Politics | European Union » A hypergraph analysis of the European Commission lobby network

Azaiez and Mandel Applied Network Science https://doi.org/101007/s41109-025-00735-6 (2025) 10:45 Applied Network Science Open Access RESEARCH A hypergraph analysis of the European Commission lobby network Amina Azaiez1* and Antoine Mandel1 *Correspondence: Amina Azaiez amina.azaiez@univ-paris1fr 1 Sorbonne Economics Centre, Paris 1 Panthéon-Sorbonne University, Paris, France Abstract We use transparency data published by the European Commission (EC) to perform a quantitative analysis of the structure and dynamics of stakeholder consultation in the EU policy-making process. We analyze the data through the prism of network theory by constructing a hypergraph whose nodes are EC officials and stakeholders, and hyperedges connect entities that participate in the same meetings. Our analysis highlights the presence of a hierarchical core-periphery structure, with a few well-connected entities that occupy the center of the network and enjoy a stable integration in the EC policy-making

process. Examination of the core composition reveals that companies and trade associations maintain closer relationships with the EC. A regression analysis shows that lobbying efforts and company size are significant predictors of company centrality, independent of other objective characteristics. Our findings provide quantitative evidence supporting the perception of lobbying as a tool dominated by well-connected actors, while also revealing heterogeneous lobbying strategies across stakeholder groups. Keywords Network analysis, Lobbying, European Commission, Hypergraphs, Rich-club phenomenon Introduction Lobbying is often perceived as a tool for entities with large financial or social capital to influence policymaking, potentially sidelining ordinary citizens’ interests. However, large-scale quantitative studies confirming or rejecting this perception are lacking. Indeed, most existing studies survey the actions and the interactions of small samples of stakeholders or policy-makers

(Dür et al. 2015; Heaney 2014; Pappi et al 1999; Knoke 2021). This paper leverages on a comprehensive dataset of face-to-face interactions between high-level European Commission (EC) representatives and stakeholders to provide a detailed analysis of the EC lobbying network. More specifically, we examine whether there exist a highly interconnected core of actors that could dominate the network. Indeed, network theory is particularly well suited for studying power concentration in lobbying contexts. First, topological measures, like Rich-club coefficients (Colizza 2006; Larsen and Ellersgaard 2017; Vitali et al. 2011), offers concrete evidence for the existence The Author(s) 2025. Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 40 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source,

provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit ​h​t​t​p​:​/​​/​c​r​e​a​​t​i​v​e​c​o​​m​m​o​n​​s​​o​r​g​​/​l​i​c​e​​n​s​ e​s​/​b​​y​-​n​c​​-​n​d​/​4​.​0​/ Azaiez and Mandel Applied Network Science (2025) 10:45 of a hierarchical structure within the network. Second,

it permits the identification of central actors and of the determinants of centrality (Cruz et al. 2017; Knoke 2021) Third, integrating higher order interactions offers insights into coalition formation and collaborative lobbying strategies. Accordingly, we model these interactions by constructing a hypergraph where the nodes represent EC officials and stakeholders, and a hyperedge connects entities that participate in the same meetings. This hypergraphic perspective allows us to analyze both the micro and the macro structures of interactions. We apply this approach both to the full network containing stakeholders and policy-makers and to the network of stakeholders only. Our analysis highlights the presence of a core-periphery structure, with a few well-connected entities that occupy the center of the network and enjoy a stable integration in the EC policy-making process. Examination of the core composition reveals that companies and trade associations maintain closer relationships

with the EC, while NGOs show increased core presence in the stakeholder-only network. This suggests different engagement strategies among stakeholder groups, with NGOs potentially focusing more on coalition-building and shared meetings. A regression analysis of company centrality identifies several determinants The number of full-time equivalent staff dedicated to lobbying activities and company size are strongly associated with higher centrality. Companies with EU-based headquarters and those with global or European levels of interest also exhibit significantly higher centrality. These findings provide quantitative evidence supporting the perception of lobbying as a tool dominated by well-connected actors, while also revealing heterogeneous lobbying strategies across stakeholder groups. Our findings can be used to assess the performance of the EU consultation process with respect to its objectives of increasing the legitimacy of the European policy process and of maintaining an open,

transparent and regular dialogue with stakeholders. Our work relates to the trend of literature that has tried to quantitatively assess the influence exerted by stakeholders on the policy-making process. In Europe, several studies have focused on textual analysis to assess the lobbying success of interest groups Dür (2008); Klüver (2013); Bunea and Ibenskas (2015). These studies aim to measure influence and to identify its determinants by comparing the recommendations or preferences of interest groups and the final legislative text. This method, however, has a number of limitations, including the quality of the results obtained with text processing algorithms Bunea and Ibenskas (2015), the black boxing of the influence underpinning process Dür (2008), and the inability to capture the influence exerted at other stages of the policy process such as the agenda-setting phase, and by other means such as direct interaction with policy-makers. Another strategy consists of measuring the

reputation of influence This is usually done by conducting surveys of organizations and/or policy-makers and asking them to select from a list of entities they consider influential about a certain issue Heaney (2014); Pappi et al. (1999); Dür and De Bièvre (2007) Our network-based perspective is complementary to these existing contributions as it provides a global map of the lobbying landscape and allows one to study the distribution of power among interest groups, e.g, through centrality measures, the existence of a core-periphery structure Larsen and Ellersgaard (2017); Laumann (1987), and rich club phenomena Colizza (2006). Network-based approaches have similarly been used to analyze the US labor policy sphere through the network of inter-organizational Page 2 of 19 Azaiez and Mandel Applied Network Science (2025) 10:45 communication Knoke (2021) or the role of political family networks in electoral outcomes in the Philippines Cruz et al. (2017) We represent more specifically

