Several studies have been conducted on rationalising urban and resilient communities with quantitative measures and frameworks, such as, Caputo et al. (2015); Nist (2015); Sharifi (2016); Jankovic (2018); Nguyen and Akerkar (2020) provided a critical overview on tools that can be used to assess the resilience of communities, defined as CRA (Community Resilience Assessment) tools. Sharifi analysed 36 different tools with different scope and geographical application, with 15 out of 36 developed for the north American context (Sharifi, 2016:634). One of the relevant considerations resulting from Sharifi’s work is that CRA tools in general do not consider the impact of change and the dynamic nature of resilience (as a reaction from communities) across time and geographical scales. Furthermore, that resilience is a moving target that should be viewed alongside a degree of uncertainty (Sharifi. 2016:644). A positive contribution toward the points raised by Sharafi’s work has been made by Petrescu and colleagues. Their R-Urban, a framework generated to “mobilize the agency of citizens and grassroots organizations via a network of self-managed collective hubs in a transition to dynamic sustainability” (Petrescu et al., 2020:2). This framework proposes a way to evaluate the resilience in communities and their capital worth, quantifying impact and economic growth in urban communities.

In general terms, the majority of the studies to date refer to a framework (or a combination of them) where many aspects, from socio-cultural to economic, are considered. Successful examples of such frameworks developed by global corporations and charities include the Oxfam GB Multi-Dimensional Approach to Measuring Resilience (Hughes and Bushell, 2013), the City resilience Index (CRI) (ARUP and Rockefeller Foundation, 2014), and the United Kingdom Department for International Development’s Measuring Resilience report (Sturgess, 2016). Such frameworks based on indictors and approaches are very useful to understand and contextualise specific geographical, socio-political and urban questions. However, in more practical terms, there is no consensus to date on a unique or holistic method to evaluate the level of resilience in communities quantitatively. With this work, we propose a method to quantitively assess the level of resilience of urban communities. In the case presented here, we suggest an equation that builds on existing work, adapting it to a case study.

Various successful attempts have been made to produce new ways of calculating urban resilience computationally (Cimellaro et al., 2015; Leykin et al., 2018; Yu and Baroud, 2019 among others). In his Designing Resilience of the Built Environment to Extreme Weather, Jankovic (2018) employed a random grammar approach (Kauffman, 1996), whereby a network model is applied to resilience at regional level. Jankovic’s model considers binary strings (0/1) to “represent raw materials, as well as transformations of these materials into processed resources. These transformations are carried out by models of artificial agents and the connectivity between the agents and the transformations of binary strings are governed by random grammar rules” (Jankovic, 2018:4).

A simple definition of resilience (R) is given by Bruneau et al. (2003) as “the ability of the system to reduce the chances of a shock, to absorb a shock if it occurs (abrupt reduction of performance) and to recover quickly after a shock (re-establish normal performance)” (Bruneau et al., 2003:736). Its mathematical formulation can be expressed as: R = ∫ t 0 t 0 + t 1 Q t t 1 d t (1) Where Q_t relates to the quality of the infrastructure of a community which varies with time (t₀ and t₁) where an event against which a system reacts with resilience. The notation 1) refers to the version presented by (Koliou et al., 2020:135).

Santos and colleagues (2018) elaborated a method to model several factors that compound community resilience into a computational framework. Within this framework, they suggest a number of functions that model, by approximation, “particular hypotheses about the system’s resilience” (Santos et al., 2018:186). This method is particularly relevant to our aim, as it provides a way to “use social theories to compose these individual resilience functions into an umbrella resilience function, while providing qualitative analysis” (Santos et al., 2018:186). Santos and colleagues have modelled the problem of representing the cultural and social aspects (cultural fragments) as factors in resilient communities using Bayesian Knowledge Bases: a directed graph with instantiation nodes and support nodes, characterised by weighted correlations and conditional probability rules (Santos et al., 2018:188). Whilst Bruneau and colleagues have expressed the notion of a measurable resilience through a simple formula, Santos et al. have introduced a series of functions per each aspect that contribute to a more elaborated idea of resilience within a framework, and ultimately combined individual functions into a composite function (Santos et al., 2018:191).

