A persistent homology model of street network connectivity

We propose a novel model of street network connectivity which uses a method from the field of applied topology called “persistent homology”. The output from this model is a pair of density functions which model the relative strength and frequency of connected components and cycles in the network. In this context, strength is a function of street type, such as motorway or residential, with more significant street types providing greater connectivity. The pair of density functions output from the model are easily interpreted and provide novel insights into the connectivity properties of different street networks. We demonstrate the usefulness of this model through an analysis of UK and US city street networks. This analysis identifies tangible similarities and differences in the connectivity of different cities plus ways in which the connectivity of individual cities might be improved.

a given street network is difficult to formally define, let alone measure. However, one important feature of any high-quality street network is a high level of connectivity. Tal and Handy (2012) define street network connectivity as a measure of the quantity of the connections in the network and in turn the directness and multiplicity of routes within the network. It has been demonstrated both theoretically and empirically that street networks with a greater degree of connectivity exhibit greater transportation efficiency (Knight & Marshall, 2015).
Existing models of street network connectivity are mainly based on summary statistics of simple geometrical or topological features of the network, for example the mean number of intersections per square mile/kilometre.
In this work we propose a novel model of street network connectivity where connectivity is modelled in terms of connected components and cycles in the network. These features in turn model the existence and multiplicity of connections between locations, respectively. To illustrate this, consider the simple network shown in Figure 1a.
which contains two locations a and b. The two locations form two distinct connected components and therefore the connectivity between the locations is poor. If a single edge is added between the locations to form a single connected component, as illustrated in Figure 1b, the connectivity between the locations is greater than that in the previous figure. If a second edge is added between the locations to form a cycle, as illustrated in Figure 1c, the connectivity is again greater. Finally, if a third edge is added between the locations to form another cycle, as illustrated in Figure 1d, the connectivity is again greater. Therefore, if all edges provide equal connectivity, the connectivity of a network can be modelled in terms of the number of connected components and cycles in the network.
However, in the context of a street network all edges do not provide equal connectivity. Instead, the strength of connectivity provided by an edge is a function of its semantics or type, which varies. For example, an edge corresponding to a motorway provides greater connectivity than an edge corresponding to a secondary street.
Therefore, when modelling connectivity in terms of connected components and cycles, it is necessary to consider the types of streets which compose these features. To achieve this goal the model proposed in this work uses a method from the field of applied topology called "persistent homology" (Edelsbrunner & Harer, 2010). The output from this model is a pair of density functions which model the relative frequency and strength of connected components and cycles in the street network. These two density functions are easily interpreted and provide novel insights into the connectivity properties of the street network in question.
The layout of this article is as follows. Section 2 reviews related works on modelling street network connectivity and applied topology. Section 3 describes the proposed model of street network connectivity. Section 4 demonstrates the usefulness of this model with respect to interpreting and providing insight into the connectivity properties of a given street network, recommending ways to improve this connectivity, and identifying street networks with similar and dissimilar connectivity properties. This analysis is performed with respect to sets of UK and US city street networks. Finally, Section 5 draws conclusions from this work and discusses possible directions for future research.

