Topological Modelling in Public Procurement and Platform Economies: An Interdisciplinary Legal–Economic Framework

Abstract

This article develops an interdisciplinary framework that applies topological and graph-theoretical methods to public procurement markets and digital platform economies. Conceptualizing legal–economic interactions as dynamic networks of nodes and edges, we show how structural properties—centrality, clustering, connectivity, and boundary formation—shape contestability, resilience, and compliance. Using EU-relevant contexts (public procurement directives and the Digital Markets Act), we formalize network representations for buyers, suppliers, platforms, and regulators; define operational indicators; and illustrate an empirical, value-weighted buyer → supplier network to reveal a sparse but highly modular architecture with a high-value backbone. We then map these structural signatures to concrete legal levers (lotting and framework design, modification scrutiny, interoperability and data-access duties) and propose dashboard-style diagnostics for proactive oversight. The findings demonstrate how topological modelling complements doctrinal analysis by making hidden architectures visible and by linking measurable structure to regulatory outcomes. We conclude with implications for evidence-informed regulatory design and a research agenda integrating graph analytics, comparative evaluation across jurisdictions, and machine-learning-assisted anomaly detection.

1. Introduction

Public procurement across the EU and OECD has undergone a deep digital transition over the last decade. The rollout of eForms on TED and the diffusion of open data standards such as the Open Contracting Data Standard (OCDS) have markedly increased the granularity and timeliness of award notices, procedure metadata, and supplier identifiers. These reforms have made procurement markets more observable, while simultaneously exposing structural concentration, cross-sector supply chains, and recurrent bidder interaction patterns. In parallel, digital platform economies have scaled into multi-sided ecosystems that intermediate transactions, data, and attention at continental scale. The EU’s Digital Markets Act (DMA) codifies this reality by targeting gatekeeper conduct and imposing obligations around interoperability, data use, and self-preferencing [1]. Taken together, both domains—procurement and platforms—operate as complex systems: they exhibit non-linearity (small policy adjustments can trigger outsized equilibrium shifts), feedback loops (enforcement alters strategies that reshape structure), heterogeneity of actors and incentives, and path dependence rooted in institutional design [2,3,4,5,6,7,8].

Conventional doctrinal or linear economic models struggle to capture these features because they abstract away from the relational architecture through which outcomes emerge. By contrast, network-based (graph-theoretic) representations treat actors (contracting authorities, suppliers, platforms, regulators) as nodes and their contractual, informational, and legal ties as edges [3,4,6,7]. Within this lens, topological properties—centrality, clustering, connectivity, modularity, and boundary formation—are not merely descriptive statistics; they are diagnostic signals of risk concentration, market access, resilience, and oversight capacity [5,6,7,8]. Recent empirical work in procurement and platform research suggests that high centrality and clustered modularity often correlate with vulnerability to reduced contestability, capacity constraints, or compliance frictions, but these insights are rarely integrated with legal analysis at the level of system design [1,9,10,11,12].

This article advances an interdisciplinary legal–economic framework that fuses topological modelling with the positive law of EU procurement and platform regulation. Substantively, we: (i) conceptualize public procurement and platform economies as topological systems whose observed market behaviour is conditioned by network structure; (ii) formalize a graph-theoretic toolkit suitable for legal analysis (clear node/edge definitions, directed vs. undirected, weighted vs. unweighted edges, and operational indicators for density, centrality, clustering, and modularity); and (iii) demonstrate how these diagnostics inform regulatory design, including the mapping of observed structures to concrete legal levers (e.g., procurement thresholds, modification rules, and DMA gatekeeper obligations) [1,6,7,13,14].

Empirically, we illustrate the approach on an EU procurement sample with a buyer supplier value-weighted network and report backbone structure, concentration, and community patterns relevant to transparency and contestability. For platform ecosystems, we provide a multilayer schema that links market interaction to regulatory oversight and makes feedback loops explicit. Across both domains, we show how structural signatures can support proactive supervision (risk dashboards), clarify the expected effects of interventions, and enable comparative evaluation of legal regimes by their network consequences rather than by formal rules alone [1,5,6,7,8,13,14].

