title: Collective frequency variation in network synchronization and reverse PageRank.
abstract: A wide range of natural and engineered phenomena rely on large networks of interacting units to reach a dynamical consensus state where the system collectively operates. Here we study the dynamics of self-organizing systems and show that for generic directed networks the collective frequency of the ensemble is not the same as the mean of the individuals' natural frequencies. Specifically, we show that the collective frequency equals a weighted average of the natural frequencies, where the weights are given by an outflow centrality measure that is equivalent to a reverse PageRank centrality. Our findings uncover an intricate dependence of the collective frequency on both the structural directedness and dynamical heterogeneity of the network, and also reveal an unexplored connection between synchronization and PageRank, which opens the possibility of applying PageRank optimization to synchronization. Finally, we demonstrate the presence of collective frequency variation in real-world networks by considering the UK and Scandinavian power grids.
authors.affiliation: Department of Mathematics, Trinity College, Hartford, Connecticut 06106, USA.
authors.forename: Per Sebastian
authors.initials: PS
authors.lastname: Skardal
authors.affiliation: Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA.
authors.forename: Dane
authors.initials: D
authors.lastname: Taylor
authors.affiliation: Department of Mathematics, Clarkson University, Potsdam, New York 13699, USA.|Department of Physics, Clarkson University, Potsdam, New York 13699, USA.
authors.forename: Jie
authors.initials: J
authors.lastname: Sun
authors.affiliation: Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain.
authors.forename: Alex
authors.initials: A
authors.lastname: Arenas