Organisational hierarchy constructions with easy Kuramoto synchronisation

Abstract

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-augmentedhierarchy 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.

Sparse Network Optimization for Synchronization

“…the spectral ratio of the graph G. As alluded in the introduction, for a somewhat different class of dynamical systems, which the Kuramoto model approximates when linearised, graphs with small Q L seem to result in highly synchronised graphs, in that the graph is connected and σ is small [18]. For the Kuramoto model such an effect has been observed in [22].…”

“…In this section we briefly recall the mathematical concepts we will use in our models, including the Tree-expander Construction approach presented in [22].…”

“…To conclude this summary we mention that [22] have provided a construction for the Kuramoto model that yields modified trees -which are therefore sparse and well-balanced -that are provably expanders, with enhanced synchronization. They also showed that a computational search over random variations of the links on the leaf nodes of this construction (maintaining the degree distribution) to find instances with lowest possible spectral ratio did not necessarily give graphs of better synchronization.…”

“…We illustrate the fact that, when adding a moderate number of edges, a great improvement in synchronization properties of the graph can be achieved. We also make comparisons with certain graphs from the literature, in particular from [22], where graphs are obtained by appending at the bottom level of a hierarchical tree, additional matchings linking the leaf nodes. The graphs found in this way in [22] enhance synchronization compared to the original tree.…”

“…We also make comparisons with certain graphs from the literature, in particular from [22], where graphs are obtained by appending at the bottom level of a hierarchical tree, additional matchings linking the leaf nodes. The graphs found in this way in [22] enhance synchronization compared to the original tree. We emphasise the key result: that we can generate graphs with optimal sparseness, balanced load, and good synchronization properties starting from a random frequency and initial phase input.…”