An equity-oriented rethink of global rankings with complex networks mapping development

Nowadays, world rankings are promoted and used by international agencies, governments and corporations to evaluate country performances in a specific domain, often providing a guideline for decision makers. Although rankings allow a direct and quantitative comparison of countries, sometimes they provide a rather oversimplified representation, in which relevant aspects related to socio-economic development are either not properly considered or still analyzed in silos. In an increasingly data-driven society, a new generation of cutting-edge technologies is breaking data silos, enabling new use of public indicators to generate value for multiple stakeholders. We propose a complex network framework based on publicly available indicators to extract important insight underlying global rankings, thus adding value and significance to knowledge provided by these rankings. This approach enables the unsupervised identification of communities of countries, establishing a more targeted, fair and meaningful criterion to detect similarities. Hence, the performance of states in global rankings can be assessed based on their development level. We believe that these evaluations can be crucial in the interpretation of global rankings, making comparison between countries more significant and useful for citizens and governments and creating ecosystems for new opportunities for development.

countries. The evaluations can provide a quantitative basis to diagnose problematic scenarios and enact support policies. The reliability of our evaluation is strengthened by a quantitative control to check that the ranked index distributions related to different network communities are separated in a relevant way, an essential condition to validate the proposed methodology. The UN and the World Bank currently make use, for statistical purposes, of subdivisions of world countries in development groups 58 and income groups 59 , respectively. These two are prominent examples of world partitions that either focus on a very specific aspect of development or rest on weak analytical foundations 60 . Our model, instead, partitions the world UN states network in an unsupervised way, encompassing, in principle, all possible dimensions of development: community detection keeps track of relevant similarities between countries, which can be sometimes hidden, unexpected and not obtainable from merely geographical and economic considerations.
The present work is organized as follows. In "Results" section, we shall focus on the main outcomes of the proposed approach: (a) a complex network of UN states with a community structure unveiling their similarity; (b) straightforward and validated procedures, based on the analysis of ranked index distributions inside and across network communities, to reconsider country performances in rankings in view of their development status. Implications of our findings and added value of the developed methodology will be addressed in "Discussion" section, while all steps of model development, from data construction to validation of development communities, will be discussed in "Methods" section; further details on our results will be provided in the Supplementary Information.

Results
In this section, we present the main findings of our work. First, we discuss the most relevant features of the constructed UN development network and the structure of its partition in communities. Then, we quantitatively compare such partition with two established country groupings employed by the United Nations and the World Bank. Finally, we develop an equity-oriented reinterpretation of rankings, based on network communities. We investigate the possibility to apply our pipeline to five different rankings, chosen as representatives of different aspects of development, namely E-government Development Index (EGDI) 2018 63 , Environmental Performance Index (EPI) 2018 64,65 , Global Gender Gap Index (GGGI) 2020 66 , Healthcare Access and Quality Index (HAQI) 2016 67 , and SDG Global Index Score (SDGGIS) 2019 68 . We also introduce rating systems, determined by the index distributions across the network communities, and discuss their application to the considered rankings. WDI network communities. Using World Development Indicator (WDI) values 17 , we design a complex network model to capture similarities and differences among the 193 UN Member States (UNMS), listed in Supplementary Tab. S1 together with their current officially assigned ISO 3166-1 alpha-3 codes 61 . The choice of working with WDIs to construct our network is determined by the need for a representation of development as multidimensional as possible. Actually, WDI values include a wide variety of data; in our work, we consider indicators from the categories Environment, Economic Policy and Debt, Education, Financial Sector, Gender, Health, Infrastructure, Private Sector and Trade, Social Protection and Labor, covering essentially all aspects of a country's condition. We will show in the following how such a multifaceted representation provides advantages in the interpretation of international rankings. In our network, each node corresponds to a UN country, which can be compared to the others through a proper similarity metrics. The network structure is therefore complete, namely each nation is connected to all the others by an edge. These edges are weighted with a measure of similarity between the countries they connect, based on pairwise Pearson correlation between their WDI values. In the present study, we consider networks based on the 2018, 2015 and 2012 WDI data, with missing entries integrated by the values from the 2 years before each reference year. Technical aspects and details about indicator selection, data filling and network construction are discussed in "Methods" section and in Supplementary Sec. S2. The upper panel of Fig. 1 shows the most relevant connections between countries in the 2018 WDI network, which therefore contains integrations from 2017 and 2016.
