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All networks can be analyzed at multiple scales. A higher scale of a networkis made up of macro-nodes: subgraphs that have been grouped into individualnodes. Recasting a network at higher scales can have useful effects, such asdecreasing the uncertainty in the movement of random walkers across the networkwhile also decreasing the size of the network. However, the task of findingsuch a macroscale representation is computationally difficult, as the set ofall possible scales of a network grows exponentially with the number of nodes.Here we compare various methods for finding the most informative scale ofpreferential attachment networks, discovering that an approach based onspectral analysis outperforms greedy and gradient descent-based methods. Wethen use this procedure to show how several structural properties of thesenetworks vary across scales. We describe how meso- and macroscalerepresentations of networks can have significant benefits over their underlyingmicroscale in terms of information transmission, which include properties suchas increase in determinism, a decrease in degeneracy, a lower entropy rate ofrandom walkers on the network, an increase in global network efficiency, andhigher values for a variety of centrality measures than the microscale.

Finding the right scale of a network: Efficient identification of causal emergence through spectral clustering