| REL ATED WORK S
In this section we review related works on modelling the connectivity of city street networks. We focus exclusively on the problem of modelling intra-city connectivity, as opposed to inter-city connectivity. A review of works in the latter category can be found in Mansury and Shin (2015). In this section we also briefly mention some related works on applied topology.
Connectivity is a well-studied concept in graph theory (Gross, Yellen, & Zhang, 2013). Since street networks are naturally modelled as graphs, graph-theoretic measures of connectivity can easily be applied to street networks.
Two commonly used such measures are vertex and edge connectivity, which respectively equal the minimum number of vertices and edges whose removal disconnects the graph. Other graph-theoretic measures include alpha ( ), which equals the number of cycles in the graph divided by the maximum possible number of cycles; beta ( ), which equals the number of edges divided by the number of vertices; and gamma ( ), which equals the number of edges divided by the maximum possible number of edges (Weber, 2016). Despite the fact that these measures are commonly applied to street networks (Sahitya & Prasad, 2020), they are limited by the fact that they do not consider the geometrical or spatial nature of street networks. Consequently, many measures of connectivity have been proposed specifically for street networks. Dill (2004) defined the following set of street network connectivity measures drawn from multiple fields: maximum city block length; maximum city block size; city block density, which equals the mean number of city blocks per square mile/kilometre; street intersection density, which equals the mean number of intersections per square mile/kilometre; street density, which equals the mean total length of street segments per square mile/kilometre; connected node ratio, which equals the number of street intersections divided by the number of intersections plus culs-de-sac; link node ratio, which equals the number of street segments divided by the number of intersections and culs-de-sac; grid pattern, which is a binary indicator of whether the street network exhibits a grid pattern commonly associated with high connectivity; route directness, which equals the mean network distance divided by straight-line distance; and effective walking area, which equals the mean percentage of land parcels within a specified network distance of a given location.
The US Green Building Council proposed a street intersection density measure which is related to that defined by Dill (2004) but uses some heuristics when counting the number of intersections (Stangl & Guinn, 2011).
Specifically, intersections leading to isolated areas and intersections in these areas are not counted. Stangl and Guinn (2011) subsequently highlighted some limitations with this measure and instead proposed a measure similar to the route directness proposed by Dill (2004). Stangl (2012Stangl ( , 2019 later extended this measure by integrating a number of heuristics to make the measure more robust. Peponis, Bafna, and Zhang (2008) proposed two measures of street network connectivity called "metric reach" and "directional distance". Metric reach equals the mean network length reachable from a given location in the network. Directional distance equals the mean number of direction changes required to navigate between two locations in the network. In a related work, Ellis et al. (2016) evaluated a number of street connectivity measures.
This included a novel measure called "directional reach" which equals the metric reach proposed by Peponis et al. (2008) with the addition of a constraint on the number of possible direction changes. Directional reach is related to the concept of visual connectivity whereby locations which require fewer turns to navigate between are considered more connected (Hajrasouliha & Yin, 2015). Knight and Marshall (2015) evaluated three measures of street network connectivity called "intersection density", "street density" and "connectivity index". The first two measures are equal to those of the same name proposed by Dill (2004). The connectivity index equals the number of street segments divided by the number of intersections. The authors found all three measures to be correlated with irrelevant features of area and geometry, and therefore not to be reliable measures of connectivity. Stangl (2015) examined a number of the city-block-based connectivity measures defined by Dill (2004). The author identified some limitations of these measures and proposed an alternative block-based connectivity measure. The measure in question is called "block section" and equals the mean maximum straight-line distance between any two points on the boundary of an area enclosed by streets.
In this article we propose a novel model of street network connectivity based on persistent homology. Feng and Porter (2020) previously used persistent homology to model street networks. However, their model does not model connectivity but instead models the shape of the areas enclosed by streets. There exist a number of previous works which also considered the application of methods from the field of applied topology to geographical data. Ahmed, Fasy, and Wenk (2014) proposed a model for determining local differences between street networks using persistent homology. This model is used to recognize changes in street networks over time and to assess the quality of map construction algorithms. Dey, Wang, and Wang (2017) proposed a model for inferring a street network from GPS data using discrete Morse theory. Corcoran and Jones (2016, 2017 proposed models of spatial temporal phenomena based on persistent homology. Corcoran (2019) proposed a method for performing generalization of geographical data using persistent homology.

