The emergence of collective behavior on social and biological networks
Wilander, Adam Troy Charles
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In this thesis, we broadly examine collective behaviors in various social and biological contexts. Aggregation, for instance, is a natural phenomenon that occurs in a variety of contexts; it is observed in schools of fish, flocks of birds, and colonies of bacteria, among others. This behavior can be found in some agent-based models, where it is typically assumed every pair of individuals interact according to a simple set of rules. In the first half of this thesis, we study a particular, well-understood aggregation model upon relaxation of the assumption that every individual interacts with every other. We review prior results on this topic -- when the underlying structure of interactions is an Erdos-Renyi graph. Seeking to incorporate community structure into the network, we establish the analogous problem under a class of networks called stochastic block graphs; a particular aspect of the system's metastable dynamics is explored upon varying the graph's connection densities. Finally, we evaluate the potential to leverage this system's dynamics in order to recover community structure (given a known graph as input). In the second half of this thesis, we similarly explore the aggregate behaviors of synchronization and desynchronization, appearing in diverse settings such as the study of metabolic oscillations and cell behaviors over time, respectively. Previous studies have leveraged a model in which repressilator entities are connected by a diffusive quorum sensing mechanism; these have shown (numerically) that the complex composition of observable behaviors depends upon the insertion point of the upregulating protein in the feedback loop. We rigorously prove a version of this; for negative feedback, negative signaling (Nf-Ns) systems we find only a unique stable equilibrium or a stable oscillation is possible. Additionally, we observe (numerically) the complex multistable dynamics that arise when a positive signal is included in the feedback loop and characterize this shift as a saddle node bifurcation of a cubic curve.