Moreover, the recent work of Jankovic (2021) is also significant. Jankovic proposed a formulation of urban resilience that relates to the notion of redundancy R where: R = 1 −   N m i n N m i n +   Δ N (2) In Eq. 2 N indicates the number of nodes in a network system. N_min is the minimum number of nodes necessary to run a certain operation, while ΔN indicates the number of additional nodes in the system (Jankovic, 2021:2–3). This formula is used in our experiment to attribute values to each category of physical elements that contribute to the resilience of urban communities, for it allows us to quantitatively evaluate the redundancy level represented by each category. The Equation 2 has been used to in the evaluation of the quality scores (Table 1; Table 3). To note that we eventually derived 3) as a simplified version of Jankovic’s Equation 2 to adapt it to our specific case.

Over the past 30 years, the city of Copenhagen has experienced a significant upward transformation. In the 1990s the city was in decline and experienced 17.5% unemployment, a disappearing manufacturing sector, an out-immigration population, and an annual deficit of $750million. Today however, Copenhagen is considered and typically portrayed as a thriving and captivating, and considered one of the best cities in Europe to live in (Copenhagen resilience and liveability, 2018:30). Thanks to several innovative policy programmes which include social, climate, and economic policies, the declining neighbourhoods such as Nordhavn, Østerbro and Refshaleoen now offer its residents varied urban living with a clear sense of social fabric. Indeed, the city has managed to reverse the dangerous trend where declining financial resources and high social and infrastructure maintenance expenditure would result in both urban sprawl and overly dense and chaotic urbanisation (Sorensen and Jacob 2020:218). Copenhagen’s revitalisation programme which focused on “building a liveable city” is according to Veolia “aligned with what is generally meant by resilience” (“Copenhagen: resilience and liveability” 2018:30).

Copenhagen was able to achieve transformative resilience through a complex layering of infrastructure and planning initiatives. For example, the city transferred vast amounts of public land to a new publicly owned, privately managed corporation and with it, was able to reassign land — primarily in the old harbour and an undeveloped area between the airport and the downtown — for residential and commercial use. The revenues projected by smart zoning and asset management — not taxes — to finance cross-city transit infrastructure spurred the regeneration of core areas of the city. A vibrant, multi-purpose waterfront, world-class transit system which includes an envious reputation as a bicycle and pedestrian city with clean air, less noise, more space and healthier citizens (Baykal, 2018:5). Thousands of housing units have also been built for market and social purposes in accordance with energy-efficient standards. The most recent effects of this smart institutional model can be seen in the ongoing transformation of North Harbor (Nordhavn), only one of multiple sites under the corporation’s control (Katz and Noring, 2017). Furthermore, Copenhagen has been much-praised for its sustainability planning: it was awarded European Green Capital in 2014, it was recognized as a Green Economy Leader by the London School of Economics in 2014 and is the self-proclaimed Capital of Sustainable Development.

In order to develop a quantitative model that can be applied to any urban area, the following eight stages were followed. We designed the workflow below which allows to evaluate the level of resilience and crucially, suggest how to improve it:

1) Identify reliable frameworks for urban resilience in existing literature, and extract a workable framework to assess urban resilience quantitatively.

The first step of the modelling stage was to investigate the resilience framework. The main issue here is the ambiguity of the term resilience. According to Wang et al. (2018), resilience is the holistic capacity of an urban complex system comprised of different subsystems to absorb, adapt, and recover from a disruptive event. The unpredictable nature of these disruptions and the multitude of possible resilient reactions prompted us to describe this problem through a complex framework.

Before describing the framework in detail, the decision was taken to divide a complex system into multiple categories and sub-categories. This approach has been adopted by many researchers in order to identify the layers/categories before breaking them down into components (Da Silva and Morera, 2015; Wang et al., 2018). The choice to carry out such a study through a graphic approach was fostered by our final goal to communicate visually in a more effective, direct and synthetic way.

The comparative analysis of various proposed framework led us to identify our 4 main categories: Social Environment, Economic Environment, Physical Environment and Management, as shown in Figure 1. A comprehensive approach to any resilient community demands the investigation all of four categories, as their compresence and balance results in the foundation of resilience itself (cf. Figure 2).

However, as for the complex nature of resilience, we can observe how different case studies have a different balance of the four categories: performing strongly in one could compensate for issues in another one.

An optimal concept of resilience would consider these four categories performing at their maximum. Nevertheless, any comprehensive analysis will not lead to a binary result: resilient, not resilient. Rather, it would lead to a resilient index wherein different case studies, categories and sub-categories have a different impact. Hence, the need for a complex modelling framework which is able to output detailed results.

By breaking down the categories in sub-categories, the framework of a complex structure becomes more evident, highlighting its multidisciplinary. As for the main categories, the identification of sub-categories has been carried out through a comparative analysis of different research and case studies. Weighting every component focusing of its localisation was useful to describe the influence on the final resilient index; while tracing cross-connections gave us back an organic and interconnected framework (Table 1).