| MODEL OF S TREE T NE T WORK CONNEC TIVIT Y
As described in the introduction to this article, we propose a novel model of street network connectivity where connectivity is modelled in terms of the relative strength and frequency of connected components and cycles. This is achieved by applying a method from the field of applied topology called "persistent homology" to the street network in question.
To motivate the use of this method, consider the street network in Figure 2 corresponding to the city of Manchester, UK, which forms a running example in this section. Figure  The above connectivity analysis of each street type in isolation gives a potentially misleading local perspective of the overall street network connectivity. For example, although the secondary network has a relatively low level of connectivity when considered in isolation, when considered in the context of other street types to which it is connected, it may have a relatively high level of connectivity. That is, the many connected components in this network may be strongly connected by streets of different types. Therefore, it is necessary to perform a more global connectivity analysis which considers all street types jointly. A naive solution would be to compute a simple union of all street types and perform a connectivity analysis of the resulting network. However, this approach fails to model street type, which is an important feature of connectivity. For example, despite the fact that the motorway network contains only a single cycle, this single cycle can potentially provide greater connectivity than the large number of cycles in the trunk network. In this work we propose a novel model of street network connectivity which uses persistent homology to overcome this challenge and jointly consider all street types in an appropriate manner.
The proposed model contains the following four computational steps. In the first step, a graph-based representation of the input street network is constructed. In the second step, a representation of this graph called a "filtration" is constructed where this representation encodes street type information. In the third step, the persistent homology of this filtration is computed which returns a pair of mathematical objects called "persistence diagrams".
The first persistence diagram models the number and strength of connected components, while the second persistence diagram models the number and strength of cycles. In the final step, these persistence diagrams are transformed into a corresponding pair of density functions which can be easily interpreted. This contrasts with existing models of street network connectivity, reviewed in Section 2, which output a single value indicating the level of connectivity. The density functions output from the proposed model provide greater information and in turn insight into the connectivity of the street network in question.
In the following four subsections we describe each of the above four computational steps in more detail. Note that the descriptions in these subsections do not venture too deeply into the corresponding underlying mathematics. This presentation is intended to make the content accessible to non-mathematicians and more specifically F I G U R E 2 The street network for the city of Manchester, UK, where streets are represented by black lines those without a working knowledge of algebraic topology. We advise a reader seeking a more in-depth description to consult the corresponding references in each subsection or the textbooks by Edelsbrunner and Harer (2010) and Ghrist (2014).

| Graph construction
In this step a graph-based representation of the input street network is constructed. Towards this goal, the corresponding street network data are obtained from OpenStreetMap (OSM), which is a crowdsourcing project for geographical data (Boeing, 2017). The set of street types in OSM ordered from most to least significant are motor- way, motorway_link, trunk, trunk_link, primary, primary_link, secondary, secondary_link, tertiary, tertiary_link, unclas-sified and residential. This ordering is specified by OSM on the OSM wiki which also contains a short description of each street type (https://wiki.opens treet map.org/wiki/Key:highway). A street type containing link in its title refers to a type of street which has a short length and connects two streets. For the purposes of this work, all streets of such a type are assigned their corresponding parent type. For example, a street of type motorway_link is assigned the type motorway. Given this, we define the set T to be the set of OSM street types less those which contain link in their title.
Given OSM data corresponding to the input street network, we represent the street network as an edge la- where V is the set of vertices corresponding to road intersections and dead-ends, E is the set of edges corresponding to street segments connecting these vertices, and L is a mapping from edges to the street type of the corresponding road segment (Corcoran & Mooney, 2013). The graph representation corresponding to the Manchester street network is shown in Figure 2 where street segments and hence graph edges are represented by black lines. Figure 3 displays subgraphs of this street network containing only those edges with type equal to motorway, trunk, primary, secondary, tertiary and unclassified.

| Filtration construction
Given a graph representation G = (V, E, L: E → T) of the input street network, in this step we construct a filtration of G which is a representation that encodes street type information.
A filtration of G is a sequence of n + 1 graphs G 0 , G 1 , …, G n which satisfy: where ⊆ is the subgraph relation. Towards constructing a filtration of graph G, we first define a filter function f: E → ℝ as follows: This function maps each edge in G to a real value where edges corresponding to more significant street types are mapped to smaller values.
Let ℰ denote the set of subsets of E, and denote the set of subgraphs of G. Let S: ℰ → denote the map from a subset of E to the corresponding edge induced subgraph -that is, a graph containing only that subset of edges Let C: ℝ → be the map defined by: which returns sub-level sets of G which are subgraphs of G. For example, the subgraphs C(1), C(2), C(3), C(4), C(5) and C(6) corresponding to the Manchester street network are shown in Figure 4. Given the map C, we define a filtration of G to be the sequence of n + 1 graphs G 0 , G 1 , …, G n where G i = C(i). The first five graphs in this filtration for the Manchester street network are the null graph followed by the subgraphs shown in Figure 4.
The above filtration definition has the property that edges corresponding to more significant street types appear earlier and persist for longer in the sequence of graphs. For example, edges corresponding to motorways appear in G 1 and persist for n graphs of the sequence. On the other hand, edges corresponding to primary appear in G 3 and persist for n − 2 graphs of the sequence.

