A Primer on Laplacian Dynamics in Directed Graphs

We analyze the asymptotic behavior of general first order Laplacian processes on digraphs. The most important ones of these are diffusion and consensus with both continuous and discrete time. We treat diffusion and consensus as dual processes. This is the first complete exposition of this material in a single work.

Graph clustering in industrial networks

The present work investigates clustering of a graph-based representation of industrial connections derived from international trade data by Hidalgo et al. (2007, The product space conditions the development of nations. Science, 317) and confirms the existence of nine industrial clusters that are reasonably consistent with expected historical patterns of diffusion of innovation and technology. This supports the notion that technological development occurs in sequential waves of innovation. The clustering method developed in this work follows conceptual ideas introduced by Pons and Latapy (2006, Computing communities in large networks using random walks. J. Graph Alg. Appl., 10, 191–218), Lambiotte and Barahona (2009, Laplacian dynamics and multiscale modular structure in networks. arXiv.org.) and other researchers—random walks can be used to assess hierarchical structures of network communities. We, however, implement these ideas differently in order to match the physics of the problem under consideration and introduce a hierarchical clustering procedure that is progressive in time and is combined with concurrent reordering of the elements. An equivalent spectral interpretation of the clustering procedure is also given and discussed in the paper.