This study of network structure and phase transitions focuses on three systems with different dynamical rules: the Ising model with competing ferromagnetic and antiferromagnetic interactions on a 2D triangular lattice, the susceptible-infected-recovered (SIR) epidemic model on an adaptive small-world network, and the SIR model on the Saramäki-Kaski dynamic small-world network.

In the Ising model with competing interactions, we employ a novel network construction using the individual spins as nodes and links occurring between two nodes if their spin-spin correlation function exceeds a set threshold. This construction yields the emergence of multiple networks of correlated fluctuations. In the spin-glass-like phase, we find spatially non-contiguous networks of correlated fluctuations, as had been previously predicted by chaotic renormalization-group trajectory arguments, but not confirmed.

In the second part of this thesis we turn to a dynamical process, disease spreading, on an adaptive small-world network. The adaptive nature of the contact network means that the social connections can evolve in time, in response to the current states of the individual nodes, creating a feedback mechanism. Unlike previous work, we introduce a method by which this adaptive rewiring is included while maintaining the underlying community structure. This more realistic method can have significant effects on the final size of an outbreak. We also develop a mean-field theory to verify our simulation results in certain limits based on master equation considerations.

The third part of this thesis treats a dynamic small-world network, in order to utilize its computational advantages to study the critical phenomena of the disease-free to epidemic phase transition. We solve the dynamical equations for the predicted critical point, and verify this point via finite size scaling arguments. The associated critical exponents are found in a similar manner, which show this model to be in a new universality class. The relative effectiveness of vaccination and avoidance is studied, and it is shown that vaccination is always more effective, but that the difference is often negligible, leading us to conclude that avoidance is a comparable outbreak control strategy.