the EU lobbying network as an hypergraph. The existing literature Bonacich et al (2004); Estrada and Rodriguez-Velazquez (2005) emphasizes the necessity of adopting such a hypergraphic perspective to represent phenomena on social networks for which the coordination and/ or the simultaneous interactions of (more than two) actors are important determinants of the dynamics. In our analysis of the EU lobbying network, we first observe that the hypergraph model fits well with the empirical structure of interactions. Second, interactions of different sizes might have different purposes and different impact on network structure, e.g, large meeting/edges might be required to foster consensus and might play a crucial role in the overall connectivity of the hypergraph. Third, the composition of edges/meetings allows to characterize the cooperation and/or competition strategies between actors Accordingly, the key role of cooperation and competition in policy processes have led to the use of

hypergraphs in a number of recent contributions on the dynamics of legal systems Coupette et al. (2024), the negotiation of environmental treaties Boulet et al (2019), the build-up of policy coalitions Le Breton et al. (2012); Groseclose and Snyder (1996), or the analysis of governance Hébert-Dufresne et al. (2025) Results In the framework of the EU transparency policy, high-level European Commission (EC) officials, namely Commissioners, cabinet members and director-generals, are required to publish information about the meetings they hold with stakeholders. We have downloaded from the EC website a list of such meetings held between December 1, 2019 and November 30, 2024 (Period of the first Von der Layen Commission). We have enriched this database with information on stakeholders obtained from the EU transparency register on lobbying stakeholders and the firm-level Bureau van Dijk database Orbis (see methods 4.1 for details) We model this dataset as a hypergraph H(V, E) where the

set of nodes V corresponds to entities (EC members and stakeholders) and each (hyper-) edge e ∈ H is associated with a meeting, listing the participating entities. We shall also consider the sub-hypergraph restricted to stakeholders, Hstak , whose hyperedges link stakeholders participating in the same meetings. Global structure of the hypergraph We first analyze the structure of H at the macro-level to highlight the general organization of the EC consultation process. The network H is formed by a single connected component that comprises |V | = 5717 nodes and |E| = 18561 edges, including multiedges corresponding to the repetition of certain meetings in identical configurations. The hypergraph displays a complex structure (see Figure 8) with a heavy-tailed degree distribution (see Sect. 82 in SI) The most represented stakeholders are companies and groups that account for 34.58% of total stakeholder degree, trade and business associations that account for 2535%, and NGOs that account

for 23, 08% of the total degree The remaining categories (including trade unions, think tanks and research institutes, professional consultancies) account only for 17.00% of the total stakeholder degree Furthermore, the analysis of stakeholder category representation (see Figure 9 in the SI) demonstrates substantial variation across departments: Social and environmental Page 3 of 19 Azaiez and Mandel Applied Network Science (2025) 10:45 affairs departments have a diversified stakeholder base, while departments focusing on economic issues have a narrower base, primarily meeting with companies and trade associations. The majority of meetings consists of one-on-one interactions between stakeholders and EC officials (see Figure 11 in SI). Larger meetings, despite their low frequency, play a crucial role in the overall connectivity of the hypergraph, as demonstrated by the substantial correlation (0.74) between edge betweenness centrality and edge size The topology of H reflects the

thematic composition of the commission. Figure 8 in the SI reveals a community structure, with clusters centered around individual Commissioners. The clustering procedure (detailed in section 84 of the SI) yields a modularity score of 0.700, indicating a strong cohesion of the observed clusters These results suggest a relatively siloed organization of the policy process, where officials from different EC departments rarely attend the same meetings and most stakeholders primarily communicate with only a few departments of the Commission. In contrast, the Hstak network, which represents interactions among stakeholders, displays a different topology (see Fig. 1) It features a large connected component encompassing 43.32% of the nodes, alongside an extensive number of small components Additionally, we qualitatively observe a spacial division between NGOs (in orange) on the one side, and business associations and companies (pink and blue) on the other. At the microscale, the network

structure reflects a limited diversity of stakeholder types within individual meetings. This homogeneity is quantitatively supported by the low edge entropy scores (detailed in Section 13), indicating that meetings tend to gather similar types of stakeholders rather than fostering diverse, cross-sector interactions. We also evaluated how close the stakeholder-only network is to a true simplicial complex, where small stakeholder gatherings are systematically embedded inside larger ones, using the three simpliciality measures proposed by Landry et al. (2024) (see Figure 14 in the SI). All three scores are markedly higher in the empirical data than in size- and degree-preserving random hypergraphs, indicating that genuine meetings are more nested than chance would produce. Hierarchical structure We then explore the potential presence of a core-periphery structure. This structure (as defined by Borgatti and Everett (2000) and analyzed in depth in the case of hypergraphs in Tudisco and

