The EU public debt synchronization: A complex networks approach

Gkatzoglou, Fotios; Sofianos, Emmanouil; Barbier-Gauchard, Amélie Article The EU public debt synchronization: A complex networks approach Economies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Gkatzoglou, Fotios; Sofianos, Emmanouil; Barbier-Gauchard, Amélie (2025) : The EU public debt synchronization: A complex networks approach, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 13, Iss. 7, pp. 1-23, https://doi.org/10.3390/economies13070186 This Version is available at: https://hdl.handle.net/10419/329466 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Academic Editor: Angela Roman Received: 28 May 2025 Revised: 18 June 2025 Accepted: 23 June 2025 Published: 27 June 2025 Citation: Gkatzoglou, F., Sofianos, E., & Barbier-Gauchard, A. (2025). The EU Public Debt Synchronization: A Complex Networks Approach. Economies,13(7), 186. https://doi.org/ 10.3390/economies13070186 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). Article The EU Public Debt Synchronization: A Complex Networks Approach Fotios Gkatzoglou 1, Emmanouil Sofianos 2,* and Amélie Barbier-Gauchard 2 1 Department of Economics, Democritus University of Thrace, 69100 Komotini, Greece; [email protected] 2Bureau d’Economie Théorique et Appliquée (BETA), University of Strasbourg, 67000 Strasbourg, France; [email protected] *Correspondence: [email protected] Abstract This study examines the evolution of public debt among the 27 EU member states using Graph Theory tools; the Threshold Weighted–Minimum Dominating Set (TW–MDS) and the k-core decomposition method, alongside a standard network quantitative metric, the density. By separating the data into three distinct periods, pre-crisis (2000–2007), European sovereign debt crisis (2008–2015), and post-crisis (2016–2023), we examine the potential synchronization of the debt ratios among EU countries through cross-correlations of the public debts. The findings reveal that public debt correlation was at its highest level during the 2008–2015 period, reflecting the universal impact of the crisis and the subsequent synchronized fiscal and monetary policy measures taken within EU. A significantly lower network density is observed in both the preand post-crisis periods. These results contribute to the overall debate on fiscal stability and policy coordination by showing how EU countries tend to align their fiscal behaviors during periods of crisis while behaving more independently during stable times. In addition, we yield a deeper insight into how economic shocks reorganize public debt interconnections within the crisis period. Finally, this analysis highlights to what extent European economic integration strengthens connections between the fiscal positions (through public debt) of the European Union member countries. Keywords: public debt; European integration; complex networks; correlation; synchronization; graph theory JEL Classification: H63; O52; E62; C63 1. Introduction The formation of the Euro Area in 1999 was a significant milestone for the economic integration of the participating European Union nations. The introduction of the common currency was intended to promote stability and stimulate economic growth. It was a necessary—but not sufficient—condition to move from the theoretical towards a real Optimal Currency Area (OCA) by Robert Mundell standards (Mundell,1961). However, the European sovereign debt crisis that manifested in the 2010s exposed fundamental weaknesses in real economic integration, governance, and policy coordination among the member states. This crisis underscored the need for improved mechanisms to monitor and manage macroeconomic imbalances that could threaten the overall stability of the region (Alves et al.,2020). The evolution of the EU has been shaped by both economic and policy-driven challenges. The Maastricht Treaty, approved in 1992, laid the foundation for a monetary union Economies 2025,13, 186 https://doi.org/10.3390/economies13070186 Economies 2025,13, 186 2 of 23 by setting convergence criteria that countries must meet to adopt the euro. These criteria include limits on budget deficits, public debt levels, inflation, long-term interest rates, and exchange rate stability. The introduction of the euro in 1999 marked the beginning of a new monetary era, with the European Central Bank (ECB) assuming responsibility for the unified monetary policy. Following the establishment of the Eurozone, the Stability and Growth Pact (SGP) became the fiscal rule for countries belonging to the monetary union. The pact seeks to balance fiscal discipline with government flexibility. The fact that these fiscal rules have been reformed several times highlights that public debt remains a persistent challenge, especially under heightened scrutiny from financial markets. The latest reform, adopted in 2024, reaffirms the 3% GDP deficit limit and the 60% GDP debt threshold and introduces flexible adjustment pathways. Due to the geopolitical context, including the war in Ukraine, which started in 2022, and political instability in the United States since early 2025, this new rule has already been relaxed. This measure allows countries to exclude defense spending from the criteria assessed under the excessive deficit procedures for a period of four years. Nevertheless, uncertainty in the markets regarding the sustainability of public debt remains, influencing credit ratings and economic stability within the region. The financial crisis of 2008 and the following European sovereign debt crisis (which occurred in 2010) further exposed the weaknesses of a monetary union without a common fiscal policy. The crisis, triggered in part by the revelation of underreported budget deficits in certain