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Abstract
The structure of networks among individuals not only has a large impact on constant-selection evolutionary dynamics but also shapes the dynamics of contagion processes. In this dissertation, we investigate two kinds of dynamics on networks: evolutionary dynamics and epidemic dynamics.
Evolutionary dynamics describes spreading and competition of different types of individuals in a population. Prior research has revealed that the population structure, which is typically modeled by networks, is a key factor that affects evolutionary dynamics. Hypergraphs are a generalization of networks and model a set of groups in a population in which a group can involve more than two individuals who simultaneously interact, differently from conventional networks. We propose and analyze evolutionary dynamics on hypergraphs, in which each node takes one of the two types of different but constant fitness values. For the corresponding dynamics on conventional networks, under the birth-death process and uniform initial conditions, most networks are known to be amplifiers of natural selection; amplifiers by definition enhance the difference in the strength of the two competing types in terms of the probability that the mutant type fixates in the population. In contrast, we provide strong computational evidence that a majority of hypergraphs are suppressors of selection under the same conditions by combining theoretical and numerical analyses. We also show that this suppressing effect is not explained by one-mode projection, which is a standard method for expressing hypergraph data as a conventional network. These results encourage us to study evolutionary dynamics on another extension of conventional networks, i.e., multilayer networks. We then study constant-selection dynamics on two-layer networks in which the fitness of a node in one layer affects that in the other layer, under birth-death processes and uniform initialization, which are commonly assumed. We show mathematically and numerically that two-layer networks are suppressors of selection, relative to the constituent one-layer networks. In fact, many two-layer networks are suppressors of selection relative to the most basic baseline, the Moran process. This result is in stark contrast with the results for conventional one-layer networks for which most networks are amplifiers of selection.
We also investigate epidemic dynamics in temporal networks. One feature of temporal networks--the concurrency of edges, quantified by the number of edges that share a common node at a given time point, may be an important determinant of epidemic processes. We propose theoretically tractable Markovian temporal network models in which each edge flips between the active and inactive states in continuous time. The different models have different amounts of concurrency while we can tune the models to share the same statistics of edge activation and deactivation (and hence the fraction of time for which each edge is active) and the structure of the aggregate (i.e., static) network. We analytically calculate the amount of concurrency of edges sharing a node for each model. We then numerically study effects of concurrency on epidemic spreading in the stochastic susceptible-infectious-susceptible (SIS) and susceptible-infectious-recovered (SIR) dynamics on the proposed temporal network models. We find that the concurrency enhances epidemic spreading near the epidemic threshold while this effect is small in many cases. Furthermore, when the infection rate is substantially larger than the epidemic threshold, the concurrency suppresses epidemic spreading in a majority of cases. Our numerical simulations suggest that the impact of concurrency on enhancing epidemic spreading within our model is consistently present near the epidemic threshold but modest. The proposed temporal network models are expected to be useful for investigating effects of concurrency on various collective dynamics on networks including both infectious and other dynamics.
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