Higham (2023)) consists of a densely connected subgraph (the core) surrounded by loosely connected actors (the periphery) who lack full integration with the core. To identify this structure, we introduce a generalized version of the normalized rich-club coefficient for hypergraphs (detailed in Sect. 43) This coefficient measures the interconnectedness of high-degree nodes relative to a random null model. Several null models are presented in Sect. 43 Figure 2a shows the rich-club coefficient (ρ) for the hypergraph restricted to stakeholders Hstak , without multiedges. It thus represents the distribution of centrality and the hierarchical structure among stakeholders in the network. We find ρ > 1 holds for all values of k, demonstrating persistent rich-club behavior among stakeholders. Figure 2b complements this analysis by considering the rich-club coefficient for the full hypergraph H, benchmarked against an ensemble of hypergraphs where the number of EC members and stakeholders

in each edge is fixed. The rich-club phenomenon among Page 4 of 19 Azaiez and Mandel Applied Network Science (2025) 10:45 Fig. 1 Stakeholder-only network Each position in the layout derives from the underlying bipartite graph that links stakeholders to the meetings they attend; nodes shown here represent only stakeholders. Circle area scales with stakeholder degree (number of meetings attended). Labels are shown for all nodes with degree k > 55 Colors indicate Transparency-Register categories, as detailed in the legend stakeholders is less quantitatively salient in this hybrid setting but still qualitatively distinguishable (except for very large values of k where the behavior of ρ is dominated by the interaction pattern of very few organizations and thus too extreme to be of significant interest). The quantitative differences between the stakeholder and the full hypergraph can be explained by the presence of EC members whose uniformly high centrality partly shadows the

difference of centrality between stakeholders. We further examine the structure of the core by conducting a (k, m)-hypercore decomposition (Liu 2020; Mancastroppa 2023). This decomposition technique extends the concept of k-cores from traditional networks to hypergraphs by iteratively removing nodes with degree ki < k and edges of size me < m. The (k, m)-hypercore is defined as the maximal sub-hypergraph in which all nodes are of degree at least k and of all edges are of size at least m. In Section 85, we examine the connectedness of H and Hstak hypercores, considering scenarios with and without multiedges. When multiedges are included, we Page 5 of 19 Azaiez and Mandel Applied Network Science (2025) 10:45 Fig. 2 Rich-club coefficient ρ(k) a Rich-club coefficients computed on the stakeholder-only hypergraph Hstak after removing singleton edges (edges containing only one node) and multiedges. The random baseline is the degree and size-preserving hypergraph configuration

model; multiedges are likewise forbidden (see model 6 in Sect. 42) b Rich-club coefficient on the full hypergraph H with multiedges removed Here, the random baseline is a degree-preserving hypergraph model that keeps, for every meeting, the exact number of stakeholders and EC members; multiedges are forbidden (see model 4 in Sect. 42) Black circles and the solid curve plot ρ; the grey band indicates ±1 standard deviation across the random ensemble; the red dashed line marks the baseline ρ = 1. Values of ρ(k) significantly above 1 signal an enrichment of edges linking high-degree nodes relative to the null model observe a mixed structural behavior: for lower-order interactions, the network remained mostly connected, but at higher interaction orders, the core becomes fragmented at a critical k value. This fragmentation suggests that repeated collaborations tend to concentrate stakeholders within similar sectors Conversely, when multiedges are excluded, the analysis reveals a

consistent connected core structure at all orders. Excluding multiedges highlights purely topological properties, showing a core that integrates highly connected entities from diverse fields of interest. Given the more stable and connected core structures revealed by the analysis without multiedges, we focus on these versions. This approach allows us to analyze the underlying topological properties of the networks, free from the potentially distorting effects of repeated interactions. Proximity to the network center is measured using hypercoreness centrality (Mancastroppa 2023) (see Sect. 44) Table 5 in the SI shows the top 50 stakeholders with the largest hypercoreness We define Core(r) as the set of nodes with a hypercoreness value larger than a given threshold r. This definition allows for an examination of the network’s core structure at varying distances from the center. Figure 3 visualizes the core composition by showing the proportions of different stakeholder categories

within Core(r) for H and Hstak . The blue dashed line in each figure represents the relative size of Core(r) or, in other words, the complementary cumulative distribution function of hypercoreness centrality. Its rapid decay confirms a core-periphery structure, with influence concentrated among a small subset of actors. The (k, m)-hypercore decomposition reveals distinct patterns in the composition of the core for H and Hstak networks. The analysis focuses on the region where Core(r) comprises between 10% and 1% of the total number of nodes, as indicated by the red dashed lines in Fig. 3 For the H network, Fig. 3a demonstrates a pronounced increase in companies’ share within the core, with trade associations maintaining a larger average representation compared to their initial graph presence. NGOs, on the contrary, exhibit a reduced average share in this core region These findings suggest that business interests, particularly companies and trade associations, maintain closer and more

consistent relationships Page 6 of 19 Azaiez and Mandel Applied Network Science (2025) 10:45 Fig. 3 Core composition The solid curves represent the proportion of each stakeholder category within Core(r) for aH and bHstak . The colors correspond to stakeholder categories as indicated in the legend The blue dashed curves represent the size of Core(r), with values shown on the right y-axis. The vertical red dashed lines mark the values of r where the size of Core(r) equals 10% (left line) and 90% (right line) with the EC. This pattern aligns with Laurens’ (Laurens 2017) concept of “bureaucratic capital,” according to which businesses develop a deep understanding of EU bureaucracy, enabling them to navigate and influence regulatory processes more effectively. In the Hstak network, a different dynamic emerges. Companies continue to maintain a significant core presence, but the share of NGOs in the core increases notably. Trade associations, by contrast, show a reduced average