Hierarchical community detection is performed by means of two independent algorithms, explored in wide ranges of their parameter spaces (see Supplementary Sec. S5). A robust subdivision of the WDI network of UNMS in four communities (I, II, III, IV), represented in the lower panel of Fig. 1 for the 2018 case, emerges from this analysis. Interestingly, countries belonging to the the same community generally share geographical or economic proximity. Comparison with the partitions determined by the UN development groups 58 and the World Bank income groups 59 indicates a descending ordering from I to IV in terms of average development level (see Supplementary Sec. S6 for details); this observation justifies the term development communities, that will be employed henceforth. The relation between these three partitions of UN countries can be formalized by computing an index of similarity between them. A possible choice is the Normalized Mutual Information (NMI) 69 , a quantity defined in [0, 1], with 1 corresponding to maximal similarity. Using this metrics, an interesting result emerges: WDI network communities are more similar to both UN development groups ( NMI = 0.47 ) and World Bank income groups ( NMI = 0.45 ), than the two established country groupings are between them ( NMI = 0.36 ). We can interpret this result as an effect of the multidimensionality of the data on which the WDI network is based, leading to a partition in communities that successfully interpolates between groupings that focus on different and more specific dimensions of development.
Finally, we observe that the partition of the network into development communities features a noticeable stability in time. Actually, the distribution of UNMS across development communities undergoes few modifications from the 2012 to the 2018 network, as shown in the diagram in Fig. 2. These results provide the basis of a new equity-oriented interpretation of international rankings, discussed in the following, that can help policy-makers and stakeholders define their agenda in view of the 2030 target date of UN Sustainable Development Goals. www.nature.com/scientificreports/ Rethinking rankings in the framework of development communities. The communities identified from the network analysis provide a tool to group countries by development similarity, and also a way to reinterpret their positions in international rankings. Assuming that countries in each given WDI network community are characterized by essentially homogeneous development levels, it is natural to expect that, given the ranking of an index related to development, the ranked values would tend to cluster together inside each community and separate from the values of countries in other communities. Such a picture would also help to identify countries whose performance goes beyond the expectations based on their development status and countries that have the potential to reach their community peers in the ranking by increasing their efforts. However, as shown in Fig. 3, representing the distributions of EGDI (left panel) and GGGI (right panel) in the development communities, the validity of the aforementioned assumption may vary from one index to the other.  www.nature.com/scientificreports/ In the two panels of Fig. 3, each country corresponds to a point whose vertical coordinate represents the value of the considered index, while the position with respect to the horizontal axis is determined by its community membership. The distributions highlight a qualitative difference between the two rankings: in the case of EGDI, community distributions are evenly spaced, partially overlapping, and following the same ordering as expected from the development hierarchy; on the other hand, as concerns the GGGI index, the community distributions are largely overlapped, with only community I slightly separated from the others. Therefore, in a case similar to GGGI, a community-based evaluation of country performance would not provide reliable information. The qualitative difference between the two cases is motivated by the fact that the considered indexes are not equally related with development: while EGDI aims at providing an evaluation of a country's E-government capabilities (in which, intuitively, developed countries are advantaged), GGGI measures gender gap with respect to a standard that varies state by state, depending on the political, economic, health and education condition. In order to quantify the relatedness between the performance of countries in a ranking and their community membership, we introduce a quality factor R, called the resolution ratio, that is as larger as the community index distributions are more separated. The resolution ratio, whose definition is given in "Methods" section, is much larger than one when the community index distributions have a limited overlap with each other, and much smaller than one when the overlap is practically full. In the intermediate case R ≃ 1 the separation between the mean values of neighboring community distributions is similar to the typical variation of the index within each community. The value R = 1 can be assumed as a threshold to distinguish cases in which the evaluation of country performances based on development communities is either meaningful or not. Table 1 summarizes the resolution ratios obtained for the five indexes considered in our analysis. As expected from the qualitative considerations on Fig. 3, the development communities are well resolved with respect to EGDI and insufficiently separated in the case of GGGI. The highest ratios are obtained for HAQI and SDGGIS, reflecting the strong relation between these indexes and development as a whole, while the result for EPI is lower, although still considerably larger than 1. The distributions of EPI, HAQI and SDGGIS are reported in Fig. 4.

