterisation of resilience, the emergence of innovation and the modelling of coupled systems.

References

Agnew

(1997) The dramaturgy of horizons: geographical scale in the ‘Reconstruction of Italy’ by the new Italian political parties, 1992–1995. Political Geography 16(2): 99–121.

Arcaute

Molinero

Hatna

, et al. (2016) Cities and regions in Britain through hierarchical percolation. Royal Society Open Science 3(4): 150691.

Arenas

Hidalgo

Orellana

, et al. (2007) Propuesta de nuevos criterios para redefinir unidades político administrativas regionales en Chile. In Pontificia Universidad católica de Chile. Metropolitana Chile: Camino al Bicentenario. Propuestas para Chile, pp. 349–374.

Batty

(2018) Inventing Future Cities. Cambridge: The MIT Press, MIT Press.

Batty

Longley

(1994) Fractal Cities: A Geometry of Form and Function. Cambridge: Academic Press.

Behnisch

Schorcht

Kriewald

, et al. (2019) Settlement percolation: a study of building connectivity and poles of inaccessibility. Landscape and Urban Planning 191: 103631.

Brenner

(2001) The limits to scale? Methodological reflections on scalar structuration. Progress in Human Geography 25(4): 591–614.

Bretagnolle

Daude

Pumain

(2006) From theory to modelling : urban systems as complex systems, Cybergeo. Revue europeenne de geographie/European journal of geography 335: 1–17.

Bulkeley

(2005) Reconfiguring environmental governance: towards a politics of scales and networks. Political Geography 24(8): 875–902.

10.

Cavallaro

Asprone

Latora

, et al. (2014) Assessment of urban ecosystem resilience through hybrid social-physical complex networks. Computer-Aided Civil and Infrastructure Engineering 29(8): 608–625.

11.

CONARA (1974) Acta N 119-a in Actas de la Junta de Gobierno 1973-1990.(Comision Nacional de la Reforma Administrativa), Diarios de Sesiones del Congreso Nacional. Manchester: BCN.

12.

Cumming

DHM

Redman

(2006) Scale mismatches in social-ecological systems: causes, consequences, and solutions. Ecology and Society 11(1): art14.

13.

Cumming

Olsson

Chapin

, et al. (2013) Resilience, experimentation, and scale mismatches in social-ecological landscapes. Landscape Ecology 28(6): 1139–1150.

14.

Fischer

(1980) Regional taxonomy: a comparison of some hierarchic and non-hierarchic strategies. Regional Science and Urban Economics 10(4): 503–537.

15.

Fluschnik

Kriewald

Ros

AGC

, et al. (2016) The size distribution, scaling properties and spatial organization of urban clusters: a global and regional percolation perspective. ISPRS International Journal of Geo-Information 5(7): 1–14.

16.

Folke

Hahn

Olsson

, et al. (2005) Adaptive governance of social-ecological systems. Annual Review of Environment and Resources 30(1): 441–473.

17.

Fowlkes

Mallows

(1983) A method for comparing two hierarchical clusterings. Journal of the American Statistical Association 78(383): 553–569.

18.

Gallos

Makse

Sigman

(2011) A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks. Proceedings of the National Academy of Sciences of the United States of America 109(8): 2825–2830.

19.

Gallos

Barttfeld

Havlin

, et al. (2012) Collective behavior in the spatial spreading of obesity. Scientific Reports 2: 454–459.

20.

Holling

(2001) Understanding the complexity of economic, ecological, and social systems. Ecosystems 4(5): 390–405.

21.

Hudson

(2007) Regions and regional uneven development forever? Some reflective comments upon theory and practice. Regional Studies 41(9): 1149–1160.

22.

Jaccard

(1912) The distribution of the flora in the alpine zone. New Phytologist 11(2): 37–50.

23.

Jonas

(1994) The scale politics of spaliality, Environment and Planning D: Society and Space 12: 257–264.

24.