This structure clearly illustrates how the alteration of a single component has the potential to impact on others, as depicted in Figure 3.

The diagram (Figure 2) reveals the connections between the main categories; or with greater detail which sub-category is likely to influence other sub-categories.

This multidisciplinary framework requires wider expertise. However, particular researchers could be focused on part of this framework (e.g., how one of these categories, or some components, influences the resilience index). Due to the type of data available for the selected case study, we decided to focus on aspects of the physical environment.

As such, the modelling and method described above was applied to the St. Kjeld Community in Copenhagen’s neighbourhood of Østerbro as one of the highest scoring resilient communities in the analysed frameworks and world ranking (Wang et al., 2018; Bizzotto et al., 2019).

The collection of data has been carried for the whole city of Copenhagen to analyse the wider Østerbro context. While the choice of the case study was chosen based on previous research, the motivation was also to source unbiased data. Moreover, the data was to be geo-localised to allow the analysis of spatial relations. The main source of data for this research was https://kbhkort.kk.dk/spatialmap: a web GIS (Geographic Information System) developed by the municipality of Copenhagen. On the very few occasions, this detailed GIS was integrated with other secondary sources (interpolating with data from GoogleMaps and from Open Data DK) where some data we needed were missing. The collected data were organised in thematic maps to scale using the graphics editor Illustrator. Each element was identified by a geo-localised point and a transparent circle indicating its area of influence. The radius of the circle was set following the component weighting using the highest performance range tier. This is shown in Figure 4 below.

The maps provided a clear picture of the Copenhagen / Østerbro and its resilience; however, these are static. The subsequent step was to import the collected data from Illustrator to Rhinoceros and Grasshopper. As the maps were vector graphic to scale, we were able to export data from Illustrator directly to Rhinoceros, maintaining all the layers and characteristic of every single element.

The table below (Table 2) illustrates the number of service buildings and other facilities in Copenhagen and their approximate distance from the centre of Østerbro.

The data were exported in a vectorial format and imported in Rhino/Grasshopper for some initial data analysis (Figure 5).

For each of these twenty categories we retrieved GIS data and mapped all instances onto a bidimensional model in order to generate a spatial system to analyse mutual relationships (distances). The hospital category contains 13 elements (buildings), library has 21 locations, cultural centres contain 20 buildings etc. The full list is illustrated in Table 2; Figure 5.

Using the developed framework along with our definition 2), distances from the centre of the neighbourhood were used for an initial assessment of the degree of resilience as shown in Figure 6. Table 3 (below) shows the values attributed based on distances and Table 4 summarises the initial findings.

The final dataset consists of 1,098 data points (the total of all elements in the 20 categories) with their X and Y coordinates (relative to the centre point of the Østerbro community), the category, their relative distance from the centre of the community and their resilience value calculated using the following notation: R =   d ( N m i n )   γ (3) Where R is the resilience value of each element N (node) in the network, d is the distance of N from the centre of the community, and γ is a coefficient which express the quality of resilience based on distance and sparsity in the network. This coefficient should consider both number of elements of same type (e.g., hospitals) in its proximity and total number of elements in the same category. We are calling γ the resilience coefficient based on distance and sparsity in the network. Figure 7 below illustrates how Grasshopper has been used to compute 3) in a spatial environment.

This section describes how the model used some clustering methods to discover some meaningful patterns in the proximity of urban typologies. The aim of this stage is to learn insights on the proximity of urban typologies both from each other and from the centre of the community. By learning about proximities, we want to extract useful information about the physical characteristics of the community to assess the level of resilience.

As shown in Figure 4, the dataset was structured in a record table. Per each instance (building or urban type) geolocation was included as per GIS data (X and Y coordinated), along with the type of building, its distance from the centre of the community in Km, and the Resilience Value computed as per (3).

In order to simplify the calculations with Python’s Pandas Data Science library, the urban typologies (station, park, library etc.) were substitute with a natural number. The resulting remapping is illustrated in Table 4 below.

Clustering algorithms are sensitive to different feature scales, since more weight could be gained by features with larger variances and those features will impose more influence on the final cluster shape. Thus, the dataset was standardised to bring each feature’s mean to zero and standard deviation to 1.

Following preliminary testing with various clustering techniques, including DBSCAN and K-Means, the decision was made to use K-means as it was the most promising from the initial runs based on our domain knowledge and the suitability of this model for our specific case study, where proximities, distance and centres are important factors. K-Means is a clustering technique based on an unsupervised learning algorithm where several pre-determined number of clusters are given and a dataset is evaluated for proximity. The result is the evaluation of the points classified in the K given clusters, with K centroids as the centres of the clusters, as shown in Figure 8.