Eurozone countries, led to a loss of market confidence and a sharp rise in borrowing costs. In response, emergency measures such as the European Stability Mechanism (ESM) were implemented to provide financial assistance to struggling economies. In addition to these temporary measures, permanent reforms were taken to address the drawbacks associated with the lack of a centralized fiscal authority. One of the most significant was the European Semester, established in 2010, which serves as a key instrument for ensuring economic policy alignment and coordination within the Euro Area. It integrates the Stability and Growth Pact (SGP), the Broad Economic Policy Guidelines (BEPGs), the European Employment Strategy (EES), the Lisbon Strategy, and the Social Open Method of Coordination (OMC), critical frameworks designed to maintain fiscal discipline and economic stability (Verdun & Zeitlin,2017). Under this system, the European Commission assesses national economic policies, issues country-specific recommendations, and monitors macroeconomic imbalances, including labor market and credit developments. These measures aim to synchronize economic cycles across member states and mitigate the risks of future financial crises (Papadopoulos & Piattoni,2019). In addition, the implementation and the second reform of the Stability and Growth Pact (through the Six Pack (2011) and the Two Pack (2013)) strengthened the coordination of fiscal policies and improved the monitoring of member states’ fiscal positions. Similar measures were taken following the 2020 pandemic crisis for the whole EU with the Next Generation EU, the European recovery plan (a 30-year bond launched to enable European countries to address both short-term emergencies and long-term challenges), and the third reform of the Stability and Growth Pact adopted in 2024 for EMU countries. Thus, since the advent of the Eurozone in 1999, numerous measures, both temporary (implemented only during economic crises) and permanent (reforms of fiscal rules), have been taken to ensure a greater convergence of the fiscal positions, thus strengthening European economic integration. This study examines the economic network of the 27 EU countries, focusing on public debt levels over three distinct periods: 2000–2007 (Period 1), 2008–2015 (Period 2), and 2016–2023 (Period 3). By analyzing cross-correlations among member states, it seeks to identify the patterns of synchronization in public debt dynamics. The findings will Economies 2025,13, 186 3 of 23 contribute to a deeper understanding of the EU’s fiscal landscape and inform future policy decisions that promote economic resilience and integration (Mundell,1961;McKinnon, 1963;Kenen,1969). This study also contributes to the assessment of fiscal integration in the EU by examining the synchronization and interconnections of public debt trajectories among member states. The originality of this research lies in its application of complex network analysis to measure both the network’s intensity as well as the number of interconnected countries within the European Union, as already employed by Papadimitriou et al. (2022). The member states constitute a union in which they adhere to shared rules regarding debt ratios. If the ratios move in different directions, implementing a unified policy across all countries becomes quite challenging. In light of this, the study aims to examine the following: (a) how the debt ratios of the European economies have evolved over time and whether their trajectories have followed the same direction; (b) the extent to which countries are interconnected in terms of debt evolution (specifically, whether they form groups exhibiting similar behavior); and (c) how the debt ratio synchronization was affected by events that had a global effect—principally the 2007–2008 financial crisis, followed by the COVID-19 pandemic in 2020 and the outbreak of the Russo-Ukrainian war in 2022. Understanding whether public debt trajectories are intertemporally linked across countries is essential for assessing the degree to which EU member states are moving in a common fiscal direction. The rest of the paper is structured as follows: In Section 2, we present the related literature, in Sections 3and 4, we present the dataset and the methodology, respectively, and in Section 5, we analyze the results, while Section 6concludes the paper. 2. Literature Review As the EU continues to navigate the challenges of fiscal sustainability, particularly in the aftermath of recent crises, it has become increasingly important to understand how its fiscal rules influence public debt. Recent research sheds light on the effectiveness of EU-level mechanisms and national fiscal frameworks, raising important questions about what works, what does not, and where reforms are needed. The study of Kraemer and Lehtimäki (2023) examines how EU-level fiscal institutions and national fiscal rules have influenced public debt dynamics across EU member states from 1990 to 2019. Using a panel data approach with fixed effects, the authors find that the Stability and Growth Pact (SGP) significantly reduced government debt, especially after its establishment. However, later reforms to the SGP showed mixed effectiveness. National fiscal rules had varying impacts depending on the rule type and sector, with general government balance and debt rules showing more consistent debt-reducing effects. Interestingly, some rules at the central government level were linked to rising debt. The results highlight that EU and national rules are largely complementary, but their design and coordination are critical for effective fiscal governance. In a follow-up study Kraemer and Lehtimäki (2024), assess the impact of European integration and the EU’s fiscal