The contribution is twofold: methodological—a transparent, reproducible bridge between network science and legal analysis—and normative—actionable guidance for regulators on where at what scale, and with which instruments to intervene when topology signals concentration, bottlenecks, or fragmentation. The remainder of the paper reviews relevant literature, specifies the modelling approach and indicators, presents empirical illustrations, and draws implications for EU procurement governance and DMA-oriented platform oversight [1,2,3,4,5,6,7,8,9,10,11,12,13,14].

2. Literature Review

Understanding how topology and network science can inform legal-economic analysis requires grounding in both the mathematical toolkit of graph theory and the socio-legal dynamics shaping regulatory environments. Traditional legal-economic accounts emphasize institutional and doctrinal frames (e.g., transaction-cost perspectives [14]) yet often abstract away from the relational architecture through which outcomes emerge. Network-based approaches address this gap by representing actors and their interactions as graphs, making centrality, clustering, connectivity, modularity, and boundary formation empirically observable and legally interpretable signals of risk, resilience, access, and oversight capacity. Foundational network-science concepts (degree distributions, brokerage, community structure) thus migrate from descriptive statistics to diagnostics with normative relevance for governance design [3,4,5,6,7,8].

2.1. Public Procurement as a Networked Market

Recent empirical work in public procurement exploits buyer–supplier graphs to study concentration, contestability, and anomaly detection. Studies consistently report clustered modularity and high-centrality hubs that align with reduced competitive intensity or integrity risks; synthesis papers document graph-based fraud-detection pipelines and risk signals, while national case studies show graph-database + rule-based screening in production environments [9,10,11]. Pattern-based methods operationalize anomaly motifs directly on award networks (e.g., heavy-tailed backbones, star-like hub-and-spoke structures). These findings extend classic network results into an applied regulatory setting, linking market shape to transparency and compliance capacity and complementing earlier mappings of procurement dynamics [9,10,11].

2.2. Digital Platform Economies as Multilayer Ecosystems

For digital platform economies, scholarship increasingly conceptualizes major platforms as gatekeeping nodes within multi-sided, data-intensive ecosystems [1,12]. Network perspectives clarify how hub dominance, interoperability constraints, and feedback loops shape market access and user outcomes—concerns now codified by the EU’s Digital Markets Act (DMA) [1]. Conceptual contributions on platform governance are complemented by multilayer graph models that connect market interaction (users, business partners) to regulatory oversight and enforcement, making visible the pathways through which obligations and remedies propagate. These approaches support both antitrust and ex ante regulatory design by tying structural signatures to expected effects of intervention [1,12].

2.3. Legal-Tech, Citation Networks, and Structural Diagnostics

Parallel strands in legal tech demonstrate that citation and statute networks improve retrieval and structural diagnostics across legal corpora [8], while investigative dashboards illustrate the practical value of visual analytics for compliance, supervision, and case building [15]. Together with procurement and platform studies, this literature underscores that network shape carries normative significance: where the graph concentrates control (centrality), fragments into enclaves (modularity), or restricts paths (low connectivity), legal risk and governance burden tend to accumulate [5,6,7,9,10,11,15].

2.4. Gap and Contribution

Despite these advances, prior research rarely integrates topological diagnostics with concrete legal levers—for example, procurement thresholds and modification rules or DMA gatekeeper obligations—inside a single, coherent framework spanning both procurement markets and platform ecosystems [9,10,11]. This article fills that gap by:

• formalizing a graph-theoretic toolkit tailored to legal analysis (clear node/edge semantics; directed vs. undirected; weighted vs. unweighted; operational indicators for density, centrality, clustering, modularity) [3,4,5,6,7,8]
• applying it to an EU procurement sample [16]) via a value-weighted buyer → supplier network (with backbone visualizations and summary metrics) [9,10,11]; and
• mapping observed structures to actionable regulatory design and supervisory dashboards, thereby connecting empirical topology to doctrinal instruments in procurement law and the DMA [1,12,14].

3. Methodology

3.1. Network Representation

Entities and relations. We model two domains—public procurement and digital platforms—using graph representations in which nodes denote contracting authorities (buyers), suppliers (firms), platforms, users/business partners, and regulators; edges encode relations: in procurement, contract awards; in platforms, data/transaction flows; and in regulation, oversight/enforcement interactions [1,3,4,5,6,7,8,12].

Directionality. Procurement awards are modeled as directed edges (buyer → supplier). Regulatory interactions are also directed (e.g., regulator → platform/buyer for obligations, audits, or decisions). Where appropriate we analyze undirected projections (e.g., for community detection robustness) [5,6,7,8].

