Localization in random bipartite graphs: numerical and empirical study

TL;DR

This paper studies localization phenomena in bipartite graphs with power-law degree distributions, analyzing theoretical properties and empirical data, revealing how the mobility edge depends on graph parameters and differs in real-world networks.

Contribution

It establishes the dependence of the mobility edge on degree distribution and graph size ratio, and compares theoretical predictions with empirical Amazon reviewer-item network data.

Findings

Mobility edge position depends on degree distribution power and size ratio.

Localization vanishes at the jamming threshold where parts are equal.

Empirical Amazon network shows disappearance of the mobility edge.

Abstract

We investigate adjacency matrices of bipartite graphs with power-law degree distribution. Motivation for this study is twofold. First, vibrational states in granular matter and jammed sphere packings; second, graphs encoding social interaction, especially electronic commerce. We establish the position of the mobility edge, and show that it strongly depends on the power in the degree distribution and on the ratio of the sizes of the two parts of the bipartite graph. At the jamming threshold, where the two parts have the same size, localization vanishes. We found that the multifractal spectrum is non-trivial in the delocalized phase, but still near the mobility edge. We also study an empirical bipartite graph, namely the Amazon reviewer-item network. We found that in this specific graph mobility edge disappears and we draw a conclusion from this fact regarding earlier empirical studies of the Amazon network.