framework on public debt levels across the member states. Using the Synthetic Control Method, the authors construct counterfactual scenarios to estimate what debt trajectories might have looked like without EU membership or the SGP. Their findings reveal that, for most EU countries, actual government debt levels were significantly lower than those in the synthetic alternatives, suggesting a clear debtrestraining effect. This effect persists even in cases of limited rule compliance, highlighting the anchoring influence of EU fiscal structures. However, important heterogeneities are also underscored, particularly after the global financial crisis. Economies 2025,13, 186 4 of 23 In a complementary perspective, Arnold et al. (2022) argue that the European Union’s current fiscal rules have fallen short in ensuring debt sustainability and macroeconomic stabilization. The authors propose a risk-based framework linking fiscal targets to debt sustainability analysis, while maintaining the 3% deficit and 60% debt-to-GDP thresholds. A strengthened role for national fiscal institutions and medium-term frameworks is emphasized, alongside the creation of a central EU fiscal capacity to support public goods and counteract asymmetric shocks. The paper calls for institutional reforms, including an independent European Fiscal Council and more robust national fiscal councils. The goal is to design rules that are both effective and realistically implementable, especially in a high-debt, post-crisis context. Beyond fiscal rules, the broader questions of economic integration and convergence remain central to the viability of the Eurozone. From a theoretical standpoint, the concept of economic synchronization has been widely discussed in the literature, particularly in relation to the Optimum Currency Area (OCA) theory (Mundell,1961;McKinnon,1963; Kenen,1969). The OCA framework posits that regions adopting a common currency should exhibit synchronized business cycles and economic homogeneity to ensure effective monetary policy transmission. However, empirical studies suggest that persistent structural disparities among Eurozone economies pose challenges to achieving full convergence (Frankel & Rose,1998;Rogoff,1985). Recent empirical studies have continued to uncover persistent structural disparities among the Eurozone economies. For instance, Coutinho and Turrini (2020) analyzed how economies in the Euro Area have been converging compared to other EU and non-EU countries. They found that, overall, the patterns are similar across groups. However, the countries that initially adopted the euro have shown weaker convergence—a trend that seems to have started even before the financial crisis, likely due to the persistent differences in productivity levels. Similarly, Franks et al. (2018) examined the economic convergence among Euro Area countries and concluded that while nominal convergence occurred in areas such as inflation and interest rates, the real convergence of per capita income levels has not been achieved among original Euro Area members following the introduction of the common currency. Methodologically, network analysis has gained significant traction in recent years, as it provides a robust framework for evaluating economic linkages and offering valuable insights into economic synchronization. Previous studies have applied this approach to evaluate financial contagion, trade dependencies, and sovereign debt risks (Dellas & Tavlas, 2009). One such study is that of Papadimitriou et al. (2014), which focuses on analyzing financial stability and economic convergence. The authors present two empirical examples: one based on a banking network of 200 US banks and another based on GDP growth rates among 22 European countries. Their findings emphasize the effectiveness of the T-MDS methodology in analyzing complex economic systems. By leveraging techniques from complex networks, researchers can identify synchronization patterns that significantly contribute to stability. Within the framework of our research, policy coordination is most effective when synchronization is widespread and encompasses the maximum possible number of member states. In another study, Papadimitriou et al. (2016) highlight the utility of network-based approaches in assessing economic cohesion within the Eurozone. The research examines the synchronization of economic cycles through complex network metrics, demonstrating how central and peripheral countries interact within the Euro Area’s financial framework. The findings suggest that while certain core economies exhibit strong interconnections, peripheral nations often display greater volatility and weaker ties to the network center, impacting overall economic stability. Economies 2025,13, 186 5 of 23 Further, Dias (2012) examines the sovereign debt crisis in the European Union, employing minimum spanning trees within a rolling window framework. The study uses data from 19 EU countries concerning daily 10-year government bond yield rates, spanning from April 2007 to October 2010. The results highlight a lack of synchronization between the four most impacted countries—Greece, Ireland, Portugal, and Spain—and the more resilient nations in the Euro Area, such as Germany and the Netherlands. Kantar et al. (2014) examine the relationships between European countries by constructing networks based on debt-to-GDP ratios. To analyze the topological properties over the period of 2000–2011, they develop hierarchical and minimum spanning trees. Their study covers both the entire period (2000–2011) and two specific sub-periods ( 2000–2004 and 2005–2011 ). The analysis reveals distinct clusters of countries grouped by their debt levels. Matesanz and Ortega (2015) examine the evolving network dynamics of