Insights on structure and influence from the adjacency and Laplacian eigenspectra of intersecting ring networks

TL;DR

This paper explores how adjacency and Laplacian eigenspectra reveal hierarchical influence and community structures in intersecting ring networks, enhancing understanding of network topology and node influence profiles.

Contribution

It demonstrates how adjacency and Laplacian eigenspectra provide complementary insights into network structure and influence, especially in complex intersecting ring networks.

Findings

Adjacency reflects network structure and community sizes.

Laplacian detects nodes' influence profiles.

Different spectral methods reveal distinct network features.

Abstract

A network's community structure commonly impacts its functions. For instance, networks seeking synchronisation will see this process follow the topology's hierarchical and community structuring. Herein, the interplay of network adjacency and Laplacian eigenspectra is shown to uncover hierarchical influence and community structure. Ring networks embedded with hubs of high connectivity are first analysed to characterise the differing insights of the adjacency and Laplacian eigenspectra. This understanding is then transferred to the case study of satellite networks composed of intersecting rings. From these scenarios, it is found that where the adjacency reflects a network's structure, the Laplacian detects nodes' influence profiles. The adjacency identifies the number and relative sizes of salient network structures, for example hubs of high connectivity. In contrast, the Laplacian decomposes these structures in sets of nodes with equivalent influence profiles, where influences emerges due to variations in nodes' connectivity.