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Highlights:
- Epidemiological models typically assume that the dynamics of social mixing takes one of two extremes, neither of which is particularly realistic. At one extreme, they assume that the social mixing dynamics are much faster than the spreading process, and hence that mass-action models can be adopted in which a continuum approximation can be used to generate differential equations based on calculus. This tends to be the extreme adopted by mathematicians and physicists and engineers, since it unlocks the power of calculus and the vast spectrum of known properties of differential operators. At the other extreme, is the limit which tends to be adopted by the social science community, in which the heterogeneity of social links is retained at the expense of assuming that the social network is static on the timescale of the spreading. Despite their attractions for analytic and computational reasons, neither limit is realistic for most social spreading processes such as rumors and colds. The few works that have attempted to allow dynamics into networks, and yet retain heterogeneity, have focused on the limit in which very few links are broken every timestep and rewiring occurs.
- In this thesis, I address for arguably the first time, problems in which both the social network and the process are evolving — and may be doing so on similar timescales. In particular, I address three complementary topics with this common characteristic.
- Although necessarily toy-like, the model of social dynamics that I consider is known to produce realistic power-laws with exponent 2.5 which mimics the distribution of group sizes observed in a range of social pheonomena, from markets to conflicts. Adding a virus on top of this process, generates a rich playground of phenomena. Specifically, in the model that I introduce, both the timescale at which the population moves and the timescale at which the virus moves are similar.
- The observation of multiple resurgent peaks and abnormal decay times is qualitatively reproduced within the model simply by varying the time scales for group coalescence and fragmentation.
- In spite of the simplicity of our model, we find that the profiles produced bear a striking resemblance to a wide variety of real-world examples from social, financial, and biological domains.
- Simulations show that plasticity during an agent’s lifetime changes the path of evolution toward interesting behaviors that better approach those observed in humans. As plasticity is increased, the average offers and acceptance threshold also increase.
In the remaining two complementary projects
- Our results so far show that plasticity has a beneficial role in the evolution of strong reciprocity, by increasing the diversity of behaviors displayed among the population. Furthermore, plasticity is evolutionarily stable as (depending on the mechanism by which it is passed on to offspring) it does not disappear from the population if it is genetically determined.
- Patterns in the network structure change, as we vary observation time. This allow us to conclude that the measurement methodology, in particular the observation time—that we use to construct the networks—change the network structure.
Taken overall, this thesis presents one of the first — if not the first — works in which no assumption is made about the timescale of social mixing. Though much remains to be done, the works contained within arguably pave the the way for future explorations in this mathematically difficult — but completely realistic — regime in which both the social structure and the spreading process co-exist on arbitrarily similar timescales. Neil Johnson