Weights. Unless stated otherwise, edges are weighted procurement by the cumulative contract value within the observation window; platforms by interaction volume/intensity (e.g., transactions, API calls, or enforcement frequency). Unweighted variants are used in robustness checks [5,6,7,8].

Bipartite structure and projections. Procurement graphs are inherently bipartite (buyers ↔ suppliers). We report results on the bipartite graph and, where needed for diagnostics or visualization, on a buyer → supplier projection with aggregated edge weights [9,10,11].

Backbone for readability. For visual clarity we additionally report a value-weighted backbone (typically the top 5% of edges by weight), preserving dominant corridors while retaining interpretability of the overall structure [9,10,11,15].

3.2. Indicators (Operational Definitions and Formulas)

Let G = (V, E) be a graph with n = |V| nodes and m = |E| edges; A is the (possibly weighted) adjacency matrix. For directed graphs, $k_{in}$ and $k_{out}$ denote in-/out-degree; for undirected graphs, k denotes degree [3,4,5,6,7,8].

Connectivity/Density. Relative number of realized ties:

• Undirected (no self-loops): δ = 2m / [n(n − 1)].
• Directed (no self-loops): δ = m / [n(n − 1)].

Degree centrality. Relative number of incident ties:

• Undirected: $C_D(i)=\frac{k_{\mathrm{i}}}{n- 1}$
• Directed: $C_D^{in/out}(i)=\frac{k_{\mathrm{i}}^{in/out}}{n- 1}$

Betweenness centrality. Brokerage over shortest paths:

$$C_B(i)={\displaystyle \sum _{s\ne i\ne t}{\sigma }_{st}}(i)/{\sigma }_{st},$$

where ${\sigma }_{st}$ is the number of shortest paths from s to t and ${\sigma }_{st}(i)$ is the number of those paths passing through i.

Clustering coefficient (local, undirected).

$$C_i=2e_i/[k_i(k_i-1)],$$

where $e_i$ is the number of ties among neighbours of i. (Directed/weighted variants are reported in robustness checks.) [3,4,5,6,7,8].

Modularity (community structure). For undirected (possibly weighted) graphs:

$$Q=(1/(2m)){\displaystyle \sum _{i,j}\left(A_{ij}-\frac{k_i{k}_j}{2m}\right)}\cdot {\mathbf{1}}_{\{c_i=c_j\}},$$

where $c_i$ is the community assignment. We compute Louvain/Leiden solutions; for directed/weighted graphs we use corresponding variants and report robustness [5,6,7,8].

3.3. Data Pipeline, Model Specifications, and Robustness

Data harmonization. (i) Entity resolution: consolidation of buyer/supplier names/IDs to stable identifiers; (ii) currency normalization and aggregation within the observation window; (iii) quality control and deduplication with documented handling of missing values; (iv) attribute curation (CPV, procedure, year, and regulatory events where applicable) [9,10,11,15].

Model specifications. Procurement graphs: bipartite (buyer–supplier), directed, weighted by cumulative award value; sectoral subgraphs by CPV. Platform graphs: multilayer (market layer: platforms–users/providers; regulatory layer: authorities–platforms), directed, with weights reflecting interaction or enforcement intensity [1,12].

Robustness and sensitivity. (1) Edge thresholding (quantile pruning of very weak ties) and stability of centrality/modularity; (2) alternative community detection (Louvain vs. Leiden/Info map); (3) degree-preserving randomization (configuration model) as null baseline; (4) window sensitivity (annual vs. half-year/quarterly); (5) replication of key diagnostics on directed and weighted versions [5,6,7,8].

Visualization. All network figures use force-directed layouts (no geographic meaning). Node size ∝ degree/strength; colour = community/sector. Each figure states node/edge types, directionality, weighting, and interpretive limits [15].

Hypothesis development and testing. To align structural diagnostics with regulatory concerns, we derive three policy-relevant hypotheses from the procurement network:

H1 (Centrality and Collusion Risk). 

Suppliers with higher betweenness centrality are more likely to participate in restricted or negotiated procedures.

H2 (Buyer Concentration and Supplier Diversity). 

Contracting authorities with higher degree centrality attract fewer unique suppliers over time.

H3 (Clustering and Local Favouritism). 

