Graph Theory for Prosecutors.

• Author: Crickard, Paul

Elvis Presley was in Viva Las Vegas with Cesare Danova, who was in Animal House with Kevin Bacon. This means that Elvis has a Bacon number of 2. (1) If this makes sense to you, you are probably familiar with the game Six Degrees of Kevin Bacon. If not, the game starts by naming an actor and trying to link them back to Kevin Bacon in as few connections as possible by using their co-star network. The game is based on graph theory, in particular the small world experiments by Stanley Milgram and later by Duncan Watts. These experiments attempted to prove that people were connected by a small number of connections.

Just as actors can be linked by movies, offenders can be linked by incidents. When two individuals commit a burglary, they have created a relationship. If we look at other individuals that have been involved with these two and add the relationships, we build a co-offender network. Co-offender networks can provide context to a single case. In a case management system, the connections between two offenders may be clear--they may be listed as co-defendants--but the relationships between them and other offenders in other cases is invisible. What if you could see that your misdemeanor possession defendant had a connection to a defendant in a murder case and could possibly offer information? Network graphs can make this possible. This article will provide an overview of graph networks and how they can, and are, used in criminal justice.

WHAT IS A GRAPH?

If you were not a math major, a graph was the plot of a function in algebra or calculus. You would use an x and y axis to plot a line for the equation of say y=2x+5 and that line would be the graph. A graph in the field of graph theory is comprised of a set of vertices (nodes in computer science) and edges. A set is a collection of objects with no duplicates. The set of vertices is a non-null set--otherwise the graph would be empty. The set of edges can be null, but is usually an unordered pair of vertices that are elements, or a subset, of the set of vertices. This means that each edge has to start and end at an existing node. This is often written as G=(V,E).

Networks can be undirected or directed and this is an important distinction. If Paul follows Yolanda on Twitter, that is a directional graph. In an undirected graph the reciprocal relationship is always true--Paul follows Yolanda and Yolanda follows Paul. Directed and undirected become important when traversing the graph. If you are trying to find the shortest path between nodes, you can walk along any edge in an undirected graph. In a directed graph, you must follow the direction of the connection--think about a one-way road.

CO-OFFENDER NETWORKS

Co-offender networks are common in criminal justice because crime is usually a multi-participant event and because the data is one of the easier to compile. To build a co-offender network, start...