Mathematical models can be used to detect, disrupt terror networks

a directed acyclic graph, with one source (the top of the volcano) and many sinks representing the lake. The parameter, “seepage,” represents the amount of contamination, and the “green number” corresponds to the number of agents required to halt it.

The release notes that a previous study modeled terrorist cells as partially ordered sets (a special kind of DAG), which are often used in mathematics to analyze an ordering, sequencing, or arrangement of distinct objects.  In such a system, terrorist plans are formulated by nodes at the top of the hierarchy, which represent the leaders or maximal nodes of the set. The plans are transmitted down to the nodes at the bottom: these represent foot soldiers in a terror network, or minimal nodes in the set, who would be presumed to carry out these plans. The assumption is that one messenger is sufficient for reception and execution of the plan. Thus, if the partially ordered set represents a courier network for a terrorist organization, the intention would be to block all routes from the maximal node to the minimal nodes by capturing or killing a subset of agents.

In this paper, the authors utilize the similarities in the previous terrorist cell model to Seepage, where greens try to prevent the sludge from moving to the sinks by blocking nodes. A number of different winning strategies employed by both players are explored when played on a DAG. The seepage and green number for disrupting a given hierarchical social network are analyzed.

The primary difference from the previous study’s model is that the Seepage model is dynamic: greens can move and choose new sets of nodes over time. The authors determine that Seepage is a more realistic model of counterterrorism, as the agents do not necessarily act all at once, but over time.

The analysis is made in two types of terrorist network structures, as Pralat explains, “We consider two extreme profiles: one where the network is regular, where every agent has about the same number of connections. The second profile is power law, where some agents have many connections, but most have very few.” This is analyzed by considering the total degree distribution of nodes in the DAG.  In regular DAGs, each level of the DAG would have nodes with about the same out-degree (number of outgoing edges emanating from a node), while power law DAGs would have many more low-degree nodes and a few with high degrees.

Mathematical analysis allows the authors to determine what point in a network would be most effective for disrupting messages. “Our mathematical results reinforce the view that intercepting the information or message in a hierarchical social network following a power law is more difficult close to levels near the source. For regular networks, it does not matter as much where the message is disrupted,” says Pralat.

“Future work could look at more complex profiles of networks, along with developing effective algorithms for disrupting the flow of information in a DAG using our game-theoretic approach.”

— Readmore in Anthony Bonato et al., “Vertex-Pursuit in Random Directed Acyclic Graphs,” SIAM Journal on Discrete Mathematics 27, no. 2 (16 April 2013): 732-56 (DOI: 10.1137/120866932)