representation compared to their initial graph positioning. This comparison suggests that central NGOs employ distinctive interaction strategies with EC representatives. Their increased core presence suggests a tendency toward shared meetings and coalition-building, a phenomenon documented in political science literature. Scholars like Beyers and De Bruycker (Beyers and De Bruycker 2018) have highlighted NGOs’ propensity to collaborate, pooling resources and expertise to enhance their collective influence. Consistent with this view, our micro-scale analysis meeting patterns (see Figure 12 in the SI) reveals that NGOs are strongly over-represented in the largest meetings but appear far less often in the smallest ones. We observe the opposite trend for trade associations: they are enriched in small meetings and underrepresented in large gatherings. Interestingly, Hanegraaff and Pritoni (2019) suggest that NGOs coalition formation is a sign of fear rather than a sign of strength as they

often face financial and influence vulnerabilities (Dür and De Bièvre 2007), which they mitigate by uniting forces. However, to definitively assess the mechanisms of NGOs’ collective action, further investigation into meeting initiation dynamics would be necessary Page 7 of 19 Azaiez and Mandel Applied Network Science (2025) 10:45 Page 8 of 19 The persistent absence of think tanks, research institutes, and academic institutions from the core of both networks is particularly striking, especially considering the EC’s proclaimed commitment to evidence-based policymaking. Microeconomic analysis of companies To better understand the determinants of centrality for companies, we conduct an ordinary least squares (OLS) regression. The model takes the following form: Ri = α + βMi + ∑ n γn Lni + δCi + ζAi + ∑ ηn Sin + ϵi n where Ri represents the hypercoreness of a company i, Mi denotes the full-time equivalent (FTE) staff employed by a company i, and Lni are binary

variables indicating whether a company has a global, European, or national level of interest. The variable Ci is a binary indicator of whether a company s head office is located within an EU country, Ai represents the assets, and Sin is a categorical variable capturing the sector. The baseline category for sectors is set to “C - Manufacturing,” as it represents the largest proportion of companies in the network Details on data processing are provided in Sect 45 of the SI. Table 1 presents the regression results for both H and Hstak , with normalized coefficients. Our analysis reveals several key findings. First, there is a strong and consistent positive association between hypercoreness and the number of FTE members employed by a company. This suggests that stakeholders with more full-time staff dedicated to lobbying activities in Brussels tend to occupy more central positions in the network. Table 1 Determinants of company centrality. OLS coefficients (standardized) from two

models explaining hypercoreness of firms in the full hypergraph (H) and in the stakeholder-only hypergraph(Hstak ) Members FTE Assets Level European Level Global Level National Head of office in EU country B - Mining and quarrying D - Electricity, gas, steam and air conditioning supply G - Wholesale and retail trade, repair of motor vehicles and motorcycles H - Transportation and storage J - Information and communication K - Financial and insurance activities M - Professional, scientific and technical activities N - Administrative and support service activities const Observations R-squared AIC BIC H 0.464* 0.253* 0.067* 0.113* −0.041 0.097* 0.064* 0.057* −0.013 −0.033 0.055* −0.007 0.038 0.014 −0.0 1178 0.398 2776.099 2852.173 Hstak 0.395* 0.157* 0.009 0.08* −0.018 0.197* 0.054 0.017 −0.018 −0.087* 0.007 −0.101* −0.005 −0.002 0.0 702 0.298 1773.811 1842.12 Stars denote significance (* p < 0.05, * p < 0.01, * p < 0.001) Full-time-equivalent staff

(Members FTE) and assets are the strongest predictors in both networks. Global level if interest is positively associated with centrality Firms whose head office is inside the EU are substantially more central. Sector dummies are relative to the baseline C Manufacturing Mining and quarrying, Electricity and gaz supply, and Information and communication show significant positive deviations from the baseline in H Azaiez and Mandel Applied Network Science (2025) 10:45 Second, company size, as measured by assets, shows a significant positive relationship with hypercoreness in both regressions. This underscores the importance of financial resources in determining network centrality, as larger organizations are better equipped to maintain sustained interactions with policy-makers. Moreover, from the EC’s perspective, backing from economically powerful actors helps secure support for its proposals from national governments and the European Parliament (Klüver 2013) However, this raises

critical questions about representativeness (Kone and Farnhill 2019), as large firms constitute merely 0.02% of EU businesses and employ only 38% of the European workforce. Although our models reveal consistent, statistically significant links between stakeholders’ resources (assets and FTE members) and centrality, these correlations should not be read as one-way causal effects. Centrality and resources can mutually influence each other. On the one hand, greater resources make it easier to gain deeper access within EU institutions. On the other hand, well-connected actors have incentives to exploit their centrality position by investing in lobbying efforts in order to steer policy outcomes toward their interests. Third, companies with their head office located within an EU country exhibit significantly higher hypercoreness. While this centrality allows domestic firms to effectively represent their interests and contribute to policy discussions, it can also lead to “aggressive