quarterly public debt-to-GDP ratios across 29 European countries from 2000 to 2014. Utilizing a rolling window approach, they construct hierarchical trees and minimum spanning trees while evaluating various standard metrics associated with complex networks. Their findings reveal that the period of the financial crisis exhibits the highest level of connectivity among countries within the network. Papadimitriou et al. (2022) explore the synchronization levels of a range of macroeconomic variables in Europe, including the general government debt-to-GDP ratio. Using correlation-based networks and Graph Theory metrics, their analysis spans three distinct periods: 1999–2004, 2005–2010, and 2011–2019. Their findings reveal significant synchronization of the debt-to-GDP ratio during the 2005–2010 period, a trend that is disrupted in the subsequent interval. The present study builds on this literature by examining public debt dynamics in the EU over three distinct periods. By analyzing the evolving structure of the network, this research aims to contribute to the ongoing discourse on fiscal stability and policy coordination within the European monetary union. Furthermore, from a methodological standpoint, the potential of k-core analysis remains largely unexplored in this context. 3. The Dataset The dataset consists of government consolidated gross debt for the general government sector, expressed as a percentage of GDP 1 , in quarterly frequency (Figure 1), spanning from 2000Q1 to 2023Q4 2 . It concerns the 27 countries (Table 1) that are currently EU member states, of which 18 also belong to the Euro Area and use the common currency. Figure 1 illustrates the progression of debt-to-GDP ratios for the 27 EU member states from the first quarter of 2000 to the first quarter of 2024. The figure highlights substantial variation among countries, with ratios spanning a wide range from low to high levels. Descriptive statistics for each country and each period are presented in Table A1 in Appendix A. Table 1. The 27 member states of the European Union in the dataset (alphabetical order). 1 Austria 15 Italy 2 Belgium 16 Latvia 3 Bulgaria 17 Lithuania 4 Croatia 18 Luxembourg 5 Cyprus 19 Malta 6 Czechia 20 Netherlands 7 Denmark 21 Poland 8 Estonia 22 Portugal Economies 2025,13, 186 6 of 23 Table 1. Cont. 9 Finland 23 Romania 10 France 24 Slovakia 11 Germany 25 Slovenia 12 Greece 26 Spain 13 Hungary 27 Sweden 14 Ireland  Figure 1. Evolution of the 27 EU member states debt-to-GDP ratio from 2000Q1 to 2024Q1. Around 2008Q2 and 2020Q2, a subtle line appears to be forming, indicating a rise in the ratios. Data source: Eurostat. The GDP of each country, expressed in current prices (millions of euros), is utilized to assign a corresponding weight to that country (the specific details are thoroughly explained in Section 4). The data source for government debt and GDP is Eurostat, ensuring consistency and comparability across all EU member states. The objective of this research, as previously stated, is to examine the temporal evolution of the debt-to-GDP ratio among the member states of the European Union and to evaluate the extent to which their movements align over time, suggesting possible synchronization. To achieve this and uncover possible structural shifts in public debt dynamics, we divide the initial 2000–2023 dataset into three consecutive intervals: 2000–2007, 2008–2015, and 2016–2023. The partitioning is based on significant macroeconomic events that shaped fiscal policy and debt dynamics in the EU. More specifically, the intervals are the following: •2000–2007 (Period 1): A pre-crisis period characterized by relative economic stability, with no major economic shocks. Moreover, the largest expansion occurred during this period, with ten countries joining the EU in 2004 (Czech Republic, Estonia, Cyprus, Latvia, Lithuania, Hungary, Malta, Poland, Slovakia, and Slovenia), followed by two more in 2007 (Bulgaria and Romania). This brought the total number of member states to 27 (EU27). Economies 2025,13, 186 7 of 23 • 2008–2015 (Period 2): The period encompassing the global financial crisis and the subsequent European sovereign debt crisis, along with the fiscal measures and policy responses implemented to stabilize EU economies for both crises. • 2016–2023 (Period 3): A post-crisis period marked by two major external shocks: the COVID-19 pandemic and the war in Ukraine, both of which had significant macroeconomic repercussions, including energy price fluctuations and inflationary pressures. The temporal partitioning of time periods enables the network analysis techniques to assess the evolving interconnections, in terms of government debt levels. By segmenting the data, we aim to capture shifts in the fiscal relationships across EU member states, identifying key structural changes in debt dynamics before, during, and after economic crises. 4. Methodology The analysis is conducted separately for each defined time period, yielding three successive networks that reflect the temporal evolution of inter-country relationships. Each node represents an EU country for a total of 27 nodes. The edges that connect the nodes denote the relationship between each pair of countries in terms of the selected variable. In our empirical analysis, the variable selected is the Pearson correlation of the debt-to-GDP ratio for each pair of countries. The Pearson correlation coefficient ri,jis defined as ri,j≜COVZi,Zj qVAR (Zi)VARZj(1) where Zi , Zj are pairs of observations for the variable under examination for member states iand j. The correlation coefficient value, in the range of [ − 1, 1], quantifies the strength and direction of the linear relationship between the two countries. Values close to the upper and lower bounds