Beyond non-backtracking: non-cycling network centrality measures

Walks around a graph are studied in a wide range of fields, from graph theory and stochastic analysis to theoretical computer science and physics. In many cases it is of interest to focus on non-backtracking walks; those that do not immediately revisit their previous location. In the network science context, imposing a non-backtracking constraint on traditional walk-based node centrality measures is known to offer tangible benefits. Here, we use the Hashimoto matrix construction to characterize, generalize and study such non-backtracking centrality measures. We then devise a recursive extension that systematically removes triangles, squares and, generally, all cycles up to a given length. By characterizing the spectral radius of appropriate matrix power series, we explore how the universality results on the limiting behaviour of classical walk-based centrality measures extend to these non-cycling cases. We also demonstrate that the new recursive construction gives rise to practical centrality measures that can be applied to large-scale networks.

Multiple scattering of waves in random media: a transfer matrix approach

Transfer matrices address multiple scattering of waves in dense media, by relating properties of a thick slice of medium to the corresponding properties of a slice thin enough for multiple scattering to be absent, hence scattering properties can easily be calculated. We present a new method for deriving a whole hierarchy of transfer matrices which can be applied to studying statistics of waves in random media under conditions of extreme multiple scattering. The new method, based on matrix power series expansions, confirms early results, and carries them forward into new areas, allowing us to study density of states, fluctuations, and transport phenomena.