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Non-linear interlinkages and key objectives amongst the Paris Agreement and the Sustainable Development Goals

The United Nations' ambitions to combat climate change and prosper human development are manifested in the Paris Agreement and the Sustainable Development Goals (SDGs), respectively. These are inherently inter-linked as progress towards some of these objectives may accelerate or hinder progress towards others. We investigate how these two agendas influence each other by defining networks of 18 nodes, consisting of the 17 SDGs and climate change, for various groupings of countries. We compute a non-linear measure of conditional dependence, the partial distance correlation, given any subset of the remaining 16 variables. These correlations are treated as weights on edges, and weighted eigenvector centralities are calculated to determine the most important nodes. We find that SDG 6, clean water and sanitation, and SDG 4, quality education, are most central across nearly all groupings of countries. In developing regions, SDG 17, partnerships for the goals, is strongly connected to the progress of other objectives in the two agendas whilst, somewhat surprisingly, SDG 8, decent work and economic growth, is not as important in terms of eigenvector centrality.


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1 Inter-linked human and natural worlds

The state-of-the-art in sustainability is described by two United Nations (UN) landmark agendas, the Paris Agreement (UN, 2015a) and the Sustainable Development Goals (SDGs) (UN, 2015b). Whilst the former focuses on preventing a global climate crisis with far reaching consequences by limiting global warming to 1.5 to 2°C above pre-industrial levels, the purpose of the latter is to end poverty, protect the planet and ensure that all people enjoy peace and prosperity by 2030. Any action for the progress on either agenda often has an influence on the other (UN Climate Change, 2019), reflecting the complexity of the human and natural worlds.

This inter-linked nature gives rise to opportunities for the creation of synergistic interventions: civil, corporate and institutional actions can efficiently create impact across both agendas, thereby improving the world profoundly. On the other hand, this inter-linked construct can also be subject to trade-offs between objectives, i.e., progress towards one agenda constrains progress towards the other. In this work, we aim to discover how climate change, as measured by local temperature rises, and the 17 SDGs are inter-linked by learning the structure of undirected graphs over these variables from their (conditional) dependencies.

Adding climate change as an 18 variable is motivated by the observation that temperature rises (or any other direct metrics of climate change) are not actually tracked within SDG 13 (climate action). Indicators of SDG 13 only track inputs (such as investment), means (such as plans and strategies), and impacts (number of people affected by disasters), but they do not account for outputs, such as changes in temperature or green house gas emissions.111Only recently (and after performing the present analysis) have ”total greenhouse gas emissions” been added as an output-quantifying indicator (13.2.2).

We use distance correlation (Székely et al., 2007) as a measure of non-linear dependence between variables of possibly varying dimensions. To account for possible interactions, each pair of variables is conditioned on any subset of the remaining variables, and the minimum resulting distance correlation is taken as the weight on an edge between these two variables. Subsequently, the weighted eigenvector centrality of every node is calculated to measure its importance within the network.

In summary, the contributions of this paper include: first, the application of a non-linear measure of (conditional) dependence to SDG data, thereby relaxing the linearity assumption on the nature of interlinkages between the SDGs, compared to the work of Lusseau and Mancini (2019); and secondly, the use of eigenvector centrality as a relative measure which also takes the importance of a node’s neighbours into account, as opposed to simple degree centrality as used by McGowan et al. (2019).

2 Methodology

We use data provided by the World Bank (2020b) and the UN (2020) in form of time-series for various indicators, which measure progress towards their associated SDGs, in conjunction with temperature recordings (World Bank, 2020a).222For detailed descriptions of indicators, see In total, these three sources provide 379 time-series, which are available on a country-level with annual measurements from 2000 to 2016333

We impute missing values (especially for the time 2000-2005) using a weighted average across countries (where data is available) with weights inversely proportional to the Euclidean distance between indicators.

