Dynamical systems taking place on networks, such as opinion dynamics, synchronization, consensus or random walks, reveal a lot about their structure. In particular we show, through a dynamical reinterpretation of well-known concepts, how centrality measures (such as pagerank, eigencentrality, etc.)  and community detection quality functions (such as modularity, Potts, model, stability, etc.) are intimately related. The dynamical interpretation allows to design new centrality or community detection measures tailored for every particular application.

In this talk, I will present recent results from 2 recent papers. i) Random teleportation is a necessary evil for ranking and clustering directed networks based on random walks. Teleportation enables ergodic solutions, but the solutions must necessarily depend on the exact implementation and parametrization of the teleportation. For example, in the commonly used PageRank algorithm, the teleportation rate must trade off a heavily biased solution with a uniform solution. Here we show that teleportation to links rather than nodes enables a much smoother trade-off and effectively more robust results. We also show that, by not recording the teleportation steps of the random walker, we can further reduce the effect of teleportation with dramatic effects on clustering. ii) The traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static weighted network. This implicitly assumes that the edges are governed by Poisson processes, which is not typically the case in empirical temporal networks. Consequently, we examine the effects of non-Poisson inter-event statistics on the dynamics of edges, and we apply the concept of a generalized master equation to the study of continuous-time random walks on networks. We show that the equation reduces to the standard rate equations when the underlying process is Poisson and that the stationary solution is determined by an effective transition matrix whose leading eigenvector is easy to calculate. We discuss the implications of our work for dynamical processes on temporal networks and for the construction of network diagnostics that take into account their nontrivial stochastic nature.