UC Berkeley

Reaction networks are a powerful tool for modeling the behavior of a wide variety of real-world systems, including population dynamics and chemical processes, as well as algorithms for sampling combinatorial objects. While many such systems have well-understood equilibrium states, the long-standing conjecture that these states will always be achieved remains open. This thesis presents the class of simplicial reaction networks, which includes a wide variety of natural combinatorial examples of use in theoretical computer science. It shows how simplicial structures can be used to understand and control the equilibrium behavior of the network as a whole, and discusses related progress towards the Global Attractor Conjecture. Finally, this thesis presents additional work exploring combinatorial approaches to the Inverse Eigenvalue Problem on graphs, including the randomized Zero Forcing algorithm and a lower bound for the Minimum Rank problem.