. Apart from measurements for the 17 SDGs, we introduce climate change as an additional variable which we define by annual average temperature per country. We consider these 18 variables as the set of nodes of an undirected graph . We learn the graph structure by computing partial distance covariances (Székely et al., 2014) between any pair of nodes, given any subset of the remaining 16 nodes. This yields a sparsely-connected undirected graph with weighted edges capturing non-linear dependencies between variables. Using these weights, we compute weighted eigenvector centralities (Newman, 2018, p.159; Appendix A.2) to find the most important nodes. Code to reproduce our findings and visualisations of networks may be found online at

2.1 Distance covariance

Let and

be two random vectors with finite first moments, i.e.,

. The distance covariance between and , denoted by , is a measure of dependence between and with the following important properties: (i) , with equality if and only if and are independent, i.e., it is a non-parametric measure that—unlike, e.g., standard correlation—is able to pick up complex non-linear dependencies; (ii) , i.e., it is symmetric; and (iii) unlike many other dependence measures is well-defined even for . This last point makes it particularly useful for our setting where, due to the different numbers of indicators per SDG, dimensionality varies considerably between variables.

Formally, the distance covariance between and is defined as


where , and where the characteristic function

of a random variable

is denoted as with .

The corresponding distance correlation is the normalised distance covariance, computed by


Properties of include: (i) ; and (ii) if and only if there exists a vector , a non-zero real number

, and an orthogonal matrix

such that .

Since and therefore

are defined in terms of the underlying joint distribution of

which is usually not known, we require a way to estimate them from data. Definitions of biased and unbiased estimators, referred to as

and , can be found in Appendix A.1.1 and A.1.2.

2.2 Partial distance covariance

As we deal with graphs of 18 nodes, any pairwise covariance may occur through the remaining 16 nodes. Thus, we condition any pair of nodes on any subset of the remaining 16 nodes. The pairwise distances and the distance matrix for are computed equivalently to and for and as explained in Appendix A.1.1. For any number of samples from , we define a Hilbert space over distance matrices computed on these points, with inner products as defined in Appendix A.1.2 (Székely et al., 2014). With this, we can compute partial distance covariances for random vectors of varying dimensions as follows.

Let , and be elements of the Hilbert space corresponding to the distance matrices computed using the samples , , and , respectively. The projection of onto and the complementary orthogonal projection are defined by


respectively. The sample partial distance covariance is then defined as


Finally, we can normalise these covariances to arrive at the sample partial distance correlations


which serve as weights on edges between any two nodes.

3 Results

We apply this methodology to the data set of the aforementioned 379 indicators for various groupings of countries, for which countries are assumed to be independent samples. This assumption allows us to see the indicators’ non-stationary time-series as

-dimensional probability distributions, where

. Whilst we only describe the networks of a few groupings in this section, we would like to refer to Appendix B for results on all groupings.

Global South Global North
SDG 6 0.48 SDG 6 0.43
SDG 4 0.42 SDG 4 0.40
SDG 7 0.38 SDG 9 0.33
SDG 17 0.37 SDG 3 0.32
SDG 3 0.26 SDG 17 0.29
SDG 15 0.25 SDG 7 0.27
Western Asia Northern Europe
SDG 6 0.48 SDG 4 0.38
SDG 4 0.36 SDG 3 0.35
SDG 17 0.34 SDG 6 0.30
SDG 3 0.33 SDG 16 0.30
SDG 16 0.32 SDG 7 0.29
SDG 7 0.26 SDG 9 0.28
Table 1: Comparison of eigenvector centralities between the Global South and the Global North (left), and between Western Asia and Northern Europe (right). Results for all groupings can be found in B.2.
Figure 1: Networks with weighted edges of (left) the Global South and (right) the Global North. The minimum partial distance correlations between the two adjacent nodes and , given any subset are weights on edges.

Firstly, we compare the Global South and the Global North (see Figure 1). The accompanied eigenvector centralities are shown in Table 1. In both groupings, SDG 6, clean water and sanitation, followed closely by SDG 4, quality education, are the most central objectives of the 18 variables. In the Global South, temperature rises are more strongly dependent on variables than in the Global North, which broadly aligns with King and Harrington (2018) who find that geographical areas in the Global South are more vulnerable to climate change than regions in the Global North. Further, SDG 1, no poverty, is strongly linked to SDG 14, life below water, in the Global South. This may be explained by the dependence of small island developing states (SIDS)—all of which lie in the Global South—on marine life to provide for their citizens’ living.

