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A wide range of natural and engineered phenomena rely on large networks ofinteracting units to reach a dynamical consensus state where the systemcollectively operates. Here we study the dynamics of self-organizing systemsand show that for generic directed networks the collective frequency of theensemble is {\it not} the same as the mean of the individuals' naturalfrequencies. Specifically, we show that the collective frequency equals aweighted average of the natural frequencies, where the weights are given by anout-flow centrality measure that is equivalent to a reverse PageRankcentrality. Our findings uncover an intricate dependence of the collectivefrequency on both the structural directedness and dynamical heterogeneity ofthe network, and also reveal an unexplored connection between synchronizationand PageRank, which opens the possibility of applying PageRank optimization tosynchronization. Finally, we demonstrate the presence of collective frequencyvariation in real-world networks by considering the UK and Scandinavian powergrids.

Collective frequency variation in network synchronization and reverse PageRank