indicate a strong relationship; when near +1, it signifies a strong positive relationship, while values close to − 1 indicate a strong negative relationship. Values near zero suggest no linear association between countries iand j. The following part introduces some key concepts of network analysis used in the empirical part of this paper, aiming to provide a clearer understanding of the concept. Graph/network: A graph G= (V,E) consists of a set of nodes Vand a set of edges E through which the nodes are connected (i.e., an edge e i,j ∈ Elinks the nodes iand j ∈ V). Each edge e i,j ∈ Econnecting the nodes iand jcarries a value which is calculated by the Pearson’s correlation coefficient ri,j (Figure 2a). This parameter measures the temporal similarity between the nodes iand j. In addition, the set of nodes associated with node iis defined as the neighborhood Biof node i, i.e., Bi=j∈V:∃eij ∈E. Isolated node: A node i ∈ Vis considered isolated when there are no edges connecting it to any of the rest nodes of the network. Interconnected node: A node i ∈ Vis termed interconnected when it is connected to at least one other node in the network via an edge. In its initial arrangement, the network of the present research is complete, meaning that there is an edge connecting every single pair of nodes, and hence no isolated nodes exist. Dominating Set: A Dominating Set DS ⊆ Vis a subset of the interconnected nodes in the network, characterized by the following property: every node ithat is not part of the DS is connected to at least one DS node jthrough an edge e i,j i.e., ∀i/∈DS,∃j∈DS :eij ∈E(Figure 2b). Economies 2025,13, 186 8 of 23  (a)(b)  (c)(d) Figure 2. (a) Presentation of a graph paradigm where each edge’s value is measured by the Pearson’s correlation coefficient ri,j ; (b) a Dominating Set of the network (red nodes); (c) a Minimum Dominating Set of the network (red nodes); (d) a Threshold Minimum Dominating Set of the network (red nodes— a threshold of 0.8 was implemented). The edges portrayed with dashed lines did not survive the thresholding step. For each of the nnodes of the network, a binary parameter x i ,i= 1, 2, . . . nis considered to symbolize its DS membership status, such that xi=(0, i f i /∈DS 1, i f i ∈DS (2) Formula (3) mathematically displays the DS concept: xi+∑ j∈Bi xj≥1, i=1, 2, . . . n(3) An interconnected node ihas to be either a node of the DS or adjacent (i.e., connected) to at least one such node. However, this is not a mutually exclusive relationship; there may be nodes that meet both conditions (Figure 2b). When the edges in a network reflect high similarity, as is the case here, the dominance attribute produces an intriguing effect: the behavior of the entire network can be effectively represented by the behavior of the DS. Minimum Dominating Set: The Minimum Dominating Set (MDS) is the DS with the smallest cardinality. min n ∑ i=1 xi!(4) The Minimum Dominating Set (MDS) is identified by applying condition (4) while complying with the constraints outlined in condition (3) (Figure 2c). Since the core focus of the current research is the temporal similarity between nodes, the standard Minimum Dominating Set (MDS) algorithm is not well-suited for this purpose due to an inherent limitation. In a correlation-based network, such as the one examined here, edges do not necessarily indicate similarity. Given that correlation values range from Economies 2025,13, 186 15 of 23 is probably the result of the heightened debt of almost all EU countries and coordinated policy measures such as the ESM. The presence of a large, densely connected core confirms that public debt dynamics were closely interrelated across member states. In contrast, the initial (2000–2007) and last (2016–2023) periods display significantly lower levels of connectivity and, thus, synchronization. Period 1 features a network of weaker interconnections between EU member states. However, despite lower overall density, the presence of a highly connected core during this period (a 9-core with a density of 1 and a 7-core with a density of 0.97 for 0.7 and 0.8, respectively) indicates the existence of a subnetwork of countries exhibiting highly similar behavior. The network in Period 3, even with slightly greater density than in Period 1, is composed differently. The absence of isolated nodes suggests higher synchronization across the network, and it is likely to be a sign of the influence of the EU-level coordination of policies due to external pressures such as the COVID-19 pandemic, the war in Ukraine, and the proactive policy of the 2008 crisis. Several factors can contribute to post-crisis divergence. The drivers of divergence stem from structural heterogeneities linked to structural differences in an economic, demographic, institutional, and political nature that exist in Europe. Such structural heterogeneities help explain why European economies respond differently to economic shocks (see Guerini et al.,2018;Campos et al.,2020). Differences in productive specialization remain significant within the EU and contribute to varying reactions to economic shocks, as each economy relies to different extents on certain sectors, sectors that themselves respond differently to changes in demand, price fluctuations, supply shocks, etc. The sociodemographic characteristics of countries (e.g., the proportion of over-65s in the population and exposure to the risk of poverty or social exclusion) and institutional diversity (e.g., parliamentary vs. semi-presidential systems), as highlighted by Lopez-Gomez (2024), also play an important role. A qualitative examination of the networks (Tables A2 and A3 in Appendix B) uncovers several noteworthy findings. While the TW-MDS algorithm tends to favor countries with higher GDPs