Contrarily, the Global North strongly depends on SDG 9, industry, innovation and infrastructure, to maintain its citizens’ high levels of living standards and to further progress towards other SDGs, as well as climate change mitigation and adaptation. Moreover, SDG 7, clean and affordable energy, is closely related to SDG 15, life on land, which could result from the increasing area of biodiverse land populated by wind turbines, solar panels, or water dams (e.g., Hernandez et al., 2015).

Figure 2: Networks with weighted edges of (left) Western Asia and (right) Northern Europe. The minimum partial distance correlations between the two adjacent nodes and , given any subset are weights on edges.

Next, we compare two geographical regions, Western Asia and Northern Europe, shown in Figure 2 with accompanied eigenvector centralities in Table 1. In Western Asia, SDG 6 together with SDG 4 are again the two most central nodes, but SDG 16, peace, justice and strong institutions, is also important, likely to be associated with the unstable political circumstances in this area during the period of recorded measurements. Additionally, SDG 5, gender equality, is strongly linked to SDG 17, partnerships for the goals, which coincides with the remarkably low percentage of women in managerial positions in Western Asia.444In Saudi Arabia, for example, only 5 to 9% of managerial positions were held by women from 2000 to 2015, whereas this number fluctuated between 32 and 36% in the United Kingdom in the same period (UN (2020), indicator 5.5.2)

In contrast, Northern Europe does not see a remarkable difference between the centralities of SDGs 6 and 4 to all others, but finds SDGs 4, 6, 3, and 17 with almost equivalently high centralities. As in the Global North, industry, innovation and infrastructure are of particular importance to progress towards the SDGs, and we fine that clean and affordable energy is closely linked to life on land.

We note, however, that most edges found in our network analysis are not statistically significant at , using the test of Székely et al. (2014). This is likely linked to the high dimensionality of the data and the short recording period. The present work is thus only a first step, and further analysis is needed to better understand non-linear interlinkages between the SDGs and climate change.

4 Conclusions

We report findings of our work in progress towards discovering dependencies amongst the Sustainable Development Goals (SDGs) and climate change. As a first step, we compute partial distance correlations between the 17 SDGs and climate change, as measured by indicators associated to the SDGs and annual average temperature, respectively. Using these measurements of non-linear dependence as edge weights in a network over these variables, we determine eigenvector centralities to unveil which variables are of particular importance, given the available data. Our results indicate that SDG 6, clean water and sanitation, together with SDG 4, quality education, are the most central nodes in nearly all continents and other groupings of countries. In contrast to many contemporary policies, our preliminary results suggest that economic growth, as measured by SDG 8, appears not to play as central of a role for sustainable development or mitigating climate change as other SDGs.


Appendix A Appendix A

a.1 Distance covariance estimators

a.1.1 Biased estimators

Suppose that we have access to a sample of pairs . First, define the pairwise distances: and . Next, define the corresponding distance matrices, denoted by and , as follows:




Having computed these, the sample distance covariance can be estimated by


which converges almost surely to the population distance covariance as (Székely et al., 2014).

a.1.2 Unbiased estimators

Unbiased estimators of the distance covariance are denoted as . Firstly, we must redefine our distance matrices and , which we call and as




Finally, we can compute the unbiased estimator for as the dot product :


a.2 Eigenvector centrality

For any graph , let be the adjacency matrix of graph with equal to the weight on the edge between node and . The eigenvector centrality of node is a measure relative to all other nodes in , defined as



is the greatest eigenvalue in the eigenvector equation

, subject to . Consequently, this centrality measure is an extension of the widely used degree centrality by considering the centrality of its neighbours besides its own.