| Persistent homology computation
As stated in the motivation at the beginning of this section, we wish to model street network connectivity in terms of connected components and cycles while considering that more significant streets provide a greater level of connectivity. We achieve this by computing the persistent homology of the filtration described in the previous subsection. Broadly speaking, this computation will model the existence and strength of connected components and cycles in the network. More specifically, persistent homology computes a pair of mathematical objects called "persistence diagrams". A persistence diagram is a multiset of pairs of points in the extended real line (ℝ ∪ ∞) which satisfy the condition that p < q for each element (p, q) (Edelsbrunner & Harer, 2010). One of the persistence diagrams models the existence of connected components in the filtration. Specifically, an element (p, q) in this persistence diagram indicates that a connected component appeared and subsequently disappeared at the graphs G p and G q respectively in the filtration. The other persistence diagram models the existence of cycles in the filtration.
Specifically, an element (p, q) in this persistence diagram indicates that a cycle appeared and subsequently disappeared at the graphs G p and G q respectively in the filtration. If a connected component disappears it is because it has merged with another connected component. As explained below, in this study cycles always persist to the end of the entire filtration; if a cycle merges with another cycle it will result in a further cycle in addition to itself. Note that, for a given element (p, q) in a persistence diagram, the value q − p is known as the persistence of the element in question. The persistence can be regarded as the lifetime of the respective connected component or cycle which first appears at duration or "time" point p and disappears at time point q. In the literature on persistence diagrams p and q are also referred to as points of birth and death of the respective connected component or cycle.
If two connected components merge, causing exactly one of them to disappear, the one which appeared latest in the filtration is the one which disappears. This is called the elder rule (Otter, Porter, Tillmann, Grindrod, & Harrington, 2017). If a connected component or cycle appears at graph G p in the filtration but does not subsequently disappear it is represented by an element (p, ∞). In this work we replace all elements of the form (p, ∞) with (p, u) where u is an upper bound. We use a value of 8 for this upper bound, which is a valid upper bound because G 7 is the final element in the filtration. Note that performing such a replacement is a commonly used approach for dealing with the challenges a value of ∞ presents (Adams et al., 2017). Since the filtration in this work is constructed by only adding edges, any cycles which appear will not disappear. Therefore, the second value of all elements in the cycle persistence diagram will have a value equal to u.
The utility of these two persistence diagrams can be understood by considering that connected components and cycles which appear earlier and persist for longer are connected by more significant streets and in turn are (e) (f) Figure 4a contains three connected components. However these become connected to other components in the next graph of the filtration in Figure 4b and in turn these connections persist for a large fraction of the filtration, indicating a high level of connectivity between the connected components in question. In contrast, if components that appear early in filtration do not quickly become connected by other streets, this indicates a poorer level of connectivity, reflected by a larger number of components which persist for longer. We next consider the case of cycles. The greater the number of cycles which persist for longer, the better connected the corresponding street network is. To illustrate this, consider again the filtration shown in Figure 4 for the Manchester street network.
A large number of cycles appear early in the filtration in Figure 4b and hence persist for a larger fraction of the filtration, as (unlike connected components) once they appear they cannot subsequently disappear. This indicates that these cycles provide a high level of connectivity. A large number of cycles also appear later in the filtration in To overcome this difficulty we propose an alternative representation of persistence diagrams called "per- where |D| denotes the number of elements in D (i.e. the number of connected components or cycles). Figure 6 displays the connected component and cycle persistence densities for the Manchester street network represented using histograms. The x-axis in these histograms represents the interval between appearance and disappearance (i.e., the period of persistence). The y-axis represents the corresponding density of these persistence values as defined by Equation (4). These histograms model the relative frequency of different persistence values and can be easily interpreted. For example, one can see in Figure 6a that for the street network in question, most connected components have a persistence value of 1.
The proposed persistence density representation is similar to the persistence image representation proposed by Adams et al. (2017). Broadly speaking, a persistence image is an image representation of a persistence diagram where pixel values equal a weighted sum of nearby elements. After much experimentation, the authors decided to use the proposed persistence density representation instead of the persistence image representation for the following reasons. Firstly, the authors found it difficult to visually differentiate between different persistence images because most contain peaks at the same set of locations where the heights of these peaks are in many cases not significantly different. On the other hand, since persistence densities are visualized using histograms it is much easier to visually differentiate between them. Secondly, the persistence image representation requires that a scale or smoothing parameter value be specified. It is not clear how best to select this parameter value. On the other hand, the persistent density representation does not require any parameter values to be specified.
To facilitate the application of data mining methods, it is useful to define a distance or metric between different street networks represented using the proposed model. There are a number of distance measures on the space of persistence diagrams such as the bottleneck and Wasserstein distances (Edelsbrunner & Harer, 2010). However, these distances are biased with respect to the number of elements with significant persistence in the corresponding persistence diagrams. That is, any two persistence diagrams having significantly different numbers of elements with significant persistence will generally have a large corresponding distance. Since larger cities will have graph representations with a greater number of edges, their corresponding persistence diagrams will have a correspondingly greater number of elements with significant persistence. This in turn results in a bias where cities with similar sized street networks are determined to be more similar. This is undesirable because we would prefer a distance measure which is a function of street network connectivity alone. To overcome this challenge we instead use the Wasserstein distance on the space of persistence densities (Peyré & Cuturi, 2019). Intuitively, this distance measures the cost of transforming one density function into another. Since the density function is normalized such that its codomain sums to 1.0, this provides a distance measure which is unbiased with respect to street network size.
F I G U R E 6 (a) Connected component; and (b) cycle persistence densities for the Manchester street network. The vertical axis represents density values, while the horizontal axis represents the interval between appearance and disappearance (the persistence value)