lobbying” (Kone and Farnhill 2019), where firms leverage their advantageous position to influence policies that may disadvantage competitors. This strategy is exemplified in cases such as the proposed Carbon Border Adjustment Mechanism (CBAM), where EU industry might gain a disproportionate advantage over non-EU producers (Rayner et al. 2023, Chap 16) Similarly, when business associations advocate for new technical standards under the guise of environmental protection or consumer needs, the issue of competition with firms from outside the EU often underlies these efforts (Laurens 2017, Chap 6). Once such standards are adopted, these associations may highlight the noncompliance of their non-European competitors, potentially leading to anti-dumping measures. Additionally, potential support from national governments enables them to employ multilevel lobbying strategies (Pappi et al. 1999) Fourth, companies with a global level of interest demonstrate significantly higher hypercoreness

compared to those without this level of interest across both regressions. In the first regression, companies with a European level of interest also show higher hypercoreness than those without. These findings suggest that broader geographic scopes of interest are associated with greater centrality within the network. This reflects EU-level policies, which predominantly affects multinational corporations whose global operations are heavily sensitive to EU-level policies (Kone and Farnhill 2019; Laurens 2017). Fifth, companies in sectors like B - Mining and quarrying and D - Electricity, gas, steam, and air conditioning supply are significantly more central than those in C - Manufacturing in the full network. This might reflect the European Commission’s focus on resource management, energy policies, and climate issues. Over time, climate policy has become constitutionally embedded in EU law (e.g, Article 191(1) of the TFEU) and has intensified under the first Von der Leyen Commission

(Rayner et al. 2023, p1–p22) Initiatives such as the European Green Deal COMMUNICATION FROM THE COMMISSION (2019), the Climate Law Regulation (2021), and REPowerEU, further underscore Page 9 of 19 Azaiez and Mandel Applied Network Science (2025) 10:45 this priority. Similarly, companies in J - Information and communication show higher centrality than manufacturing firms due to growing regulatory interest in digital policies like the Digital Services Act and AI Act. The remaining sectors are comparable to the baseline sector. In the stakeholder-only network, sectors like B - Mining and quarrying, D - Energy supply, C - Manufacturing, J - Information and communication, G - Wholesale trade, M - Professional services, and N - Administrative services show statistically comparable centrality. Other sectors such as H - Transportation and storage and K - Financial and insurance activities exhibit lower centrality than manufacturing firms. These variations in sectoral access probably

reflect both economic priorities and political contexts, but may also result from differing capacities among businesses to implement effective corporate political strategies (Kone and Farnhill 2019). Finally, similar conclusions can be drawn using the multiedge versions of the hypercoreness computation (see Section 8.6) Additionally, we extend this analysis to include all stakeholders (not just companies) in Section 8.6 The results confirm that FTE members, global level of interest, and head office location within an EU country remain significant determinants of hypercoreness across all stakeholder types Discussion Our analysis of the EC’s consultation process during the first Von der Leyen Commission period (2019–2024) reveals a complex and hierarchical structure within the EC lobbying network. The core-periphery structures identified in both the full network and the stakeholderonly network demonstrate a concentration of access and influence among a restricted group of

stakeholders. Companies and trade associations show a strong presence in the core of the full network, suggesting that business interests are deeply integrated into the policy-making process. Although network centrality does not directly measure influence, previous research has shown a correlation between central network positions and influence, in particular in the case of core-periphery networks (Larsen and Ellersgaard 2017; Knoke 2021, p70). Notably, Logeart (2020) demonstrates that entities with frequent meetings with policymakers experience increased lobbying success. We further observe that NGOs occupy core positions in the stakeholder-only network. Although this may reflect a strategic approach to enhance collective influence, recent research suggests it could also signal vulnerability rather than strength (Hanegraaff and Pritoni 2019). Logeart (Logeart 2020) also highlights that, despite similar levels of access to EC representatives, the lobbying efforts of NGOs are less

efficient than that of the business sector. Our results raise critical questions about the effectiveness of the EC consultation process in achieving its stated objectives. While public consultations are intended to enhance the legitimacy of EU regulations (Drake 1997; Tsakatika 2005; Schmidt 2012; Thomas 2009), the Treaty on the European Union mandates broad consultations to ensure coherent and transparent Union actions and emphasizes an open, transparent and regular dialogue with representative associations and civil society. Consolidated Version of the Treaty on European Union (2016). Specifically, the consultation process must consider a diverse range of perspectives, ensuring balance and fairness in their representation. Schmidt (2012); European Governance - A White Paper (2001) However, Page 10 of 19 Azaiez and Mandel Applied Network Science (2025) 10:45 our findings suggest that this principle is not fully realized in practice. The over-representation of business

organizations in the core of the network, the limited access of civil society to discussions on economic issues and the emergence of lobbying efforts as a predictor for stakeholder centrality all point to a consultation process that is potentially biased and unequal. The compartmentalization of consultations across EC departments, further exacerbate these concerns, potentially creating informational asymmetries that undermine the intended inclusivity and balance of the process, hindering the development of a truly “reinforced culture of consultation and dialogue” (European Governance - A White Paper 2001). Having set out these substantive policy?relevant findings, we conclude with a brief reflection on the methodological choices that we believe, represent one of the paper s main original contributions. Our results indicate that representing meetings as higherorder interactions is more than a minor modeling choice: the hypergraph representation lets us probe micro-scale meeting

patterns (see SI 8.3) and, by avoiding the link explosion of clique projections, yields a hypercoreness measure that more accurately captures embeddedness (see 4.4) Building on this foundation, an interesting direction for future research is to embed the hypergraph in a multiplex framework Lotito et al. (2024) to capture the variety of linkages binding organizations. For instance, integrating dimensions such as corporate membership in professional associations, funding connections with think tanks and NGOs, interlocking directorates, or coalition formations could provide a richer understanding of organizational influence. Such a multiplex representation could extend the analytical framework beyond a single-layer network, enabling the examination of more complex relationships, as proposed in studies by Heaney (2014) and Zeng and Battiston (2016). Finally, the scope of this analysis centers on the lobbying dynamics within the EC, with limited consideration of lobbying directed at other