Lebel

Anderies

Campbell

, et al. (2006) Governance and the capacity to manage resilience in regional social-ecological systems. Ecology and Society 11(1): art19.

25.

Lukas

Flitner

(2019) Scalar fixes of environmental management in Java, Indonesia. Environment and Planning: Nature and Space 2(3): 565–589.

26.

MacKinnon

(2011) Reconstructing scale: towards a new scalar politics. Progress in Human Geography 35(1): 21–36.

27.

Marshall

Gil

Kropf

, et al. (2018) Street network studies: from networks to models and their representations. Networks and Spatial Economics 18(3): 735–749.

28.

Marston

(2000) The social construction of scale. Progress in Human Geography 24(2): 219–242.

29.

Marston

Paul

Iii

, et al. (2016) Human Geography without Scale 30(4): 416–432.

30.

Moore

(2008) Rethinking scale as a geographical category: from analysis to practice. Progress in Human Geography 32(2): 203–225.

31.

Newman

MEJ

Jensen

Ziff

(2002) Percolation and epidemics in a two-dimensional small world. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 65(2): 021904.

32.

OECD (2017) Making Decentralisation Work in Chile, OECD Multi-level Governance Studies. Paris, France: OECD.

33.

Phipps

(1971) Dendrogram topology. Systematic Zoology 20(3): 306–308.

34.

Piovani

Molinero

Wilson

(2017) Urban retail location: insights from percolation theory and spatial interaction modeling. PLoS One 12(10): e0185787.

35.

Pumain

(2006a) Alternative explanations of hierarchical differentiation in urban systems. In: ‘Hierarchy in Natural and Social Sciences. Berlin, Germany: Springer Science and Business Media, 169–222.

36.

Pumain

(2006b) Hierarchy in Natural and Social Sciences. Berlin, Germany: Springer Science and Business Media.

37.

Rangan

Kull

(2009) What makes ecology ’political’ ? rethinking ’scale’ in political ecology. Progress in Human Geography 33(1): 28–45.

38.

James Rohlf

(1982) Consensus indices for comparing classifications. Mathematical Biosciences 59(1): 131–144.

39.

Rose

(1999) Governing. Cambridge: Cambridge University Press, 15–60.

40.

Rozenblat

(2009) European urban polycentrism: a multiscale typology. Geographica Helvetica 64(3): 175–185.

41.

Rozenblat

(2022) Cities’ systems and networks’ proximities: toward a multiplex approach. In: Handbook of Proximity Relations. Cheltenham: Edward Elgar Publishing, 1–26.

42.

Rozenfeld

Rybski

Andrade

, et al. (2008) Laws of population growth. Proceedings of the National Academy of Sciences of the United States of America 105(48): 18702–18707.

43.

Rozenfeld

Rybski

Gabaix

, et al. (2011) The area and population of cities: new insights from a different perspective on cities. The American Economic Review 101(5): 2205–2225.

44.

Saracli

Dogan

(2013) Comparison of hierarchical cluster analysis methods by cophenetic correlation, Journal of Inequalities and Applications 2013: 203–208.

45.

Shannon

(1948) A mathematical theory of communication. Bell System Technical Journal 27(3): 379–423.

46.

Simon

(1962) The architecture of complexity. Proceedings of the American Philosophical Society 106(6): 467–482.

47.

Simon

(1977) The organization of complex systems, models of discovery. Boston Studies in the Philosophy of Science 54: 245–261.

48.

Smith

(1992) Geography, Difference and the Politics of Scale. London: Palgrave Macmillan UK, 57–79.

49.

Sokal

Rohlf

(1962) The comparison of dendrograms by objective methods. Taxon 11(2): 33–40.

50.

Strano

Nicosia

Latora

, et al. (2012) Elementary processes governing the evolution of road networks. Scientific Reports 2: 296–298.

51.

Terlouw

(2012) From thick to thin regional identities? Geojournal 77(5): 707–721.

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