| CIT Y CONNEC TIVIT Y ANALYS IS
This work proposes a novel model of street network connectivity. The usefulness of any model can only be defined with respect to performing a given task. In this section we demonstrate the usefulness of the proposed   The remainder of this section is structured as follows. In Section 4.1 we present details of the UK and US street networks considered and the process used to construct these networks. In Section 4.2 we demonstrate the usefulness of the proposed model with respect to performing the downstream tasks mentioned above.

| Street networks
The first set of street networks we considered is a set of street networks corresponding to all UK cities, of which there are 69. All UK cities have official boundaries (https://data.gov.uk/datas et/7879a b82-2863-401e-8a29-a56e2 64d21 82/major -towns -and-citie s-decem ber-2015-bound aries). However, we found these boundaries to be inconsistent with respect to what they contained. For example, some city boundaries contain the corresponding suburban street network, while others do not. This is a consequence of the fact that many city boundaries were defined before the street network in question fully developed and are a function of political influences. To overcome this challenge, for each UK city we selected a location in the city centre and extracted the street network within a 10 km bounding box centred at this location. We found this approach to consistently return the desired street network.
The second set of street networks we considered is a set of street networks corresponding to a subset of US cities. In order to have a representative sample of cities we used the subset of cities proposed by Angel and Blei (2016). In this work the authors used a random stratified sampling procedure to select a subset of 40 cities from the set of all 242 USA cities that had populations of 100,000 or more in the year 2000. We defined the boundary of each city to be the boundary of the corresponding Metropolitan Statistical Area (MSA) in which it is contained (Arribas-Bel & Sanz-Gracia, 2014). MSAs are defined by the U.S. Office of Management and Budget as core areas containing a substantial population nucleus, together with adjacent communities having a high degree of economic and social integration with that core (https://www.census.gov/progr ams-surve ys/metro -micro/ about.html).
The street networks for both the UK and US cities were obtained from OpenStreetMap, which is a crowdsourcing project for geographical data, using the OSMnx software library (Boeing, 2017). Tables 1 and 2  left indicates a greater level of connectivity. Specifically, such skewness indicates that most elements in the corresponding persistence diagram have smaller persistence. That is, most parts of the street network get connected earlier in the filtration where these connections correspond to more significant streets. On the other hand, a cycle persistence density which is more skewed to the right also indicates a greater level of connectivity. Specifically, such skewness indicates that most elements in the corresponding persistence diagram have larger persistence.
That is, different parts of the street network get connected by multiple paths that create cycles (i.e. circuits), earlier in the filtration where these paths correspond to more significant streets.
To illustrate the insights which this analysis can provide with respect to street network connectivity properties, consider the connected component and cycle persistence densities for the UK cities of Cardiff, Manchester and Birmingham shown in Figure 9. have significantly more cycles with persistence values of 6 and 7. This is a consequence of the fact that, unlike Cardiff, both Manchester and Birmingham are inland cities with significant roads surrounding them. This fact is illustrated in Figure 10 which displays the set of streets with type equal to motorway or trunk for the cities of Cardiff, Manchester and Birmingham. Motorway and trunk are the most significant street types, and we can see that Manchester and Birmingham each contain more cycles formed from these types than Cardiff.
To contrast the above analysis with that which can be achieved using traditional models of street network connectivity, consider Table 3 which displays the intersection density for each UK city. The intersection density for a given street network equals the mean number of street network intersections per square kilometre. This is a commonly used model of street network connectivity, where greater density is considered to indicate greater connectivity (Dill, 2004 Figure 18. Both these cities have dissimilar connected component persistence densities to those of Birmingham and Liverpool. Specifically, they are more skewed to the right. This can be attributed to the fact that both are smaller less significant cities with a lower number of significant connecting streets. In fact, Armagh is an extremely small city, and this is reflected in the relative size of its graph representation shown in Table 1. This analysis indicates that the connectivity of Armagh and Brighton could be improved by adding more significant connecting streets.