Evaluation of country performances in rankings.
In the case R > 1 , reasonable predictions on the performance of countries in the considered ranking may be done on the basis of community membership, and  www.nature.com/scientificreports/ deviations from the expected behavior can be critically evaluated. The attention can be focused on countries whose performance in a given ranking is either above or below expectations determined by the results of countries in the same community and in other communities. Specifically, we can identify top-of-the-class countries in the ranking as those whose ranked index values fall, at the same time • above the 75th percentile of the community they belong, • above the 25th percentile of at least one more developed community.
Following a similar criterion, we define as room-for-improvement countries the ones whose index values are located both • below the 25th percentile of the community they belong, • below the 75th percentile of at least one less developed community.
Top-of-the-class countries can be seen as reference cases that could inspire the action of similar states to improve their status in the ranking. To further quantify the mismatch between the performances of such states and expectations based on their community membership, we award a top-of-the-class country with a symbol " ↑ " for each 25th percentile of a more developed community that is overcome by its index value. On the other hand, room-forimprovement countries can be interpreted as states which have the potential, in terms of development, to achieve better results in the considered ranking and close the gap with similar countries. In this case, we mark a country with a symbol " * " each time the index value falls below the 75th percentile of a less developed community.
States with the highest indexes in community I and with the lowest indexes in community IV are not covered by the above definitions, due to the lack of more and less developed communities, respectively. Therefore, for a comprehensive mapping of remarkable performances in ranking, we introduce the categories of benchmark and trailing countries. Benchmark countries are states belonging to community I, characterized by an index value above the 75th community percentile; they play a crucial role in policy design, being regarded as best-practice by the rest of the world. Trailing countries belong to community IV, with their index values falling below the 25th community percentile; these states are noteworthy because they could need specific support to improve their condition in the development areas related to the ranking. In the following, referring to Figs. 3 and 4, we shall assess country performances in each community for the rankings with R > 1 analyzed in this work.

E-government Development Index 2018.
The EGDI measures the effort and ability of all the 193 UNMS in using ICTs to provide public services at the national level, through the assessment of three key-aspects: the development of telecommunication infrastructure, the capacity-building in human capital and the quality of online service delivery 63  The assessment identifies 13 top-of-the-class and 14 room-for-improvement countries, corresponding to the 8.0% and 8.6% of ranked countries, respectively.
A community-based rating scheme. To complete the analysis of country performances in rankings in view of their development status, we introduce a rating criterion, based on crossing information on the position of each UNMS in the overall ranking with its membership to a community of the WDI network. This framework, whose results in the case of EGDI are illustrated in Table 2, is made of the following steps: 1. The range of the ranked index is partitioned in the quartiles ( Q 1 , Q 2 , Q 3 , Q 4 ), in descending order of values. Each country receives a primary rating, consisting in a capital letter (A, B, C, D), according to the quartile in which its index value falls. 2. Countries in a given quartile are assigned as many small-case versions of the primary-rating letter as the number of more developed communities with representatives in the same quartile.
The secondary rating enforces equity by rewarding states that reach the same quartile as at least one member of a more developed community. Ratings with respect to other indexes with R > 1 are reported in Supplementary Tabs. S9-S11.