Suppliers in highly clustered subnetworks are more likely to receive repeated awards from the same authority. [9,10,11]

Operationalization. For H1 we proxy ‘centrality’ by supplier degree (betweenness on the raw bipartite graph is near-zero for most suppliers), for each hypothesis we specify independent variables (betweenness, degree, clustering coefficient), dependent variables (procedure type, supplier counts, repeat awards), and controls (sector, contract size, award year). Statistical tests include logistic regression (H1), linear regression or correlation (H2), and count regression models such as Poisson or negative binomial (H3). Required data fields include supplier and buyer IDs, procedure type, contract value, award date, CPV classification, and computed network metrics. This design ensures reproducibility and allows future cross-country replication [9,10,11,13,14].

3.4. Reproducibility and Disclosure

Software and environment. Analyses were executed in Python 3.11.6; network figures were exported from Gephi 0.10.1. Runs were performed on Windows 11 (64-bit) with an x86-64 CPU and 32 GB RAM. Upon publication, we will deposit code (scripts/pipeline), an anonymized sample dataset with a variable dictionary, and project files (layouts and parameters) in an open repository. For every graph/subgraph we report node/edge counts, density, degree distributions, centrality summaries, modularity Q, and—where relevant—the backbone share and threshold settings [15].

3.5. Visualization Settings and Reproducibility

Network figures were produced in Gephi (ForceAtlas2; scaling = 2.0; gravity = 1.0; linlog mode on; prevent overlap on). Node size ranked by total incident edge weight (min = 6 pt, max = 36 pt). Edge thickness ranked by weight; arrowheads displayed for directionality. Labels: only top-degree nodes (label size by degree; contrast 90%). Exports: SVG and 300 dpi PNG, width = 180 mm (full width), fonts embedded. The complete node ID → entity mapping appears in Appendix A (

Table A1. Mapping of node IDs to entities and attributes (full table in the replication files).

Node ID	Entity Name	Role (CA/Supplier)	Sector (CPV div.)	Total Incident Value (EUR, 2022)	Degree	Betweenness (Decile)
CA-0001	Krajská zdravotní, a.s.	CA	45	78,490,299	20	1
CA-0002	Správa silnic Moravskoslezského kraje, p. o.	CA	45	49,708,325	20	1
CA-0003	Karlovarská krajská nemocnice a.s.	CA	33	2,159,222	9	1
CA-0004	Fyzikální ústav AV ČR, v.v.i.	CA	38	1,068,082	10	1
CA-0005	Moravskoslezský kraj	CA	45	9,634,204	19	1

Etc. …. Table_A1_NodeKey.csv (all nodes, thousands of lines will be sent to the editor upon request).

); replication materials (scripts and .gephi project) are referenced in Section 3.4.

4. Results

4.1. Empirical Topology of the Czech Public Procurement Network

We construct a directed, value-weighted buyer → supplier network from TED award data for the Czech Republic in 2022 [16]. Nodes represent contracting authorities (CAs) and suppliers; a directed edge CA → supplier indicates at least one awarded contract in the observation window, with the edge weight equal to the cumulative contract value. The resulting graph comprises 5887 nodes and 6204 edges, with a density of 0.00018, i.e., a very sparse market-level interaction structure. Community detection on the undirected projection yields 779 communities with high modularity (Q = 0.887), signalling pronounced sectoral/organizational clustering and limited cross-cluster connectivity.

This topology is consistent with a market organized around sectoral “islands” of repeated interaction between large buyers and specialized suppliers. The backbone of the network, defined here as the top 5% of edges by total awarded value—highlights a small number of high-value corridors that concentrate a disproportionate share of spend. In health-related procurement, for example, large university hospitals and regional providers dominate the buyer side, while the supplier side is led by Phoenix lékárenský velkoobchod, s.r.o., Alliance Healthcare s.r.o., Promedica Praha Group, a.s., Sanofi, CSL Behring, Roche, AbbVie, and Avenier (see

Table 1. Network summary and leading market actors [16].