As seen in Table 5, there are five dimensions to evaluate in this experiment. Clustering algorithms such as K-Means do not perform accurately when dealing with data of high dimensionality, namely with several features (Trunk, 1979). To determine cluster memberships, a Euclidean distance formula is used, and with a high number of dimensions the algorithm obtains similar distances and does not result in meaningful clusters. To address this issue, a dimensionality reduction technique, namely T-Distributed Stochastic Neighbour Embedding (t-SNE) (Van der Maate and Hinton, 2008) was employed, which is an appropriate algorithm for reducing dimensions with non-linear relationships.

It is necessary to specify a number of clusters in order to use K-Means. The Elbow method is a popular statistical technique for determining the optimal number of clusters, for it calculates the sum of squared distances from each point to its assigned centroid for each iteration of K-Means using a progressive number of clusters. The selected number of clusters should be at the elbow of the line chart, or with the fewest sum of squared distances at the lowest number of clusters. This results in the data inside each cluster are more tightly grouped. According to the results in Figure 9 the optimal number of clusters for our dataset is 4.

The K-Means model was simulated with four clusters, and the results are shown in Figure 10 coloured by Resilience Value (red = more resilient and blue = less resilient). Figure 10 summarises some statistics from resulting clusters (type distributions are based on encoding found in Table 6).

The plot in Figure 11 depicts a non-trivial distribution of points and their clustering that makes use of all features. Different values of resilience are distributed in different clusters within the plot. In particular, the highest levels of resilience are distributed across three clusters (highlighted in bright orange/red colour) and are located far from the centroid in each of the three clusters. Concentrating on the highest-values cluster (Figure 12), Cluster two contains 6 metro stations, 3 nurseries, 1 library, 2 playgrounds, 3 primary schools, 12 skateparks, 4 sport halls, and a large number of green areas (50–100).

The combination of urban typologies in a highly resilient cluster is also illustrated in Figure 13, where the majority of types is no. 4 (green areas >80), followed by no. 6 (kindergarten >60) and no. 15 (skateparks >20). Similarly, the other high-scoring clusters (number 1 and 3) have high concentration of green areas, kindergarten and, on a lesser extent, hospitals. The 3 clusters (1, 2, 3) show also a high concentration of schools (both primary and secondary) and skateparks. Conversely, cluster 0 includes all points with R values lower than 4; yet with high number of instances. Cluster 0 contains a large number of points that score low in the resilience scale.

The results suggest that communities that score higher in our study (considering Cluster 2 in particular) have a concentration of transportation facilities (metro stations in this case), education (libraries, nurseries and primary schools) and structures for sport and entertainment (skateparks, playgrounds, sport halls and green areas). Interestingly, the number of sport and entertainment facilities is significantly higher (83%) than the other 2 categories (8% for transport and 9% for education), as shown in Figure 14. Within the 83%, the majority of elements are green areas (50 over 66 in total).

This suggests that a high score in resilience can be directly related to the number and density of green areas in the local reach of the community. This is not surprising, considering that the city of Copenhagen has on average 25% of green areas (Cö mertler, 2017). However, from the data used in this study, we can infer that there is a strong correlation between the number and extension of green areas and the resilience value of a community.

The proposed model suggests a way to quantitatively evaluate the resilience of urban communities based on physical characteristics. Other components of existing theoretical frameworks (e.g., social and economic aspects) should be considered in future works. This model uses Østerbro as one of Copenhagen’s (and world’s) most resilient communities as a base case. The initial correlations between urban typologies, their concentration, distance from the community centre should not be considered as indications or design guideline for resilience. With this method, we simply laid the foundations for a more ambitious plan, where meaningful correlations will be drawn as the result of multiple iteration of this model and applications to different contexts.

The intention of this publication is to present initial progresses in the development of this workflow.

As an initial case study, the analysis of the resilience values of Copenhagen should be used as a base case. Our proposed method should be applied to other urban contexts and communities to evaluate their level of resilience. By comparison, our proposed method will help designers and urban scientists to: 1) evaluate the resilience of a given community and 2) find ways in which resilience can be improved (by intervening in the measures used here: position and density of urban typologies and their mutual proximity.

Future works for us will include the expanding the proposed model to include more metrics from our initial frameworks (from the Physical, Social and Economic Environments, and Management), as well as expanding the model to different communities in various global contexts.