Appendix B Appendix B

b.1 Networks of groupings

b.2 Eigenvector centralities

b.3 Groupings of countries

Northern Africa Eastern Africa Middle Africa Southern Africa Western Africa Sub-Saharan Africa Africa Caribbean Central America South America Latin America and the Caribbean North America Americas Central Asia Eastern Asia South-eastern Asia Southern Asia Western Asia Asia Eastern Europe Northern Europe Southern Europe Western Europe Europe Australia and New Zealand Oceania (excl. AUS + NZ) Oceania (incl. AUS + NZ)
Algeria Burundi Angola Botswana Benin Burundi Algeria Antigua and Barbuda Belize Argentina Antigua and Barbuda Canada Antigua and Barbuda Kazakhstan China Brunei Darussalam Afghanistan Armenia Kazakhstan Belarus Denmark Albania Austria Belarus Australia Fiji Australia
Egypt, Arab Rep. Comoros Cameroon Lesotho Burkina Faso Comoros Egypt, Arab Rep. Bahamas, The Costa Rica Bolivia Bahamas, The Greenland Bahamas, The Kyrgyz Republic Korea, Dem. People’s Rep. Cambodia Bangladesh Azerbaijan Kyrgyz Republic Bulgaria Estonia Bosnia and Herzegovina Belgium Bulgaria New Zealand Papua New Guinea New Zealand
Morocco Djibouti Central African Republic Namibia Cote d’Ivoire Djibouti Morocco Barbados El Salvador Brazil Barbados United States Barbados Tajikistan Japan Indonesia Bhutan Bahrain Tajikistan Czech Republic Finland Croatia France Czech Republic Solomon Islands Fiji
Tunisia Eritrea Chad South Africa Gambia, The Eritrea Tunisia Cuba Guatemala Chile Cuba Cuba Turkmenistan Mongolia Lao PDR India Cyprus Turkmenistan Hungary Iceland Greece Germany Hungary Vanuatu Papua New Guinea
Ethiopia Congo, Rep. Ghana Ethiopia Burundi Dominica Honduras Colombia Dominica Dominica Uzbekistan Malaysia Iran, Islamic Rep. Georgia Uzbekistan Poland Ireland Italy Liechtenstein Poland Micronesia, Fed. Sts. Solomon Islands
Kenya Congo, Dem. Rep. Guinea-Bissau Kenya Comoros Grenada Mexico Ecuador Grenada Grenada Myanmar Maldives Iraq China Moldova Latvia Malta Luxembourg Moldova Palau Vanuatu
Madagascar Equatorial Guinea Liberia Madagascar Djibouti Haiti Nicaragua Guyana Haiti Haiti Philippines Nepal Israel Korea, Dem. People’s Rep. Romania Lithuania Montenegro Netherlands Romania Kiribati Micronesia, Fed. Sts.
Malawi Gabon Mali Malawi Eritrea Jamaica Panama Paraguay Jamaica Jamaica Singapore Pakistan Jordan Japan Russian Federation Norway Portugal Switzerland Russian Federation Samoa Palau
Mauritius Sao Tome and Principe Mauritania Mauritius Ethiopia Puerto Rico Peru Puerto Rico Puerto Rico Thailand Sri Lanka Kuwait Mongolia Slovak Republic Sweden Serbia Slovak Republic Tonga Kiribati
Mozambique Niger Mozambique Kenya Trinidad and Tobago Suriname Trinidad and Tobago Trinidad and Tobago Timor-Leste Lebanon Brunei Darussalam Ukraine United Kingdom Slovenia Ukraine Tuvalu Samoa
Rwanda Nigeria Rwanda Madagascar Uruguay Belize Belize Vietnam Oman Cambodia Spain Denmark Tonga
Seychelles Senegal Seychelles Malawi Venezuela, RB Costa Rica Costa Rica Qatar Indonesia Estonia Tuvalu
Somalia Sierra Leone Somalia Mauritius El Salvador El Salvador Saudi Arabia Lao PDR Finland
South Sudan Togo South Sudan Mozambique Guatemala Guatemala Syrian Arab Republic Malaysia Iceland
Uganda Uganda Rwanda Honduras Honduras Turkey Myanmar Ireland
Tanzania Tanzania Seychelles Mexico Mexico United Arab Emirates Philippines Latvia
Zambia Zambia Somalia Nicaragua Nicaragua Yemen, Rep. Singapore Lithuania
Zimbabwe Zimbabwe South Sudan Panama Panama Thailand Norway
Angola Uganda Argentina Argentina Timor-Leste Sweden
Cameroon Tanzania Bolivia Bolivia Vietnam United Kingdom
Central African Republic Zambia Brazil Brazil Afghanistan Albania
Chad Zimbabwe Chile Chile Bangladesh Bosnia and Herzegovina
Congo, Rep. Angola Colombia Colombia Bhutan Croatia
Congo, Dem. Rep. Cameroon Ecuador Ecuador India Greece
Equatorial Guinea Central African Republic Guyana Guyana Iran, Islamic Rep. Italy
Gabon Chad Paraguay Paraguay Maldives Malta
Sao Tome and Principe Congo, Rep. Peru Peru Nepal Montenegro
Botswana Congo, Dem. Rep. Suriname Suriname Pakistan Portugal
Lesotho Equatorial Guinea Uruguay Uruguay Sri Lanka Serbia
Namibia Gabon Venezuela, RB Venezuela, RB Armenia Slovenia
South Africa Sao Tome and Principe Canada Azerbaijan Spain
Benin Botswana Greenland Bahrain Austria
Burkina Faso Lesotho United States Cyprus Belgium
Cote d’Ivoire Namibia Georgia France
Gambia, The South Africa Iraq Germany
Ghana Benin Israel Liechtenstein
Guinea-Bissau Burkina Faso Jordan Luxembourg
Liberia Cote d’Ivoire Kuwait Netherlands
Mali Gambia, The Lebanon Switzerland
Mauritania Ghana Oman
Niger Guinea-Bissau Qatar
Nigeria Liberia Saudi Arabia
Senegal Mali Syrian Arab Republic
Sierra Leone Mauritania Turkey
Togo Niger United Arab Emirates
Nigeria Yemen, Rep.
Sierra Leone
  • World contains all listed countries.