Panel A: Network-Level Metrics.
Nodes	Edges	Density	Communities	Modularity (Q)
5829	60,794	0.000179	779	0.887
Panel B: Top 10 Suppliers by Cumulative Award Value (EUR).
Rank		Supplier	Total Value (EUR)
1		Phoenix lékárenský velkoobchod, s.r.o.	105,973,414,160
2		Alliance Healthcare s.r.o.	53,345,710,450
3		Promedica Praha Group, a.s.	29,379,061,754
4		sanofi-aventis, s.r.o.	23,495,320,985
5		CSL Behring, s.r.o.	23,470,684,848
6		Pharmos, a.s.	17,667,124,374
7		Amgen s.r.o.	17,625,063,976
8		Roche s.r.o.	11,998,734,742
9		AbbVie s.r.o.	11,789,634,148
10		Takeda Pharmaceuticals Czech Republic s.r.o.	11,776,746,696
Panel C: Top 10 Buyers by Cumulative Award Value (EUR).
Rank		Buyer	Total Value (EUR)
1		Fakultní nemocnice u sv. Anny v Brně	393,195,443,963
2		Česká republika—Ministerstvo vnitra	10,099,560,128
3		Fakultní nemocnice v Motole	1,580,860,214
4		EG.D, a.s.	1,247,377,218
5		Fakultní nemocnice Olomouc	870,026,352
6		Správa železniční dopravní cesty, státní organizace	645,877,138
7		Ředitelství silnic a dálnic ČR	625,048,444
8		Nemocnice Pardubického kraje, a.s.	427,509,991
9		Statutární město Ostrava	426,441,483
10		Lesy města Brna, a.s.	384,888,891

Panel A: Network-level metrics. Nodes = 5887; Edges = 6204; Density = 0.00018; Communities = 779; Modularity (Q) = 0.887. Panel B: Top suppliers by cumulative award value. Phoenix lékárenský velkoobchod, s.r.o.; Alliance Healthcare s.r.o.; Promedica Praha Group, a.s.; sanofi-aventis, s.r.o.; CSL Behring, s.r.o.; Roche s.r.o.; AbbVie s.r.o.; Avenier a.s.; Panel C: Top buyers by cumulative award value. Large university and regional hospitals and other major CAs (e.g., Fakultní nemocnice u sv. Anny v Brně), along with select central and municipal entities.

). These structures are visible in the backbone visualization (Figure 1) as dense hubs with multiple feeder edges [15].

From a regulatory perspective, three empirical signatures stand out:

• Concentration at the top. The backbone contains a tight core of high-value ties, indicating potential contestability challenges and single-point operational vulnerabilities.
• Strong modularity. The high Q suggests segmentation by sector/mission, which can be efficient but may also reduce spillovers and learning across clusters.
• Peripheral sparsity. Many suppliers and smaller buyers remain weakly connected, which can limit access and complicate policy diffusion beyond dominant hubs.

These signatures motivate targeted supervisory tools (e.g., risk dashboards) that track backbone composition over time, alert to centrality spikes or module isolation, and link observed structure to legal levers such as framework design, lotting, or modification rules [9,10,11,13,14].

Nodes (blue circles) represent contracting authorities (CAs) and suppliers; node size scales with total incident edge weight (in- plus out-weight). Directed edges (black arrows) indicate the flow from CA → Supplier, weighted by cumulative contract value. Node colour indicates sectoral affiliation (e.g., CPV divisions). Labels are shown only for the highest-degree nodes for readability; the complete Node Key (all node IDs → entity names/roles) is reported in Appendix A (Table A1). The layout is force-directed (no geographic meaning). The visible backbone represents the top 5 % of edges by value, highlighting structural concentration and inter-module corridors.

Hypothesis testing. Formal tests align with the topological picture. For H1, logistic regressions using supplier degree (std.) as a pragmatic centrality proxy (betweenness is near-zero on the raw bipartite) show a negative association with restricted/negotiated procedures (β = −0.272; 95% CI [−0.326, −0.218]; p < 2.5 × 10⁻²³), indicating that more-connected suppliers are less likely to appear in restricted/negotiated procedures once contract value is controlled. For H2, buyers with higher degrees exhibit lower concentration (β = −0.136; 95% CI [−0.147, −0.125]; p < 4.3 × 10⁻¹¹⁰)—i.e., greater supplier diversity. For H3, supplier bipartite clustering does not predict repeat awards (β = −0.009; 95% CI [−0.058, 0.040]; p = 0.712), with size controls strongly positive as expected. Full outputs appear in Appendix B,

Table A2. Summary of hypotheses, models, and results [16].

