Rich-club vs rich-multipolarization phenomena in weighted networks

TL;DR

This paper extends the rich-club phenomenon to weighted networks, revealing that weighted analyses can contrast with unweighted results and uncover local alliances or lack of cohesion among high-strength hubs.

Contribution

It introduces a weighted rich-club coefficient, discusses normalization methods, and demonstrates contrasting behaviors in real networks compared to unweighted approaches.

Findings

Weighted rich-club coefficient can differ sharply from unweighted results.

High-strength hubs may form local alliances or lack cohesion.

Weighted analysis enhances understanding of network functionalities.

Abstract

Large scale hierarchies characterize complex networks in different domains. Elements at their top, usually the most central or influential, may show multipolarization or tend to club forming tightly interconnected communities. The rich-club phenomenon quantified this tendency based on unweighted network representations. Here, we define this metric for weighted networks and discuss the appropriate normalization which preserves nodes' strengths and discounts structural strength-strength correlations if present. We find that in some real networks the results given by the weighted rich-club coefficient can be in sharp contrast to the ones in the unweighted approach. We also discuss that the scanning of the weighted subgraphs formed by the high-strength hubs is able to unveil features contrary to the average: the formation of local alliances in rich-multipolarized environments, or a lack of cohesion even in the presence of rich-club ordering. Beyond structure, this analysis matters for understanding correctly functionalities and dynamical processes relying on hub interconnectedness.