EU institutions, such as the European Council and the European Parliament. Expanding the analysis to include these additional levels could offer a more comprehensive view of the multi-level lobbying landscape in the EU and deepen our understanding of how influence and access vary across EU institutional contexts. Methods Datasets description and processing •  Commission meetings with interest representatives Members of the Commission, cabinet members and director-generals are required to publish information on all meetings with interest representatives 2014/839/EU, Euratom: Commission Decision (2014) . The original dataset includes information such as the date, location, names of EC representatives, stakeholders met, their identification code in the EU Transparency Register, the name of the cabinet or the directorate general and the subjects discussed during these meetings. We have collected all the available data on meetings held by Commissioners, cabinet members and

director-generals from the inception of the Ursula von der Leyen Commission first term on December 1, 2019 until its end Novembre 30th, 2024. The publication of meetings’ information is processed by each cabinet or directorate general separately. ’This means in practice that, if the same meeting is attended by more than one Commission representatives concerned by the publication rules, it will result in information about that meeting being published more than once. As Page 11 of 19 Azaiez and Mandel Applied Network Science (2025) 10:45 an example, information about a meeting with an interest representative held jointly by two Members of the Commission and one Director-General will be published on all three dedicated webpages to ensure proper transparency and the easy and complete information of the public about the meeting and its participants.’ (answer of the Secretariat General of the EC to our request for information on duplicated meetings.) To solve this, we select all

meetings with identical date and set of stakeholders and compute the fuzzy matching partial ratio (partial means inclusive) between meetings’ subjects. Figure 4 shows the number of meetings’ subject with a fuzzy ratio > r. The curve is essentially flat for r = 0 − 20, then decreases rapidly until r ∼ 60, after which the decline becomes gradual. Manual inspection of all pairs of matched subjects with ratios ranging from 60 to 100 shows that r ≥ 70 retain only genuine duplicates, while lowering the threshold starts introducing false positives. Conversersely, raising the threshold above 70 discards true duplicates. We therefore adopt r = 70 as the default setting for the main analysis, meaning that if the ratio ≥ 70, then the meetings are considered referring to the same meeting. In this case, we form one meeting with the corresponding stakeholder set the union of EC representative sets. This procedure merges 460 pairs. •  EU Transparency Register This is a public

database listing interest representatives that carry out lobbying activities in Brussels. For this study, we consider information on the identification code, the name, the category of registration, the head of office country, the level of interest and the full-time equivalent persons involved in lobbying activities. The register is updated every 6 months and encompasses information about the stakeholders that are registered within this period. We compiled data from January 2017 to January 2024. For stakeholders registered multiple times during this period, we retained only the most recent information. The “Identification Code” served as the primary key for merging this data with meeting data. 43 identification codes present in the meeting data were not found in the Transparency Register. 34 stakeholders Fig. 4 Duplicate detection with fuzzy-ratio cut-off The number of meeting-subject pairs whose fuzzy-matching score (fuzzy partial ratio) exceeds r Page 12 of 19 Azaiez and

Mandel Applied Network Science (2025) 10:45 had the same name in the Transparency Register but were associated with different identification codes, likely due to re-registrations. To address the issue of duplicate names, we consolidated these entities into single stakeholders, using the most recent registration information. •  Bureau van Dijk database Orbis The Orbis database is a data source for private companies available under commercial license. We extract information on the revenue, the assets and the number of employees and the NACE (the industry standard classification system in the European Union). We focus on stakeholders registered as companies or groups in the Transparency Register. The names of these companies, along with their head office countries, are uploaded to the Orbis database for matching. Matches with an A score (indicating a high confidence level) are automatically approved. For matches with lower confidence scores, a manual verification process is

conducted to ensure accuracy. Following this procedure, we achieve a 93.9% match rate between the Transparency Register and the Orbis database. Additionally, we identify 15 companies in the Transparency Register that are matched to identical entities in the Orbis database. These duplicate entries are consolidated into a single company in the meetings dataset to avoid redundancy. Hypergraph Configuration Models via MCMC The Hypergraph Configuration Model (HCM) extends the classical configuration model for graphs. In the graph setting, one samples uniformly from all graphs with a prescribed degree sequence. The HCM generalizes this by also fixing each hyperedge s size In our ensemble, each node s degree and each edge s size are held constant. We forbid degenerate edges, so no node can appear twice in the same hyperedge Chodrow s formulation studies the uniform distribution over all such simple hypergraphs Chodrow (2020) and proposes a Markov Chain Monte Carlo algorithm for hypergraph

sampling. We define six related null models on a given empirical hypergraph H and on Hstak . Here, Hstak is obtained by removing nodes corresponding to EC members and all singleton edges from H. 1. Full HCM, with multiedges We preserve the degree and the size sequence of H Identical hyperedges may appear multiple times. 2. Full HCM, without multiedges We preserve the same degree and size sequences Any proposed duplicate hyperedge is rejected. 3. Subdimensional HCM, with multiedges We split each hyperedge into two disjoint parts: stakeholders and EC members. We then fix each part s marginal edge?size distribution independently. Duplicate hyperedges are permitted 4. Subdimensional HCM, without multiedges As in (3), but proposals that create duplicates are disallowed. 5. Stakeholder?only HCM, with multiedges We remove all EC member nodes and all singleton edges from H to obtain Hstak . We then apply the full hypergraph configuration model to Hstak . Duplicate hyperedges are allowed 6.