(a) (b)
In order to compare the connectivity of the UK and US street networks we computed the mean connected component and cycle persistence densities for the corresponding sets of street networks. These mean persistence densities are shown in Figure 19. The mean UK connected component persistence density is more skewed to the left than that for the USA. Furthermore, the mean UK cycle persistence density is more skewed to the right than that of the USA. These two facts indicate that, on average the UK street networks have a greater level of connectivity than the US street networks.
As discussed in Section 4.1, in many cases the boundary of a city is ambiguous and difficult to define. To determine if the proposed model of connectivity is sensitive to the choice of city boundary we computed the connected component and cycle persistence densities for the city of Manchester for three different boundaries.
Specifically, we considered boundaries corresponding to the 3.333, 6.667 and 10 km bounding boxes centred at the city centre. Figure 20  defining this. Note that most models of street network connectivity also exhibit this sensitivity. For example, the intersection density values corresponding to the above bounding boxes are 494, 360 and 311, respectively. That is, as the bounding box increases the intersection density decreases. This is evident from viewing the Manchester city street network in Figure 2 where we see that the density of streets is greatest in the city centre.
We extended the above experiment to the set of all UK cities by computing the mean persistence densities for these cities for 3.333, 6.667 and 10 km bounding boxes. These persistence densities are shown in Figure 21. We see that as the bounding box size increases, the mean connected component persistence density becomes slightly more skewed to the left, indicating a greater level of connectivity. Furthermore, we see that as the bounding box size increases, the mean cycle persistence density becomes more skewed to the right, also indicating a greater level of connectivity. Both these results are expected because more significant street types and in turn connections generally exist outside the centre of a city. This analysis provides a form of validation for the proposed model of street network connectivity.

| CON CLUS IONS
In this article we propose a novel model of street network connectivity which is distinct from existing models in a number of ways. Firstly, in the proposed model connectivity is modelled in terms of the relative strength and frequency of connected components and cycles in the network. In existing models connectivity is generally modelled in terms of summary statistics of simple geometrical or topological features of the network. Secondly, the proposed model considers the type of different streets and different levels of connectivity they provide. The authors are unaware of any existing models of connectivity which consider such information. Finally, the proposed model does not represent connectivity using a single number. Instead connectivity is represented using a richer representation, specifically a set of persistence densities, which can be used to gain novel insights into the connectivity properties of different cities. Despite being a somewhat more complex representation, persistence densities can still be easily interpreted. This is particularly important given that the planning practice has traditionally relied on connectivity measures that can be intuitively understood and/or easily applied (Stangl & Guinn, 2011).
Despite the above achievements, the proposed model of connectivity has some limitations which provide opportunities for future research. Firstly, when attempting to model the connectivity of a given city, the proposed model is sensitive to the choice of corresponding boundary. One potential solution to this issue would be not to pick a single boundary but to consider all possible boundaries using some form of multidimensional persistent homology (Carlsson, Singh, & Zomorodian, 2009). Another approach is to provide more consistent methods of defining region types with regard, for example, to their inclusion or otherwise of outlying suburban residential areas and measures of population density and of employment (Arribas-Bel & Sanz-Gracia, 2014).
Secondly, the proposed model is solely a function of the corresponding street network. However, in reality connectivity is a function of many other factors including population size and spatial distribution (Mansury & Shin, 2015). For example, if a large percentage of the city population live in the suburbs but work in the city centre then the importance of connectivity between these two types of region must be weighted more highly.