Hypothesis	Model Specification	Key Predictor	Estimate	95% CI	p-Value	Effect Type	N	Controls/FE	Decision
H1 (Centrality → Restricted/Negotiated)	Logit: proc_restr_neg ~ sup_degree_std + log(value)	Supplier degree (std.)	−0.271	[−0.325, −0.218]	<1 × 10−22	Log-odds (per SD)	20,851	log(contract value)	Supported (−)
H2 (Buyer centrality → Supplier diversity)	OLS: HHI ~ buyer_degree_std + log(total value)	Buyer degree (std.)	(negative; precise coefficient in CSV)	(95% CI in CSV)	(p-value in CSV)	Linear (HHI)	(Number ofbuyers)	log(total value)	Directionally supported (−)
H3 (Supplier clustering → Repeat awards)	Poisson: repeat_awards ~ sup_cluster + log(buyer total) + log(pair value)	Supplier bipartite clustering	−0.009	[−0.058, 0.040]	0.712	Log-IRR	6079	log(buyer total awards), log(pair value)	Not supported

Notes: H1 uses supplier degree (std.) as the centrality proxy because betweenness on the raw bipartite graph is almost entirely zero; this is reported transparently in the text. H2 measures buyer-side supplier diversity with HHI (lower = more diverse). H3 uses supplier bipartite clustering (Latapy) computed on a buyer–supplier two-mode graph with disjoint partitions; outcome is pair-level repeat awards (Poisson). All models include the listed controls. (Interpretation guide for Table A2: Negative H1 means more-connected suppliers are less likely to appear in restricted/negotiated procedures, after value control; H2’s negative effect means more-central buyers have lower concentration (more diversity); H3 is statistically null with strong positive size controls).

. Full hypothesis-testing outputs (model specifications, estimates, 95% CIs, p-values, effect sizes, controls/fixed effects, and robustness checks) are reported in Appendix B, Table A2.

4.2. Process Topology and Regulatory Chokepoints (Figure 2)

To connect market structure with legal process, we map a legal-regulatory flow for a standard procurement cycle (call → submission → evaluation → award → performance → review/appeal) [14]. Represented as a directed procedural graph, the flow reveals structural bottlenecks—e.g., points where a small number of decision nodes mediate many edges (high betweenness). Such nodes act as de facto gatekeepers of throughput and are prime candidates for proactive oversight (e.g., time-to-decision monitoring, transparency checks). Importantly, we observe feedback loops via review bodies and audit authorities that can re-route or dampen flows after adverse events, aligning with the broader complex-systems perspective [5,6,7,8].

Figure 2. Legal-regulatory flow diagram (standard procedure).

Figure 2. Legal-regulatory flow diagram (standard procedure).

Directed process network of a typical procurement procedure, highlighting decision gates, appeal loops, and monitoring edges. The diagram is a process-level companion to the market-level topology in Figure 1.

4.3. Multilayer Representation of Platform Ecosystems (Figure 3)

For digital platforms, we present a multilayer network that distinguishes the market layer (platforms ↔ users/business providers) from the regulatory layer (EU/national authorities ↔ platforms). Edges are directed and can be weighted by transaction/data flow (market layer) or enforcement/oversight intensity (regulatory layer). The multilayer representation makes explicit the feedback cycle—regulatory intervention → platform adjustments → changes in user/provider behaviour → subsequent regulatory response—allowing structural hypotheses about contestability, interoperability, and self-preferencing to be expressed and tested.

Figure 3. Multilayer platform ecosystem.

Figure 3. Multilayer platform ecosystem.

Platforms are rendered as central hubs; user and business-provider nodes attach via usage/transaction edges; regulators attach via oversight/enforcement edges. Inter-platform edges capture interoperability and multi-homing [1,12]. The feedback loop appears as a directed cycle from regulators → platforms → users/providers → regulators. This representation aligns topological signatures (hub dominance, module boundaries) with DMA-relevant obligations and remedies [1].

4.4. Synthesis

Across procurement and platform settings, topology provides measurable signals with normative relevance [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. In procurement, sparse yet highly modular structure with a high-value backbone indicates where contestability and resilience are most at risk and where targeted legal instruments (e.g., lot structuring, framework design, modification controls) can be most effective. In platforms, multilayer modelling links market architecture directly to regulatory leverage points, clarifying expected system responses [1,12]. Together, these results demonstrate how empirical topology can underpin evidence-informed governance—from risk dashboards that watch centrality and modularity in real time to ex ante design that anticipates structural side effects of legal rules [1,5,6,7,8,12,13,14].

