Hierarchical graphs and oscillator dynamics.

Dorrian, Henry Joseph (2015) Hierarchical graphs and oscillator dynamics. Doctoral thesis (PhD), Manchester Metropolitan University.

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Abstract

In many types of network, the relationship between structure and function is of great significance. This work is particularly concerned with community structures, which arise in a wide variety of domains. A simple oscillator model is applied to networks with community structures and shows that waves of regular oscillation are caused by synchronised clusters of nodes. Moreover, we demonstrate that such global oscillations may arise as a direct result of network topology. We also observe that additional modes of oscillation (as detected through frequency analysis) occur in networks with additional levels of hierarchy and that such modes may be directly related to network structure. This method is applied in two specific domains (metabolic networks and metropolitan transport), demonstrating the robustness of the results when applied to real world systems. A topological analysis is also applied to the real world networks of metabolism and metropolitan transport using standard graphical measures. This yields a new artificial network growth model, which agrees closely with the graphical measures taken on metabolic pathway networks. This new model demonstrates a simple mechanism to produce the particular features found in these networks. We conclude that (where the distribution of oscillator frequencies and the interactions between them are known to be unimodal) the observations may be applicable to the detection of underlying community structure in networks, shedding further light on the general relationship between structure and function in complex systems.