APPLICATIONS OF NETWORK THEORY TO FRUSTRATED SPIN SYSTEMS AND TRANSITIONS IN MODELS OF DISEASE SPREAD

APPLICATIONS OF NETWORK THEORY TO FRUSTRATED SPIN SYSTEMS AND TRANSITIONS IN MODELS OF DISEASE SPREAD

By Thomas E. Stone, Jr.

Thesis Advisor:  Dr. Susan R. McKay

A Lay Abstract of the Thesis Presented

in Partial Fulfillment of the Requirements for the

Degree of Doctor of Philosophy

(in Physics)

August, 2010

            This study of network structure and phase transitions focuses on three systems with different dynamical rules:  a model of a magnetic material known as a spin glass, the susceptible-infected-recovered (SIR) epidemic model on one type of network (called a small-world network) where the links between individuals can change in response to an outbreak, and the SIR model on a different type of network (the Saramäki-Kaski dynamic small-world network) that models a mobile population.

            Spin-glass ordering is unlike that of ferromagnets, antiferromagnets, or paramagnets due to competing interactions.  These competing interactions lead to unusual predictions for the behavior of correlations occurring between spins.  We employ a novel network construction to observe these unique correlations for the first time, which are seen as spatially non-contiguous clusters of magnetic moments that are fluctuating in the same manner.  Our work lends further evidence for the existence of spin-glass behavior in two dimensions, which remains an unresolved question.

            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 theory to verify our simulation results in certain limits.

            The third part of this thesis treats a dynamic small-world network, in order to use its computational advantages to study in detail the disease-free to epidemic phase transition.  We develop and solve the equations for the critical mobility required for an epidemic to occur, and verify this point numerically.  The associated critical exponents are also found, which show that this model shares some universal features with other works.  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.