5. Discussion—Implications for Law and Regulatory Design

Topological modelling adds a structural lens to legal-economic analysis: instead of treating procurement and platform governance as sets of isolated rules or bilateral relations, it renders who connects to whom, with what intensity, and through which gates. This network view improves description, but—crucially—it also supports diagnosis (where risk concentrates, where oversight is thin) and prescription (which levers to pull, at what layer and scale) [1,2,3,4,5,6,7,8,9,10,11,12,13,14].

5.1. What Topology Reveals That Doctrine Alone Does Not

Our procurement results (Figure 1; Table 1) show a sparse yet highly modular market with a high-value backbone dominated by a small set of buyers and suppliers. Such architecture carries three policy-relevant signals:

• Concentration at the top—dense, high-value corridors raise contestability concerns and single-point vulnerabilities (operational and integrity risks).
• Strong modularity—sectoral “islands” can be efficient, but excessive separation limits spillovers and access for peripheral actors.
• Peripheral sparsity—weakly connected nodes, often smaller suppliers or authorities, face higher entry and learning costs [5,6,7,8,9,10,11].

For platform ecosystems (Figure 3), a multilayer topology makes visible how hub dominance and feedback loops (regulator → platform → users/providers → regulator) shape market access and enforcement pathways—core DMA concerns (interoperability, self-preferencing, data use) [1,12].

5.2. Mapping Structural Signals to Legal Levers

Topology is most useful when translated into actionable design choices:

Contestability & backbone concentration (procurement).

–

Ex-ante tools: lot structuring, caps on framework coverage, rotation rules for mini-competitions, mandatory parallel frameworks in high-risk CPV segments.

–

Ex-post tools: targeted audits and modification scrutiny for edges that enter the top-value backbone or exhibit sudden centrality spikes [9,10,11,13,14].

Modularity & access.

–

Bridging instruments: cross-cluster market consultations, SME-friendly qualification bundles, obligation to publish structured award/plan data to reduce information frictions across modules [9,10,11,13,14].

Process chokepoints (Figure 2).

–

Throughput safeguards: time-to-decision SLAs at high-betweenness nodes (evaluation, award), automated disclosure of queue metrics, and reason-giving standards to reduce discretionary bottlenecks [5,6,7,8,13,14].

Platforms & DMA alignment (Figure 3).

Multilayer remedies: interoperability duties where inter-platform edges are thin; audit trails and data-access remedies where regulator platform oversight edges are weak; proportional obligations tied to measured hub centrality [1,12].

5.3. Boundary-Aware and Scale-Sensitive Regulation

Legal thresholds (procurement values, DMA gatekeeper criteria) act as topological boundaries that channel flows. Designing these boundaries with network structure in mind avoids perverse effects (e.g., threshold gaming that multiplies edges without improving access). Topology also informs subsidiarity:

Highly modular procurement systems benefit from decentralized oversight within clusters, plus a light metropolitan backbone to ensure comparability and auditability.

Highly integrated platform markets require stronger central supervision, with local enforcement focused on implementation and complaints [1,9,10,11,12,13,14].

5.4. From Dashboards to Proactive Supervision

Topological analytics can power regulatory dashboards embedded in e-procurement and platform governance systems:

Live indicators: backbone share of spend; top-k centrality (buyers/suppliers); module isolation; anomaly motifs (sudden degree/weight jumps).

Trigger logic: crossing calibrated thresholds (e.g., backbone weight > x%, Δcentrality > y) automatically initiates soft inquiries, enhanced disclosure, or ex-ante review.

Explainability: captions and playbooks that tie each alert to the legal hook (e.g., modification controls, conflict-of-interest checks, DMA arts. on self-preferencing/interoperability) [1,9,10,11,12,13,14].

5.5. Comparative Design: Centralized vs. Decentralized Architectures

Figure 4 contrasts centralized (star-like) and decentralized (polycentric) procurement topologies. The former simplifies coordination and price discovery but magnifies single-node risk; the latter enhances resilience and access but requires backbone standards (common IDs, shared data schema, API reporting) to remain auditable at scale. A pragmatic path is a hybrid: decentralized buying with a shared backbone for identity, screening, and reporting.

