UCLA

Networks are a powerful tool to investigate complex systems. In this work, we apply network–theoretic tools to study criminal, educational, and ecological systems.

First, we propose two generative network models for recruitment and disruption in a hierarchal organized crime network. Our network models alternate between recruitment and disruption phases. In our first model, we simulate recruitment as Galton–Watson branching. We simulate disruption with an agent that moves towards the root and arrests nodes in ac- cordance with a stochastic process. We prove a lower bound on the probability that the agent reaches the kingpin and verify this numerically. In our second model, we propose a network attachment mechanism to simulate recruitment. We define an attachment probability based on an existing node’s distance to the leaf set (terminal nodes), where this distance is a proxy for how close a criminal is to visible illicit activity. Using numerical simulation, we study the network structures such as the degree distribution and total attachment weight associated with large networks that evolve according to this recruitment process. We then introduce a disruptive agent that moves through the network according to a self-avoiding random walk and can remove nodes (and an associated subtree) according to different disruption strate- gies. We quantify basic law enforcement incentives with these different disruption strategies and study costs and eradication probability within this model.

In our next chapter, we adapt rank aggregation methods to study how Mathematics students navigate their coursework. We first translate 15 years of grade data from the UCLA Department of Mathematics into a network whose nodes are the various Mathematics courses and whose edges encode the flow of students between these courses. Applying rank aggregation on such networks, we extract a linear sequence of courses that reflects the order students select courses. Using this methodology, we identify possible trends and hidden course dependencies without investigating the entire space of possible schedules. Specifically, we identify Mathematics courses that high–performing students take significantly earlier than low–performing students in various Mathematics majors. We also compare the extracted sequence of several rank aggregation methods on this data set and demonstrate that many methods produce similar sequences.

In our last chapter, we review core–periphery structure and analyze this structure in mu- tualistic (bipartite) fruigivore–seed networks. We first relate classical graph cut problems to previous work on core–periphery structure to provide a general mathematical framework. We also review how core–periphery structure is traditionally identified in mutualistic networks. Next, using a method from Rombach et al., we analyze the core–periphery structure of 10 mutualistic fruigivore–seed networks that encode the interaction patterns between birds and fruit–bearing plants. Our collaborators use our network analysis with other ecological data to identify important species in the observed habitats. In particular, they identify certain types of birds (mashers) that play crucial roles at a variety of sites, which are though to be less important due to their feeding behaviors.