Global North Global South LDC LLDC SIDS G20 Emerging Markets OPEC Low Income Lower middle Income Upper middle Income High Income
Albania Fiji Yemen, Rep. Afghanistan Antigua and Barbuda Australia Bangladesh Algeria Afghanistan Angola Albania Antigua and Barbuda
Austria Micronesia, Fed. Sts. Afghanistan Armenia Bahamas, The Canada Egypt, Arab Rep. Angola Benin Bangladesh Algeria Australia
Belarus Tonga Burundi Azerbaijan Barbados Saudi Arabia Indonesia Equatorial Guinea Burkina Faso Bhutan Argentina Austria
Belgium Vanuatu Angola Bhutan Belize United States Iran, Islamic Rep. Gabon Burundi Bolivia Armenia Bahamas, The
Bosnia and Herzegovina Tuvalu Benin Bolivia Comoros India Mexico Iran, Islamic Rep. Central African Republic Cambodia Azerbaijan Bahrain
Bulgaria Solomon Islands Mozambique Botswana Cuba Russian Federation Nigeria Iraq Chad Cameroon Belarus Barbados
Croatia Samoa Burkina Faso Burkina Faso Dominica South Africa Pakistan Kuwait Congo, Dem. Rep. Comoros Belize Belgium
Cyprus Papua New Guinea Niger Burundi Dominican Republic Turkey Philippines Libya Eritrea Congo, Rep. Bosnia and Herzegovina Canada
Czech Republic Palau Central African Republic Central African Republic Fiji Argentina Turkey Nigeria Ethiopia Cote d’Ivoire Botswana Chile
Denmark Kiribati Chad Chad Grenada Brazil Korea, Dem. People’s Rep. Saudi Arabia Gambia Djibouti Brazil Croatia
Estonia Bangladesh Lesotho Ethiopia Guinea-Bissau Mexico Vietnam United Arab Emirates Guinea Egypt, Arab Rep. Bulgaria Malta
Finland Bhutan Liberia Kazakhstan Guyana France Brazil Congo, Dem. Rep. Guinea-Bissau El Salvador China Cyprus
France Cambodia Congo, Dem. Rep. Kyrgyz Republic Haiti Germany Russian Federation Venezuela, RB Haiti Ghana Colombia Czech Republic
Greece China Djibouti Lao PDR Jamaica Italy India Liberia Honduras Costa Rica Denmark
Germany India Togo Lesotho Kiribati United Kingdom China Madagascar India Cuba Estonia
Greenland Indonesia Equatorial Guinea Malawi Maldives China South Africa Malawi Indonesia Dominica Finland
Hungary Lao PDR Eritrea Mali Mauritius Indonesia Mali Kenya Dominican Republic France
Iceland Malaysia Ethiopia Moldova Palau Japan Mozambique Kiribati Ecuador Germany
Ireland Myanmar Gambia Mongolia Papua New Guinea Korea, Dem. People’s Rep. Nepal Kyrgyz Republic Equatorial Guinea Greece
Italy Mongolia Madagascar Nepal Puerto Rico Niger Lao PDR Fiji Greenland
Latvia Nepal Malawi Niger Samoa Rwanda Lesotho Gabon Hungary