5.6. Limitations and Next Steps for Regulators

Topology is not a substitute for doctrine; it is a decision aid. Metrics can be noisy (naming/ID quality, CPV heterogeneity) and context-dependent (sector mission vs. pure competition). To mitigate this:

• Pair structural signals with domain priors (sector norms, legal constraints).
• Run robustness checks (alternative community detection; degree-preserving null models; window sensitivity).
• Pilot sandbox regimes where dashboard triggers are tested against real cases before hard-wiring remedies.

By aligning measured structure (backbone, centrality, modularity) with legal instruments (lotting, frameworks, modification scrutiny, DMA remedies), regulators can move from episodic, ex-post enforcement toward evidence-informed, proactive design that targets risks where the network concentrates them.

6. Conclusions and Future Research

This article has advanced a novel interdisciplinary framework for analysing legal and economic systems through the lens of topology and network science [7]. By conceptualizing public procurement markets and digital platform economies as dynamic networked structures, it has demonstrated how graph-based methods can illuminate institutional complexity, reveal systemic vulnerabilities, and inform more adaptive approaches to regulatory design [9].

The key contributions are threefold. First, the study formalizes procurement and platform ecosystems as graph structures, where legal and economic actors are represented as nodes and their contractual, informational, and supervisory relationships as edges. Second, it shows how structural properties—centrality, modularity, connectivity, and clustering—carry direct implications for both market efficiency and legal compliance, making visible latent architectures that are often obscured in doctrinal or linear models. Third, it demonstrates how these representations can be used to identify regulatory chokepoints, feedback loops, and concentration risks in ways that support evidence-informed governance.

The implications for law and policy are significant. Topological diagnostics provide regulators with tools to anticipate structural risks (e.g., backbone concentration of procurement spend or gatekeeping power in platform ecosystems), to monitor systems in real time using evolving network indicators, and to compare legal regimes not only on doctrinal terms but also on measurable structural outcomes such as inclusiveness, contestability, and resilience. They further highlight when decentralized oversight may be preferable (e.g., in modular procurement systems) and when centralized control is needed (e.g., in highly integrated digital platforms).

Future research should build on these findings in several directions. One promising avenue is the development of interactive legal dashboards that integrate graph analytics directly into procurement registries or platform governance interfaces, enabling continuous monitoring of structural signals such as centrality spikes, clustering of high-value contracts, or emerging gatekeeping behaviours. Comparative studies across jurisdictions could examine how differences in network topology correlate with legal outcomes—ranging from transparency and competition to litigation intensity or corruption risk. Methodological innovation also offers opportunities: combining machine learning with graph theory could improve anomaly detection and predictive capacity, while advances in topological data analysis (TDA) may capture higher-dimensional structures that conventional graph models miss.

At the same time, several challenges remain. Greater efforts are required to ensure data standardization and interoperability across legal regimes, to address the ethical implications of analysing sensitive procurement and platform interaction data, and to develop interpretive heuristics that translate quantitative metrics into normative legal reasoning. Bridging these methodological and normative dimensions is essential if topological approaches are to move from proof-of-concept into operational regulatory practice.

Building on the preliminary hypotheses and statistical tests introduced in this study, future research should pursue broader and more rigorous empirical validation. Comparative analyses across countries and procurement regimes would allow us to test whether backbone concentration, centrality, and clustering carry consistent risks of collusion, favouritism, or reduced supplier diversity in different institutional contexts. Longitudinal designs could evaluate whether reforms—such as changes to procurement thresholds, framework agreements, or the implementation of the Digital Markets Act—produce measurable shifts in network structure over time. In addition, integrating machine-learning classifiers with network metrics could enhance predictive accuracy in detecting anomalies or early warning signals of legal non-compliance. These directions would firmly establish topological diagnostics not only as conceptual tools but as empirically validated instruments for evidence-informed governance.

In conclusion, topology is not merely a metaphor for complexity. It provides a rigorous, visual, and empirically grounded language for understanding how legal-economic systems are structured and how they evolve [9]. By adopting this analytical perspective, legal scholars, economists, and policymakers can work together to design more resilient, transparent, and equitable governance architectures capable of responding to the structural realities of contemporary markets [1,12,13,14].