Discussion
Let us summarize the main findings of the work presented in this paper: the emergence of a robust community structure in the complex network obtained from WDI values; the quantification through a quality factor of how much the distributions of a given ranked index in different communities are separated; the introduction of a straightforward method, based on WDI communities, to reinterpret rankings. The complex network is constructed, following the proximity principle 38 , starting from correlations between sets of WDIs pertaining to different countries. The resulting network provides information on the development both on a state-specific level and on a collective level. Information on how each state is correlated to the rest of the network is synthesized in Supplementary Fig. S4 and Table S8. From the results therein, one can deduce that some countries are characterized by generally weak correlations, and therefore low affinity with other states. In particular, the following countries have an average correlation smaller than 0. 25 Table S8), and represent almost all the entries of such category: the United States are least-correlated for 85 countries, Somalia for 66, South Sudan for 30. Another interesting case is represented by China, due to its peculiarity, as confirmed by not only the average correlation, but even its maximal value (namely, 0.5719 with Russian Federation), which represents the only maximal correlation below 0.6 in the network. In general, maximal similarity relations follow geographical proximity: in 43% cases, the most correlated country is a state that shares a land border. Interesting exceptions to this rule are, e.g., the reciprocal maximal correlation between Canada and Australia, and the fact that the maximally correlated states to Japan and New Zealand are Germany and Iceland, respectively. Future research will be devoted to scrutinize the impact of the indicator selection procedure on both UNMS mutual similarities and their community membership. Knowledge of these mechanisms can provide a relevant instrument to modulate policy design in different development areas. A further perspective is represented by the use of a set of optimized indicator combinations, such as the cluster-driven composite indicators proposed in Ref. 35 , to construct the network. Such a choice, which goes beyond the scope of this work, would constitute a refined and controllable alternative to the indicator selection process described in "Data collection and preprocessing" section.
As for the value of our model on a collective level, we remark that the partition of world countries based on the WDI network largely fulfills the requirements for a good classification scheme, proposed by the Expert Group on International Economic and Social Classifications 70,71 , including determination of exhaustive and mutually exclusive categories, comparability with related standard classifications, stability in time, solid data-driven foundations and explanations, and a reasonable number of groups, reflecting reality of the field. In addition, the similarity patterns found by community detection in the WDI network are neither trivial nor redundant with those obtained from traditional approaches, e.g. geographical criteria or methods employed by UN and World Bank, based on country development and income, respectively. Nonetheless, we find a general coherence of our results with the aforementioned grouping procedures. Specifically, by interpreting the results of the NMI between the WDI communities, the UN development groups and the World Bank income groups, we are able to determine that the partition emerging from our model represents an interpolation between the two established groupings. The idea of grouping countries starting from a given set of indicators has already been considered in literature, where hierarchical clustering based on a small number of indicators was applied to problems concerning the technological gap 72 and the relation between innovation and competitiveness 73 . It is worth noticing that the use of hierarchical clustering to partition our WDI network leads to very uneven country groups, in which the less www.nature.com/scientificreports/ correlated states (see discussion above) tend to be isolated. Complex network models have also been employed to characterize countries and their mutual relations, mainly in studies with a socio-economic and political background 48,56 . These works investigate the existence of possible hierarchies, unveiled by centrality measures 57 , as well as influences of one state on another 74,75 . To the best of our knowledge, no previous study has focused on the reinterpretation of rankings through the analysis of mutual similarity relations between countries in a complex network framework. Our model provides an immediate and versatile procedure to evaluate the discrepancy between a country's performance in a ranking and its development condition. This assessment is based on comparison both inside a community and with other communities, namely with different development levels.
