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

TL;DR

This paper introduces a spectral clustering method to efficiently identify the most informative scale of networks, demonstrating its advantages over other approaches and highlighting benefits of macro-scale representations for information transmission.

Contribution

The paper presents a spectral analysis-based approach for scalable identification of optimal network scales, outperforming greedy and gradient methods in preferential attachment networks.

Findings

Spectral clustering outperforms other methods in scale detection.

Macro-scale networks improve information transmission properties.

Structural properties vary significantly across different network scales.

Abstract

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 preferential attachment networks, 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 these networks vary across scales. We describe how meso- and macroscale representations of networks can have significant benefits over their underlying microscale in terms of information transmission, 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.