Bipartite structure and dynamics of political corruption networks

Abstract

Political corruption is inherently an affiliation process linking agents to corruption cases; yet it is often studied via one-mode projections that connect co-offenders within the same scandal, implying a loss of information that potentially confounds properties of agents and cases. Here, we adopt a bipartite representation to analyze datasets of corruption scandals in Brazil and Spain spanning nearly three decades. By tracking the temporal growth of these networks, we quantify density and redundancy measures to capture partner reuse and co-occurrence across cases. Networks in both countries become progressively sparser over time, and agent redundancy is systematically higher than case redundancy, indicating a small cadre of recidivists who recombine largely with novice partners rather than forming durable co-offending ties. These networks exhibit near-exponential degree distributions, reflecting low recidivism and likely high coordination costs and secrecy constraints of large-scale scandals. Our bipartite view further reveals a moderate cross-mode disassortative degree mixing between agents and cases, with high-degree agents distributing their activity across small cases and large scandals mainly comprising low-degree participants. Finally, identifying atypical individuals within the bipartite structure reveals criminal trajectories marked by a gradual rise in network embeddedness that can appear ordinary in agent-projected networks.

Figures (6)

• Figure 1: Bipartite structure of corruption networks. Visualizations of the bipartite corruption networks for (A) Spain and (B) Brazil. In each panel, nodes on the left represent individuals and nodes on the right represent corruption scandals. Edges link individuals to the scandals in which they participated. Nodes are ordered from top to bottom by descending degree and are colored by score defined as the sum, over all incident edges, of the product of the degrees of the two endpoints; edge colors interpolate between the colors of their endpoint nodes (darker shades indicate larger values). These visualizations highlight that agents involved in many cases tend to distribute their participation across small and large cases, whereas agents involved in few cases are predominantly associated with the largest corruption scandals, yielding a structure marked by high-degree individuals involved in multiple cases and large scandals encompassing many low-degree participants.
• Figure 2: Evolution of topological properties during the growth of bipartite corruption networks. Panels show (A,B) graph density, (C,D) agent redundancy, and (E,F) corruption-case redundancy for the Spanish (top) and Brazilian (bottom) networks. Density is the proportion of realized agent–case edges among all possible. For an agent, redundancy is the fraction of pairs of its case neighbors that are also jointly linked by some other agent; the network-level agent redundancy is the mean over all agents. Case redundancy is defined analogously for each case as the fraction of pairs of its agent neighbors that also co-occur in at least one other case, averaged over cases. The persistent decline in density shows that both networks become progressively sparser over time, indicating that network expansion is primarily driven by the entry of new agents and cases rather than repeated collaborations among known actors, further suggesting that most scandals are short-lived and likely transaction-specific coalitions. Both networks consistently exhibit agent redundancy significantly higher than case redundancy, likely reflecting a small cadre of recidivists who repeatedly mobilize novice collaborators across scandals, consistent with secrecy and risk management strategies that discourage persistent criminal partnerships.
• Figure 3: Degree dynamics of agents and cases during the growth of bipartite corruption networks. Evolution of the average degree of agents (number of cases per agent) in the (A) Spanish and (B) Brazilian networks. Evolution of average case degree (number of agents per case) for the (C) Spanish and (D) Brazilian networks. Yearly cumulative distributions of the rescaled agent degree (degree divided by the corresponding yearly average) for (E) Spain and (F) Brazil. Yearly cumulative distributions of the rescaled case degree for (G) Spain and (H) Brazil. Cumulative distributions are color-coded by year, with black curves denoting the last available year. The stabilization of the average agent degree at small values reflects low recidivism among political agents. Similarly, the stabilization of the average case degree at small numbers implies that typical corruption cases are small, a scale likely enforced by coordination costs and secrecy constraints. The rescaled cumulative degree distributions for both agents and cases exhibit consistent collapse across years and present near-exponential forms on semi-log plots, indicating a lack of scale-free behavior. However, the pronounced deviations in the tails of the case degree distributions suggest that the largest scandals are generated by mechanisms distinct from those governing typical cases.
• Figure 4: Disassortative degree mixing between agents and corruption cases. Evolution of the Pearson correlation between the average degree of the cases in which each agent appears and their agent degree for the (A) Spanish and (B) Brazilian networks. Evolution of the Pearson correlation between the average degree of agents participating in each case and the case degree for the (C) Spanish and (D) Brazilian networks. Shaded bands show 95% bootstrap confidence intervals and horizontal dashed lines denote zero correlation. The small yet persistent negative correlations across most years indicate a moderate cross-mode disassortative mixing between agents and cases. This means that high-degree agents (those involved in many cases) are not concentrated in the largest scandals but tend to appear in smaller cases, whereas large scandals are populated mainly by low-degree participants. This pattern is consistent with secrecy and coordination constraints, as well as risk-spreading strategies employed by recidivists, who likely avoid co-appearing with other experienced agents in large coalitions. Structurally, this disassortativity suppresses hub–hub coalescence, favors many small scandals loosely stitched together, and delays core densification.
• Figure 5: Mining outliers in bipartite corruption networks. Scatter plot depicting agents' number of partners, average number of partners per case, and number of corruption cases for the (A) Spanish and (B) Brazilian networks. In both previous panels, results correspond to the final stage of each network. Dashed lines show the 1:1 relationship between the average number of partners per case and the total number of partners. Agents on this line participate in a single case. Cross markers indicate outliers identified with the isolation forest algorithm, and points are color-coded by their anomaly scores. The wide distribution across the three dimensions corroborates the need for a bipartite framework to disentangle recidivism, collaborative breadth, and exposure to large scandals. Visualization of agents flagged as outliers in (C) Spain and (D) Brazil. Outlier agents, their incident edges, and the cases they join are highlighted in color, while all remaining nodes and edges are in gray. This visualization confirms that atypical agents are often involved in small- and mid-sized scandals rather than being concentrated in the largest cases, reducing their average number of partners per case. Evolution of the anomaly score for each agent in the (E) Spanish and (F) Brazilian networks. Agents identified as outliers in at least one year are labeled and displayed in color, with larger markers indicating years in which they were classified as outliers. The final stage of the Brazilian network contains three outliers; two of which share identical properties and therefore overlap in panels (B) and (F). The temporal analysis further reveals that the criminal trajectories of most atypical agents are marked by a gradual ascent in the anomaly-score ranking over several years before reaching outlier status, suggesting that the emergence of atypical involvement is often a career process involving an accumulation of opportunities and know-how, rather than an isolated sequence of events.