Stakeholder?only HCM, without multiedges Same as (5), but duplicate edges in Hstak are forbidden. We sample each model using a Metropolis Hastings rewiring algorithm. Page 13 of 19 Azaiez and Mandel Applied Network Science (2025) 10:45 Page 14 of 19 1. Randomly select two hyperedges 2. Propose a swap of their vertices that exactly preserves the prescribed sizes 3. Accept the proposal with an acceptance probability of an edge swap between a pair of edges ∆ and Γ a(Ht+1 |Ht ). For the full and stakeholder-only models, a(Ht+1 |Ht ) = 2|∆∩Γ|1m∆ mΓ where m∆ is the number of multiedges parallel to ∆. For the subdimentional model, the acceptance probability of an edge swap between ˙ 2 and Γ1 ∪Γ ˙ 2 , is a(Ht+1 |Ht ) = |∆1 ∩Γ1 |×|∆12 ∩Γ2 | ∆ = ∆1 ∪∆ . 2 m mΓ ∆ 4. If ‘multiedges=False‘, reject any move that would create a duplicate hyperedge 5. Otherwise accept or reject according to the Metropolis Hastings criterion Chodrow (2020) proves

that, for the full configuration model, the resulting Metropolis Hastings chain is irreducible and reversible with respect to the uniform distribution on all hypergraphs that share the prescribed degree and size sequences and contain no degenerate edges. The sub-dimensional variant introduced here satisfies the same property: the proof follows exactly the counting argument of Chodrow s Theorem 2, applied separately to each sub-edge, and therefore likewise guarantees convergence to the uniform ensemble under the additional sub-dimensional constraints. Rich-club coefficient for hypergraphs We introduce a generalization of the normalized Rich-club coefficient for hypergraphs by associating to every degree k ∈ N, ϕ(k) = ∑ E>k i|ki >k ki where E>k is the number of edges that contain nodes of degree greater than k and the normalization of ϕ is given by the sum of the degree of these rich nodes. The normalization of the Rich-club coefficient is given by: ρ(k) = ϕ(k)

ϕrand (k) where ϕrand (k) is computed using a null mode. The coefficient ρ(k) measures the extent to which high-degree nodes preferentially connect among themselves relative to a reference generative model. For the null model, we employ variations of the Hypergraph Configuration models introduced in the previous section. Figure 5a, b present the results for models (1) and (2) that account for both stakeholders and EC members. In these cases, ρ < 1 for values of k > 10 indicates that highdegree nodes are less interconnected compared to a random hypergraph with similar degree and size distributions. However, while the highest-degree nodes correspond to EC members (as shown in Figure 8), H lacks edges composed exclusively of EC members. As such edges are present in the random ensemble of hypergraphs to which H is compared, the coefficient ρ is systematically biased downwards and a rich-club phenomenon involving EC members can not, by lack of data, be put forward in our

framework. Yet, our focus is on the distribution of centrality and the hierarchical structure of the network among stakeholders. In order to investigate these features, we restrict attention to the hypergraph Hstak in the null models (5) and (6) represented in Figures (e) and (f ). Azaiez and Mandel Applied Network Science (2025) 10:45 Fig. 5 Rich-club coefficient a Rich-club coefficient measured with model (1) (based on H including multiedges) b Rich-club coefficient measured with model (2) (based on H excluding multiedges) c Rich-club coefficient measured with model (3), (based on H including multiedges and preserving the number EC members and stakeholders in each edge in the null model) d Rich-club coefficient measured with model (4) (based on H excluding multiedges and preserving the number EC members and stakeholders in each edge in the null model) e Rich-club coefficient measured with model (5), (based on Hstak including multiedges) f Rich-club coefficient measured with model

(6) (based on Hstak excluding multiedges) There, we find ρ > 1 holds for all values of k, demonstrating persistent rich-club behavior among stakeholders. Finally, in the null models (3) and (4), we consider a hybrid setting in which we benchmark the hypergraph H to an ensemble of hypergraphs where the number of EC members and stakeholders in each edge is fixed but their ids are randomly reshuffled. The rich-club phenomenon among stakeholders is less quantitatively salient in this hybrid setting but still qualitatively distinguishable, except for very large values of k where the behavior of ρ is dominated by the interaction pattern of very few organizations and thus too extreme to be of significant interest. These quantitative differences between null models (3) and (5) (respectively (2) and (4)) can be explained by the presence of EC members whose uniformly high centrality partly shadows the difference of centrality between stakeholders. Page 15 of 19 Azaiez and Mandel Applied

