Edge Augmentation on Disconnected Graphs via Eigenvalue Elevation

TL;DR

This paper introduces an eigenvalue elevation-based algorithm for augmenting disconnected graphs by adding inter-community edges, effectively improving connectivity and community detection accuracy.

Contribution

It presents a novel spectral method for edge augmentation that elevates zero eigenvalues to connect graph components, supported by theoretical bounds and extensive simulations.

Findings

Algorithm consistently improves graph connectivity in synthetic and real networks.

Elevates zero eigenvalues to facilitate inter-community edge addition.

Achieves over 50% success rate in connecting communities across various detection methods.

Abstract

The graph-theoretical task of determining most likely inter-community edges based on disconnected subgraphs' intra-community connectivity is proposed. An algorithm is developed for this edge augmentation task, based on elevating the zero eigenvalues of graph's spectrum. Upper bounds for eigenvalue elevation amplitude and for the corresponding augmented edge density are derived and are authenticated with simulation on random graphs. The algorithm works consistently across synthetic and real networks, yielding desirable performance at connecting graph components. Edge augmentation reverse-engineers graph partition under different community detection methods (Girvan-Newman method, greedy modularity maximization, label propagation, Louvain method, and fluid community), in most cases producing inter-community edges at >50% frequency.