Liechtenstein Pakistan Mali Paraguay Sao Tome and Principe Sierra Leone Mauritania Georgia Iceland
Lithuania Philippines Rwanda Rwanda Seychelles Somalia Moldova Grenada Ireland
Luxembourg Sri Lanka Senegal South Sudan Singapore South Sudan Mongolia Guatemala Israel
Malta Thailand Sierra Leone Tajikistan Solomon Islands Syrian Arab Republic Morocco Guyana Italy
Montenegro Timor-Leste Mauritania Turkmenistan Suriname Tajikistan Myanmar Iran, Islamic Rep. Japan
Netherlands Vietnam Guinea-Bissau Uganda Timor-Leste Tanzania Nicaragua Iraq Korea, Dem. People’s Rep.
Norway Maldives Guinea Uzbekistan Tuvalu Togo Nigeria Jamaica Kuwait
Poland Grenada Comoros Zambia Vanuatu Uganda Pakistan Jordan Latvia
Portugal Dominica Sao Tome and Principe Zimbabwe Yemen, Rep. Papua New Guinea Kazakhstan Liechtenstein
Romania Barbados Zambia Philippines Lebanon Lithuania
Serbia Antigua and Barbuda Uganda Sao Tome and Principe Libya Luxembourg
Slovakia Cuba Tanzania Senegal Malaysia Netherlands
Slovenia Bahamas, The South Sudan Solomon Islands Maldives New Zealand
Spain Puerto Rico Sudan Sudan Mauritius Norway
Sweden Jamaica Bhutan Timor-Leste Mexico Oman
Switzerland Algeria Cambodia Tunisia Montenegro Palau
Ukraine Angola Bangladesh Ukraine Namibia Panama
United Kingdom Benin Haiti Uzbekistan Paraguay Poland
Canada Botswana Kiribati Vanuatu Peru Portugal
United States Burkina Faso Lao PDR Vietnam Romania Puerto Rico
Azerbaijan Cameroon Myanmar Zambia Russian Federation Qatar
Georgia Central African Republic Nepal Zimbabwe Samoa Saudi Arabia
Israel Chad Vanuatu Serbia Seychelles
Russian Federation Congo Tuvalu South Africa Singapore
Turkey Cote d’Ivoire Solomon Islands Sri Lanka Slovak Republic
Australia Congo, Dem. Rep. Timor-Leste Suriname Slovenia
New Zealand Djibouti Thailand Spain
Korea, Dem. People’s Rep. Egypt, Arab Rep. Tonga Sweden
Japan Equatorial Guinea Turkey Switzerland
Singapore Eritrea Turkmenistan Trinidad and Tobago
Ethiopia Tuvalu United Arab Emirates
Gabon Venezuela, RB United Kingdom
Gambia, The United States
Ghana Uruguay
Sierra Leone
South Africa
South Sudan
Syrian Arab Republic
Sao Tome and Principe
Bahamas, The
Costa Rica
Dominican Republic
El Salvador
Puerto Rico
Trinidad and Tobago
Venezuela, RB
Antigua and Barbuda
Iran, Islamic Rep.
Kyrgyz Republic
Saudi Arabia
United Arab Emirates
Yemen, Rep.
  • LDC: Least Developed Countries

  • LLDC: Land Locked Developing Countries

  • SIDS: Small Island Developing States

  • Emerging Markets: BRICS + N-11