Comparing the distributions of ranked indexes across the different communities, as well as the numbers of ( ↑ ) and ( * ) points awarded in each ranking, we observe that our evaluation method can feature varying sensitivities, depending on the value of the resolution ratio R. In particular, when the index results from indicators that are strongly related to the selected WDIs, the ranking is characterized by a large R and its distributions within different communities tend to separate from each other, with only few states performing differently from what expected on the basis of their development community membership. When the redundancy between the ranked index and WDIs is lower, the overlap between community distributions is larger, and a higher number of states receive a mark, either positive or negative. Finally, marks become pointless in the case R < 1 , when community distributions are mostly overlapped with each other. It is worth stressing that the assignment of ( ↑ ) and ( * ) ratings allows the comparison of countries within the same ranking, and cannot be used to compare the performance of a given country in different rankings, since higher redundancies between the index and the WDIs generally yield a smaller number of both awards ( ↑ ) and penalties ( * ). In a ranking characterized by a high R with respect to the WDI network partition, the attribution of a negative rating should thus be considered much more worrying than if the same penalty were attributed in a less WDI-related ranking. The discussed regimes are also observed in the community-based rating, described in "Evaluation of country performances in rankings" section. For strongly WDI-related rankings, index values of countries belonging to communities I, II, III and IV are concentrated in the quartiles Q 1 , Q 2 , Q 3 and Q 4 , respectively. Instead, the spreading of communities across quartiles is much more pronounced in the case of weakly WDI-related indexes, entailing the assignment of a larger number of secondary ratings.
A natural question arises as to why introducing a network-based partition to reinterpret rankings, instead of using established groupings such as UN development groups and World Bank income groups. The reasons of the first choice are both conceptual and phenomenological. From a conceptual point of view, WDI communities emerge from a network based on several sectors of the development spectrum, and are obtained through numerical criteria in an unsupervised way, without bias on the arrangement of countries in groups. On the other hand, World Bank income groups have been set focusing on a narrower aspect of development, while UN development groups and other groupings are based on a small number of indicators, chosen and aggregated according to criteria that can suffer from different kinds of bias and variation in time 60 . From a phenomenological point of view, the evaluation of country performance in a ranking, especially when deviations from the expected result occur, is as reliable as the partition is related to the ranked index values 71 . In Table 3 we compare the resolution ratios of the rankings with respect to the aforementioned partitions, focusing on the cases in which the proposed rating scheme is meaningful ( R > 1 ). It is evident that WDI network communities provide the highest resolution ratio for all the indexes, thus representing the sharpest way of the three to identify discrepancies between rankings and development. We argue that the improved R for all the rankings is due to both the multidimensionality of WDI dataset and the absence of possible biases in obtaining the partition.
The proposed methodology highlights the different facets of a country's condition, which are often the result of the actions undertaken and priorities established by its government. Development communities and rating criteria define a rigorous, transparent and reproducible procedure, that can provide a basis for policy design. The aim of our analysis is not to pass judgments, that could be simplistic, on state performances, but rather to highlight, for each country, strengths and unexpressed potentialities. This kind of information can be crucial in supporting the definition of adequate and promising development trajectories. In particular, top-of-the-class states could target a general improvement of their WDIs, starting from the specific fields in which they excel. On the other hand, countries labeled as room-for-improvement in a specific ranking have concrete growth perspective, since they are provided with the necessary resources to address policies towards a better result in that field. This information, emerging in our model, can have a pivotal value in development strategy planning, as well as to assess the effects of measures already adopted, e.g. the public policies undertaken over the years to strengthen ICT-related mechanisms for public benefit in Ghana, Rwanda and Uganda 41,63 , recognized in our scheme as community IV top-of-the-class countries in the EGDI ranking.

Methods
In the following, we shall focus on the methods employed to construct and analyze a complex network in which UN countries, representing nodes, are mutually connected with increasing strength, according to their similarity. Specifically, a community detection based on WDIs is performed and used as a touchstone to interpret various country rankings. A scheme representing the pipeline of this study is displayed in Fig. 5.
Data collection and preprocessing. World Development Indicators. The database on which the network is built is represented by the World Development Indicators (WDIs) 17 , "a compilation of relevant, highquality, and internationally comparable statistics about global development and the fight against poverty", containing time-series indicators, going back to 1960 in the best case, for 217 economies (referred to both UN and non-UN countries) and more than 40 economic or geographical country groups. We choose to focus on the 193 countries belonging to the UN 77 , listed in Supplementary Table S1 along with their official ISO 3166-1   www.nature.com/scientificreports/ exceeding the 99th percentile from above and the 1st percentile from below were replaced by the reference percentiles before performing the linear rescaling of the indicator in [0, 1].