when identifying dominant nodes, this is not consistently the case. For instance, in the case of the 0.7 threshold, Germany—despite having the highest GDP in the EU—is not identified as a dominant node in any period except for Period 3. Interestingly, even during Period 2, when network connectivity is high, Germany does not emerge as dominant. Additionally, Germany is absent from the maximum k-core in all three periods. Summarizing the behavior of the EU’s three largest economies, France appears as a dominant node in Periods 1 and 3, while Italy consistently holds a dominant status across all periods. Under the stricter 0.8 threshold, Germany is identified as dominant in Periods 1 and 3, France does not appear as dominant in any period, and Italy is identified as dominant only in Period 3. Notably, Germany remains excluded from the maximum k-core throughout all periods, regardless of the threshold. It is worth noting that Germany not only remains outside the maximum k-core but also maintains significantly fewer connections than the countries included in that core across all periods. This discrepancy is especially evident under the 0.8 threshold: in Period 1, Germany is part of the 3-core while the maximum one is the 7-core; in Period 2, it belongs to the 6-core, whereas the maximum is the 14-core and includes around 60% of the countries; and in Period 3, Germany is again in the 3-core, compared to the maximum of the 6-core (Table A4 in Appendix B). These findings suggest that Germany, despite being the EU’s largest economy, occupies a peripheral position within the network in terms of its debt ratio evolution. This has important implications, potentially complicating the policy coordination efforts related to fiscal stability. Another finding is that the behavior of the countries appears to be independent of their Euro Area membership status. No clear patterns emerge in terms of clustering into Economies 2025,13, 186 16 of 23 neighborhoods or cores or the identification of isolated status based on whether a country is part of the common currency area. Furthermore, the synchronization observed during the 2008–2015 period does not seem to be limited to Euro Area members. Finally, there is no indication of geographical divisions, such as a north–south split. Tables A2 and A3 in Appendix Bprovide relative information for the 0.7 and 0.8 thresholds, respectively. The findings presented suggest that big crises have a considerable impact on fiscal interrelations between the EU member states. During relatively tranquil periods, countries tend to behave more independently, reflecting a rather dissimilar behavior in their actions. In contrast, events that trigger global financial uncertainty—along with the resulting policy interventions (such as the ESM)—force countries to act in a more coordinated manner, following a similar trajectory. This article highlights the impact of progress made since 2008 in terms of fiscal integration on the synchronization of national public debt. Indeed, the ad hoc intervention mechanisms implemented since the 2008 financial crisis (European Stability Mechanism, European Semester, Six Pack, Two Pack, Next Generation EU, etc.) appear to have borne fruit by preventing excessive national divergence in public finance. These results confirm that when stabilization mechanisms for economic shocks that affect countries differently (i.e., asymmetric shocks, or symmetric shocks with asymmetric effects) are implemented at the EU level, the ultimate impact on national public debt is reduced, and synchronization is strengthened. In such cases, an EU-level stabilization mechanism steps in to cushion economic shocks, relieving pressure on national budgets. This suggests that the optimality of the Eurozone may be an endogenous process, driven by the gradual implementation of tools for stabilizing economic shocks within the Eurozone, but it will also have an impact on non-Eurozone countries. Looking ahead, the key challenge is to ensure the sustainability of these alternative economic stabilization mechanisms, which have so far been implemented only temporarily, to ease the constraints on national budgets. The creation of a European fiscal capacity could fulfill this role of absorbing symmetric shocks (with asymmetric effects due to structural heterogeneities) on national economies (see Burriel et al.,2020), ultimately stabilizing public debt trajectories, as already occurs in the United States, Canada, and Australia. This would allow national budgets to focus primarily on delivering public goods and services (such as education, health, social protection, security, and transport) while being subject to rigorous oversight under the reformed Stability and Growth Pact (2024), excluding national defense expenditure. Meanwhile, cyclical stabilization would be handled through a European fiscal capacity. 6. Conclusions In this paper, we explore how fiscal behavior and interconnections among EU countries evolved over time, especially in response to major economic shocks, based on public debt. By using Graph Theory tools, we examined public debt dynamics across three distinct periods: Period 1, the years before the global financial crisis (2000–2007), Period 2, the turbulent years of crisis and response (2008–2015), and Period 3, the post-crisis period shaped by new global disruptions (2016–2023). It is important to note that this study is limited to the period following the year 2000 due to data constraints. While analyzing the network of countries prior to 2000 could potentially yield valuable insights into the early stages of the union, such an analysis is unfortunately not feasible given the lack of available data. The findings show a clear pattern: when external pressures rise, so does