Network Science (2025) 10:45 Page 16 of 19 Hypercore decomposition The (k, m)-hypercore decomposition generalizes the k-core decomposition in graphs. Nodes with degree ki < k and edges with size me < m are iteratively removed. The resulting (k, m)-hypercore is the maximal sub-hypergraph where each node belongs to at least k edges and all edges are of size at leat m. We perform the hypercore decomposition for both H and Hstak . For Hstak , the decomposition is applied to its largest connected component after removing singleton edges (edges containing only one node). Additionally, we consider both the original hypergraphs and versions without multiedges In the absence of multiedges, edges are merged during the recursive process, ensuring that the resulting (k, m)-hypercore is the maximal subhypergraph where each node belongs to at least k distinct edges and all edges are of size at leat m. To measure proximity of nodes to the center of the network, we consider the hypercoreness

centrality of nodes, following the method outlined by Mancastroppa (2023). Namely, for a node i, the m-core number, denoted as Cm (i), is defined as the highest value of k such that i belongs to the (k, m)-hypercore but not the (k + 1, m)-hypercore. The hypercoreness of a node i is then defined as : R(i) = M ∑ Cm (i) m=2 m kmax nm m where kmax is the maximum value of k such that the (k, m)-hypercore is not empty and nm denotes the fraction of edges of size m. To further motivate our preference for the hypergraph representation over the traditional graph projection, we directly compare node hypercoreness calculated on the original hypergraph with the standard coreness obtained from its clique-expansion graph. Given a hypergraph H = (V, E), its clique expansion is the graph G = (V ′ , E ′ ) that keeps the same vertex set (V ′ = V ) and links every pair of nodes that co-occurs in a hyperedge, i.e, E ′ = { (i, j) | i ̸= j, i, j ∈ e, e ∈ E } We compute four centrality

scores: R Rstak C Cstak : : : : V full vertex set stakeholders only full vertex set stakeholders only model hypergraph hypergraph clique expansion graph clique expansion graph Because we focus on purely topological properties, multiedges are discarded in every case. Figure 6 reports Pearson correlations among the four measures. Scores derived within the same modeling framework are the most similar: the two clique-based values correlate at r = 0.93, whereas the two hypergraph values correlate at r = 079 Cross-framework correlations are weaker, showing that the representation changes the centrality ranking. To further investigate differences and similarities of both frameworks Fig 7 plots node coreness against hypercoreness, coloring points by their hypergraph degree. The relationship is strongly asymmetric: almost every node that is hypergraph-central is also graph-central, yet many nodes with high coreness have only modest hypercoreness. These ’inflated’ nodes typically have

low hypergraph degree but belong to a few large Azaiez and Mandel Applied Network Science (2025) 10:45 Page 17 of 19 Fig. 6 Correlation between coreness and hypercoreness Fig. 7 Relationship between hypercoreness and coreness a Full hypergraph H compared with its clique-projection graph b Stakeholder-only hypergraph Hstak compared with the corresponding stakeholder-projection graph. Each dot is a node; the x-axis shows its hypercoreness, the y-axis its classical coreness Marker color encodes the node’s degree in the hypergraph on a logarithmic scale. The red diagonal (y = x) marks perfect agreement between the two centrality measures meetings. After projection, such meetings explode into ( d 2 ) pairwise links, artificially boosting node degree in the projected graph and hence its coreness. Hypercoreness treats the meeting as a single higher-order object and thus avoids this inflation. Pairwise projection over-emphasizes actors who appear in a few crowded meetings and

blurs the distinction between genuine brokerage and simple participation in a large event. Hypercoreness therefore offers a more faithful measure of embeddedness, supporting our choice of the hypergraph model for the main analysis Data processing for regression Our pool of potential dependent variables encompasses the head office country, the NACE (the industry standard classification system in the European Union), levels of interest, revenue, number of employees, assets, and the full-time equivalent number of persons involved in lobbying activity (members FTE). Azaiez and Mandel Applied Network Science (2025) 10:45 Our initial step involves data cleaning, with a focus on addressing the skewed distribution of both continuous and categorical independent variables. For continuous variables such as revenue, number of employees, assets, and members FTE, which exhibit high skewness, we perform a log transformation to rectify this issue. Similarly, for categorical data, characterized by

certain categories having a low proportion of companies, we address this imbalance by grouping categories (e.g, considering geographical and cultural regions instead of individual countries) and removing outliers Subsequently, we conduct a correlation study among the log-transformed continuous variables. This analysis reveals a high correlation among the log-transformed financial data, leading us to keep assets as an independent variable in our OLS regression. Furthermore, we perform a correlation study among the categorical variables, which indicates no significant correlations This preliminary analysis ensures that our subsequent regression model is built on independent and meaningful variables, setting the stage for a robust examination of the factors explaining centrality of companies. Supplementary Information The online version contains supplementary material available at https://doi.org/101007/s41109-025-00735-6 Supplementary file 1 (pdf 7585 KB) Author contributions A.A:

conceptualization, software, formal analysis, data curation, methodology, writing - original draft, visualization AM: validation, writing - review & editing, supervision Funding The authors received no specific funding for this work. Data availability EU transparency register : Secretariat-General, ‘Transparency Register’, 2022 (updated 2018-11-28), accessed 2024-0110, http://data​.europaeu/​88u/dataset​/transpa​rency-register Meetings held by EC representatives: Secretariat-General, ‘European Commission - Meet- ings with interest representatives’, accessed 2024-01-10, h ​ ​t​t​p​:​/​/​d​a​t​a​.​e​u​r​o​p​a​​e​u​/​8​8​u​/​d​a​t​ a​s​e​t​/​e​u​r​o​p​e​a​n​- commission-meetings-with-interest-representatives. Bureau van Dijk database Orbis The full code and instructions to reproduce the analyses are available at:

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