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

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