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Organisational hierarchy constructions with easy Kuramoto synchronisation

We provide a graph theoretic construction enabling tree-based hierarchies to be modified into structures with enhanced connectivity and synchronisability. Specifically, the construct transforms trees into members of a family of graphs known as expanders, which we call ‘expander-augmented-hierarchies’ or trees. We show that this produces graphs with significantly enhanced synchronisation properties in the context of the Kuramoto model of phase oscillators coupled on networks. When considered as organisational structures these networks enjoy both the managability of simple hierarchies with near regular degree distribution, and low critical coupling by the addition of relatively few extra edges. For the expander augmented tree, we examine the synchronisation properties, computed through the time-averaged Kuramoto order parameter over an ensemble of natural frequencies. We compare this with a range of other networks including hierarchies augmented by random matching of the leaf nodes. For these we compute the ratio Q of smallest to largest Laplacian eigenvalues, the smallness of which has been argued to be an indicator of good synchronisability. While not the best of these in Q, the expander-augmented-hierarchy exhibits synchronisability barely distinguishable from others with lower Q within the variance over an ensemble of natural frequencies. However, the expander augmented tree has the advantage that its properties are automatically designed for as opposed to the outcome of a random search for low Q-value graphs that in itself scales very poorly.