Dynamics On and Of Complex Networks: Functional Communities and Epidemic Spreading

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The work presented in this thesis focusses on two topics: functional communities and epidemic spreading on dynamic networks. The first part of the thesis focuses on a functionally-based definition of community structure for complex networks. In particular, we consider networks whose function is enhanced by the ability to synchronize and/or by resilience to node failures. For networks whose functional performance is dependent on these processes, we propose a method that divides a given network into communities based on maximizing a function of the largest eigenvalues of the adjacency matrices of the resulting communities. We also explore the differences between the partitions obtained by our function-based method and the structure-based modularity approach. A major finding is that, in many cases, modularity-based partitions do almost as well as the function-based method in finding functional communities, even though modularity does not specifically incorporate consideration of function. We also discuss the spectral properties of the networks with community structure, relevant for the case of functional communities studied in this thesis.

In the second part of the thesis, we study a discrete time SIR model on dynamic networks. In our dynamic network model, we consider the case where the nodes in the network change their links both in response to the disease and also due to social dynamics. We assume that the individuals trying to make new connections mix randomly, and, with a certain probability, we also allow for the formation of new susceptible-infected links. We find that increasing the social mixing dynamics inhibits the disease's ability to spread in certain cases. This occurs because susceptibles who randomly disconnect from infected individuals preferentially reconnect to other susceptibles, inhibiting the disease spread. Finally, we also extend our dynamic network model to take into account the case of hidden infection. Here we find that, as expected, the disease spreads more readily if there is an initial time period during which an individual is infectious but unaware of the infection.