the fiscal synchronization across member states. Period 2 presents the highest level of connectivity among countries, with fewer isolated nodes and a large, dense core. This suggests that the Economies 2025,13, 186 17 of 23 financial and sovereign debt crises, combined with policy responses like the ESM, brought EU countries closer together in terms of fiscal behavior. In contrast, in more stable periods, such as the years preceding the crisis and the most recent interval, the overall network density tends to decrease. Still, there is nuance here. Even in those less turbulent periods, we discovered tightly connected subnetworks, pointing to smaller groups of countries with similar debt dynamics. Interestingly, the network in Period 3, while not as synchronized as in the crisis years, shows some improvement over the pre-crisis era. The absence of isolated countries and a slightly greater density suggest that past experiences and EU-wide coordination mechanisms might have laid the groundwork for closer fiscal alignment, even in the face of new shocks like the pandemic or the war in Ukraine. Thus, our results highlight the impact that the various measures implemented to support EU countries during the various economic crises have had on the intensity of connections in terms of the trajectories of public debt. By taking charge of part of the cushioning of the cyclical shocks linked first to the financial crisis (Period 2) and then to the pandemic crisis (Period 3), the European mechanisms implemented have favored the better synchronization of public debts, an intensification of the connections between national public debts, and also the disappearance of the isolated countries. To conclude, the results highlight the importance of shared challenges in shaping how EU member states manage their public finances. During periods of stability, national strategies tend to diverge, reflecting local priorities and political contexts. However, in times of crisis, there is a clear shift toward convergence, likely driven by the need for coordinated responses and shared tools. This has implications not only for interpreting past behavior but also for how EU policymakers may approach the design of future fiscal frameworks. Thus, a better synchronization of national public debts in the EU represents a real opportunity to ensure the lasting monetary and financial stability of the monetary union and could appear as the solution to the challenge posed by the surveillance of national public finances. More synchronized national public debts would make it possible to establish effective common European tools to support the cushioning of economic shocks. Looking ahead, maintaining debt synchronization may require formalizing some of the stabilization tools introduced since 2008. A permanent European fiscal capacity could help absorb economic shocks more evenly, reducing the pressure on national budgets and allowing them to focus on providing public goods and services to citizens. Over time, this could support stronger fiscal alignment across the EU and reinforce the goals of the reformed Stability and Growth Pact. Author Contributions: Conceptualization, F.G., E.S. and A.B.-G.; methodology, F.G., E.S. and A.B.-G.; software, F.G., E.S. and A.B.-G.; validation, F.G., E.S. and A.B.-G.; formal analysis, F.G., E.S. and A.B.-G.; investigation, F.G., E.S. and A.B.-G.; resources, F.G., E.S. and A.B.-G.; data curation, F.G., E.S. and A.B.-G.; writing—original draft preparation, F.G., E.S. and A.B.-G.; writing—review and editing, F.G., E.S. and A.B.-G.; visualization, F.G., E.S. and A.B.-G.; supervision, F.G., E.S. and A.B.-G.; project administration, F.G., E.S. and A.B.-G.; funding acquisition, E.S. All authors have read and agreed to the published version of the manuscript. Funding: Emmanouil Sofianos would like to acknowledge that this work is part of the Interdisciplinary Thematic Institute MAKErS of the ITI 2021-2028 program of the University of Strasbourg, CNRS and INSERM. It has received financial support from the IdEx Unistra (ANR-10-IDEX-0002), and from the Programme Investissement d’Avenir” as part of the SFRI-STRAT’US project(s) (ANR-20SFRI-0012). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Economies 2025,13, 186 18 of 23 Data Availability Statement: The original data presented in the study are openly available in Eurostat at https://ec.europa.eu/eurostat/web/main/data/database (accessed on 18 June 2025). Conflicts of Interest: The authors declare no conflict of interest. Appendix A Table A1. Descriptive statistics (mean, standard deviation, minimum, and maximum) by country and period. Mean Std Min Max Country 2000– 2007 2008– 2015 2016– 2023 2000– 2007 2008– 2015 2016– 2023 2000– 2007 2008– 2015 2016– 2023 2000– 2007 2008– 2015 2016– 2023 Germany 62.86 74.38 65.06 3.36 5.10 3.39 57.60 64.00 58.70 67.30 81.00 70.90 France 64.14 87.95 105.92 3.40 9.48 7.14 58.80 67.00 98.10 68.20 98.30 117.60 Italy 108.40 123.29 140.11 2.14 10.41 7.07 103.50 104.40 133.70 112.70 138.10 157.60 Spain 47.91 74.30 107.28 7.06 25.09 7.41 35.70 34.90 97.70 59.70 104.50 124.20 Netherlands 50.00 60.71 52.31 3.40 6.95 5.14 42.80 43.60 44.40 59.60 68.20 63.40 Sweden 47.94 39.47 37.75 3.98 3.11 3.65 39.20 35.00 30.30 57.40 45.70 43.50 Belgium 103.35 103.84 106.32 7.93 5.92 4.02 87.30 89.70 97.70 117.40 111.00 116.00 Austria 70.51 80.38 79.58 2.90 5.87 4.47 65.40 66.70 70.70 76.10 86.10 86.90 Poland 43.72 51.93 51.60 4.18 4.15 3.32 35.70 42.70 45.70 49.10 57.90 58.60 Denmark 46.37 45.08 39.77 8.59 6.60 3.83 29.50 26.40 33.60 58.20 52.60 46.40 Greece 104.31 151.74 187.98 1.89 26.03 12.62 99.90 103.70 167.50 107.80 182.30 215.90 Finland 42.25 52.14 70.33 3.27 11.14 4.13 34.80 30.70 64.80 48.60 68.80 77.10 Portugal 63.97 110.53 123.52 