E-government Development Index 2018.
This index is available for all UN countries, and is defined in the range [0, 1], with 0 and 1 corresponding to the worst and best achievable performances, respectively. Therefore, no rescaling is required for the data shown in the left panel of Fig. 3. UN states development network. The WDIs, selected and rescaled according to the procedure described in Supplementary Sec. S2, are used to compute the pairwise Pearson correlations between countries. We use these correlations to construct a complex network made of 193 nodes, representing UN countries, linked by weighted edges, whose weight coincides with the pairwise Pearson correlation. The resulting network is made of 193 × 192/2 = 18,528 links, being therefore complete. The choice to work with a complete network is motivated by a number of reasons. First, a network with a simpler topology, made of sparser connections, would require the definition of a threshold value for link weights, below which a pair of countries is considered disconnected. This procedure would lead us to different results according to the choice of the threshold, introducing arbitrariness in our analysis and loss of information; actually, weak and negative correlations between countries contribute to determine the network partition in communities. Moreover, an additional problem arising from the choice of trimming the network is the fact that the thresholding operation would act inhomogeneously on countries, erasing, in particular, the information on similarities and differences related to countries for which correlations are weak on average.

Global Gender Gap
Community detection. The identification of communities must take into account the fact that the network is complete and weighted with both positive and negative edge weights. Thus, we choose two community detection algorithms that have recently been optimized in order to handle negative weights 46 , which is a rather uncommon feature: Spin Glass, based on statistical mechanics concepts 78,79 and Leiden 80 . We perform for both algorithms hierarchical community detection by recursive partitioning, an approach already explored in Refs. 69,81,82 . In our multi-step procedure, subsequent detection is applied to partition the communities obtained at the previous stage, as long as an iteration condition is satisfied. Such condition is represented by the accordance between outputs of community detection for different runs of the algorithm at the considered step. At each step, communities are found using the algorithms of the igraph library 83 . Actually, each algorithm operates within a pipeline that is not entirely deterministic, yielding in principle different outputs when applied to the same network; however, the outcome of a robust community detection should be as independent as possible from randomness. To obtain reliable communities, we adopt the following criterion: the network is partitioned 100 times by one of the chosen algorithms; if a single outcome is found in more than 90% cases, communities are accepted and the procedure moves on to the next step; otherwise, the iteration stops, and the partition found at the previous level is returned as the final result. This method is implemented for different configurations of both community detection algorithms, through an in-depth exploration of their parameter space. Moreover, the initial condition required to launch the Leiden algorithm is changed in the different runs, randomly assigning nodes to a random Resolution ratio. In "Rethinking rankings in the framework of development communities" section, we introduced the resolution ratio R as a tool to distinguish cases in which the separation between ranked index distributions related to different communities is significant or not. Here, we define this quantity in terms of basic statistical parameters (mean and variance) of the community distributions and the overall distribution. The definition of the resolution ratio R is based on the relation between the values {x i } assigned to each element i = 1, . . . , N of a given set and the partition of that set in K disjoint groups of cardinality n c , with c = 1, . . . , K .
Given the values and the partition, one can evaluate the overall mean µ and variance D of the whole set {x i } , as well as the mean µ c and variance D c for each group c. The key observation 71 is that the overall variance D can be separated in two positive contributions Since D int coincides with a weighted average of group variances, while D ext represents the contribution determined by the discrepancy between the group means and the overall mean, a good indicator of separation of group distributions is given by R = D ext /D int .
Disclaimer. The designations employed and the presentation of the material in this paper do not imply the expression of any opinion whatsoever on the part of the United Nations concerning the legal status of any country, territory, city or area, or of its authorities, or concerning the delimitation of its frontiers or boundaries. The designations "developed" and "developing" economics are intended for statistical convenience and do not necessarily imply a judgment about the state reached by a particular country or area in the development process.
The term "country" as used in the text of this publication also refers, as appropriate, to territories or areas. The views expressed are those of the individual authors of the paper and do not imply any expression of opinion on the part of the United Nations.