Structural asymmetry as a fraud signature: detecting collusion with Heron's Information Coefficient

TL;DR

This paper introduces Heron's Information Coefficient (HIC), a geometric measure that quantifies structural imbalance between active and inactive subgraphs in dynamic networks to detect collusive patterns in public procurement. By defining HIC via the triangle area formed by distances among the original, active, and inactive networks and normalizing against an equilateral reference, the authors provide a scalable, topology-aware indicator that complements traditional robustness metrics. Applied to eight years of Brazilian health procurement data and synthetic networks, HIC demonstrates robust detection of covert structures and greater sensitivity to structural shifts than conventional metrics. The work suggests practical use for auditors and policymakers to monitor procurement integrity and advocates for integration with existing legal and economic analyses, along with future real-time monitoring and cross-metric comparisons.

Abstract

Fraud in public procurement remains a persistent challenge, especially in large, decentralized systems like Brazil's Unified Health System. We introduce Heron's Information Coefficient (HIC), a geometric measure that quantifies how subgraphs deviate from the global structure of a network. Applied to over eight years of Brazilian bidding data for medical supplies, this measure highlights collusive patterns that standard indicators may overlook. Unlike conventional robustness metrics, the Heron coefficient focuses on the interaction between active and inactive subgraphs, revealing structural shifts that may signal coordinated behavior, such as cartel formation. Synthetic experiments support these findings, demonstrating strong detection performance across varying corruption intensities and network sizes. While our results do not replace legal or economic analyses, they offer an effective complementary tool for auditors and policymakers to monitor procurement integrity more effectively. This study demonstrates that simple geometric insight can reveal hidden dynamics in real-world networks better than other Information Theoretic metrics.

Figures (19)

• Figure 1: A small network $G$ with two complementary subgraphs $G^a$ (active) and $G^i$ (inactive). The D-distance between each pair is $D(G,G^i)=0.4582513$$D(G,G^a)=0.3032006$ and $D(G^i,G^a)=0.2715595$. The largest triangle with perimeter $D(G,G^i)+D(G,G^a)+D(G^i,G^a)$ has area $0.05134147$ but the triangle with sides $D(G,G^i)$, $D(G,G^a)$ and $D(G^i,G^a)$ has $0.03964952$ and, thus, the fraction between the equilateral triangle and this second triangle $0.03964952/0.05134147=0.77$ is the Heron's coefficient.
• Figure 2: Multi-dimensional scaling map considering the $D$-distance and four small networks, $G_1$, $G_2$, $G_3$ and $G_4$. Here we visualize how non-isomorphic networks with 4 nodes form triangles with different areas based on the $D$-measure.
• Figure 3: (A) shows the average values of the Heron's Information Coefficient for 100 experiments. (B) shows the network efficiency evolution for active and inactive subgraphs, as well as the average. The average efficiency of the active and inactive subgraphs tend to plateau close to 0.75 after a spike of the maximum HIC value.
• Figure 4: Network with 10% of active links, where link activation probability depends on edge betweenness centrality ($b_e$) and a real number $\gamma$, such that $p_e \propto b_e^\gamma$ being proportional to $b_e^\gamma$. (A) shows an example of active and inactive edges and (B) the average HIC for 100 seeds across different $\gamma$ values.
• Figure 5: Sensitivity comparison of the edge removal across relevant metrics (HIC, degree robustness, betweeness robustness, clustering, and modularity) for different network types (scale-free, small-world, Erdős–Rényi, community-structured, hub-and-spoke, and a real bidding network. The real network corresponds to the 2016 Surgical Masks network, where our metric outperformed the others, as it consistently did for Brazilian bidding networks.