7.00 22.82 9.26 54.00 71.40 97.90 73.80 134.70 137.50 Ireland 30.61 87.54 59.25 5.19 29.28 10.41 23.60 26.40 42.40 45.20 123.10 75.90 Czechia 25.27 37.81 37.72 4.56 5.78 4.24 15.20 26.00 29.60 30.60 44.90 43.50 Hungary 59.53 77.77 73.27 4.74 4.75 3.91 52.20 64.40 65.00 66.80 83.80 79.70 Romania 19.57 29.33 41.01 4.76 9.61 6.27 11.60 10.50 33.50 25.90 39.10 49.70 Croatia 38.00 62.80 75.75 2.53 16.33 6.45 31.40 35.60 61.80 41.70 85.20 88.50 Luxembourg 7.58 18.85 23.26 0.40 4.37 2.47 6.80 8.30 19.60 8.30 26.80 28.30 Slovakia 42.15 44.58 54.47 7.58 10.19 4.10 29.60 26.60 48.00 51.70 57.50 60.20 Slovenia 26.58 51.32 75.52 1.32 20.89 5.36 22.90 21.90 66.00 28.60 84.70 85.50 Bulgaria 44.25 17.21 23.67 19.89 4.83 2.86 16.30 12.70 20.00 77.40 28.70 29.30 Lithuania 20.41 33.30 38.68 3.21 9.41 3.62 14.60 13.30 32.60 24.50 42.40 45.90 Cyprus 61.16 79.32 98.83 4.14 24.25 11.97 53.10 46.00 73.60 68.10 114.70 117.50 Latvia 12.94 38.10 41.77 1.84 11.50 2.86 8.60 10.40 37.70 15.30 48.80 46.50 Estonia 4.97 8.76 13.82 0.62 2.52 4.82 3.60 4.10 8.30 5.80 11.70 20.20 Malta 65.78 64.60 47.74 3.35 3.60 4.73 60.80 55.00 39.20 71.20 69.70 56.20 Appendix B Table A2. Identity of the dominant nodes (with the corresponding cardinality of the neighborhood they represent), the isolated nodes, and the entities that compose the max k-core subgraph in periods 2000–2007, 2008–2015, and 2016–2023 for the 0.7 threshold level. 2000–2007 2008–2015 2016–2023 TW-MDS Dominant nodes Neighborhood cardinality Dominant nodes Neighborhood cardinality Dominant nodes Neighborhood cardinality France 6 Italy 21 Germany 8 Italy 3 Netherlands 23 France 10 Sweden 12 Italy 7 Cyprus 2 Sweden 7 Isolated nodes Isolated nodes Isolated nodes Austria Malta No isolated Greece Luxembourg Slovenia Economies 2025,13, 186 19 of 23 Table A2. Cont. 2000–2007 2008–2015 2016–2023 k-core decomposition (max k-core) Spain Sweden Belgium Denmark Finland Ireland Romania Slovakia Bulgaria Lithuania France Italy Spain Netherlands Belgium Austria Denmark Greece Finland Portugal Ireland Czechia Romania Croatia Luxembourg Slovakia Slovenia Lithuania Estonia France Spain Finland Czechia Romania Luxembourg Slovakia Latvia Estonia Table A3. Identity of the dominant nodes (with the corresponding cardinality of the neighborhood they represent), the isolated nodes, and the entities that compose the max k-core subgraph in periods 2000–2007, 2008–2015, and 2016–2023 for the 0.8 threshold level. 2000–2007 2008–2015 2016–2023 TW-MDS Dominant nodes Neighborhood cardinality Dominant nodes Neighborhood cardinality Dominant nodes Neighborhood cardinality Germany 4 Denmark 14 Germany 6 Spain 10 Slovenia 15 Italy 3 Poland 4 Netherlands 4 Cyprus 1 Denmark 3 Latvia 1 Finland 8 Isolated nodes Isolated nodes Isolated nodes Italy Malta No isolated Austria Greece Croatia Luxembourg Slovenia k-core decomposition (max k-core) Spain Sweden Belgium Denmark Ireland Romania Slovakia Bulgaria Lithuania France Italy Spain Netherlands Belgium Austria Greece Finland Portugal Czechia Romania Croatia Luxembourg Slovakia Lithuania Estonia France Finland Romania Luxembourg Slovakia Latvia Estonia Economies 2025,13, 186 20 of 23 Table A4. The highest k-core to which each EU member state belongs across all three time periods, under both the 0.7 and 0.8 threshold levels. 2000–2007 2008–2015 2016–2023 country threshold 0.7 0.8 0.7 0.8 0.7 0.8 highest k-core Germany 4 3 12 6 6 3 France 4 3 16 14 7 6 Italy 3 0 16 14 6 2 Spain 9 7 16 14 7 5 Netherlands 8 6 16 14 5 2 Sweden 9 7 7 3 5 2 Belgium 9 7 16 14 6 3 Austria 0 0 16 14 6 3 Poland 4 3 12 7 6 3 Denmark 9 7 16 10 5 3 Greece 0 0 16 14 3 1 Finland 9 6 16 14 7 6 Portugal 4 3 16 14 5 3 Ireland 9 7 16 12 4 2 Czechia 3 1 16 14 7 3 Hungary 4 3 6 2 6 3 Romania 9 7 16 14 7 6 Croatia 2 0 16 14 5 3 Luxembourg 0 0 16 14 7 6 Slovakia 9 7 16 14 7 6 Slovenia 0 0 16 13 6 3 Bulgaria 9 7 7 3 5 1 Lithuania 9 7 16 14 6 3 Cyprus 2 1 15 13 5 3 Latvia 4 1 12 7 7 6 Estonia 8 3 16 14 7 6 Malta 2 1 0 0 5 3 Appendix C The figures below 6 illustrate the network topology as derived from the TW-MDS technique, showcasing its transformation across the three periods under examination (applying a 0.7 threshold). Nodes in green highlight the dominant entities, while red marks those that remain isolated. Gray nodes signify those that are interconnected but not dominant. All connections of a dominant node are colored green, whereas links between non-dominant interconnected nodes appear in gray. Economies 2025,13, 186 21 of 23  Figure A1. The 2000–2007 EU network (threshold level of 0.7). Figure A2. The 2008–2015 EU network (threshold level of 0.7). Economies 2025,13, 186 22 of 23  Figure A3. The 2016–2023 EU network (threshold level of 0.7). Notes 1 Germany’s data in Eurostat data base is marked as “p” (provisional, indicating values that are subject to revision) for the period from 2018Q1 to 2023Q4, while for the Netherlands, this provisional status applies from 2023Q1 to 2023Q4. According to Eurostat’s revision calendar, the most recent update to these data was on 22 April 2025. 2 At the start of the examined period (2000Q1), 15 countries were part of the European Union, and 11 of them had adopted the common currency. 3 The density calculation is directly affected by the threshold level. Although in its initial setup the network of the current study is complete, after the threshold imposition on the edge’s values, several edges are removed from the network. Thus, Et counts the edges that survived the thresholding step. 4Being aware of the k-core sub network structure for any k, it is possible to calculate the density of the subgraph. 5 For instance, setting a threshold of 0.7 allows the survival of correlations ranging between 0.7 and 1. 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