How Graph Theory Reveals Hidden Criminal Networks: Centrality Measures Explained

When watching detective TV shows, you often see a board covered with photos of suspects linked by lines; investigators use such diagrams to analyze and plan.

The core idea is to use these relational maps to locate the most important roles in the network, a task that traditionally relies on expert judgment.

Mathematically, the network can be represented as a graph, enabling systematic analysis through graph theory.

Graphs and Networks

A graph (or network) models relationships between objects with a set of nodes and a set of edges.

For example, the figure below shows a graph with five nodes and five edges.

In a crime‑network model, nodes represent suspects or related persons, and edges represent their connections.

To identify “key” nodes, we apply the concept of centrality , which measures the relative importance of each node.

Centrality

Centrality metrics detect important nodes; different measures can yield different results when searching for pivotal criminals.

Degree Centrality

Degree centrality evaluates importance by counting how many connections a node has, useful for spotting popular figures in a network.

Applying the formula to the example graph gives the degree centrality of each suspect based on its three connections within a total of seven nodes.

Betweenness Centrality

Betweenness centrality measures how often a node lies on the shortest paths between other nodes, reflecting its role in information flow.

Shortest paths (computed by observation in small graphs or by Dijkstra’s algorithm in larger ones) are counted, and each node on a path receives an equal share of one point. Summing these points yields the node’s betweenness centrality; the highest score identifies the most critical intermediary.

Normalization by the total number of nodes does not change the ranking.

Closeness Centrality

Closeness centrality assesses how easily a node can be reached from all other nodes, highlighting actors that can quickly disseminate information.

To compute it, count the number of nodes at distance 1, distance 2, etc., up to the longest shortest‑path distance, then apply the standard formula.

Using the example graph, the calculated closeness values identify which suspect is most centrally positioned.

Different centrality measures can lead to different definitions of “most important” criminal; the most connected individual is not always the key player.

Economic Crime Case Study

Graph theory also applies to financial networks of organized crime. For a large drug‑trafficking syndicate, transaction data can be transformed into a graph where nodes are individuals or accounts and edges are financial transfers.

Analyzing this graph helps pinpoint critical accounts to disrupt the money flow.

Degree Centrality Analysis : Identify the most frequently transacting accounts.

Method: Compute degree centrality for each node.

Result: Highlight accounts with the highest number of connections.

Betweenness Centrality Analysis : Find accounts that act as vital intermediaries in fund movement.

Method: Compute betweenness centrality.

Result: Reveal nodes that facilitate money laundering or distribution.

Closeness Centrality Analysis : Locate the most centrally positioned accounts in the financial network.

Method: Compute closeness centrality.

Result: Identify accounts that can efficiently coordinate and control cash flow.

These examples demonstrate three main advantages of graph‑theoretic analysis in crime investigation: systematic modeling of large datasets, multi‑perspective identification of key nodes via different centrality measures, and efficient processing of complex relationship networks even at scale.

Mathematics, especially graph theory, plays a vital role in detecting and combating organized crime. By mathematically modeling criminal networks, investigators can more systematically and scientifically identify and track key individuals, improving investigative efficiency and enhancing law‑enforcement capabilities. This showcases the power of mathematics in real‑world applications and aims to inspire interest in graph theory and its vast potential.

Reference: Gomez, M. (2021, November 22). Solving crimes with maths: Busting criminal networks. In Marianne (Ed.), Share this page.