Convergence criteria for self-consistent measures in bipartite networks

TL;DR

This paper derives explicit convergence criteria for self-consistent measures in bipartite networks, providing methods to improve convergence through node management and regularization schemes.

Contribution

It introduces a novel explicit convergence criterion for iterative measures in bipartite networks and proposes two approaches to enhance convergence stability.

Findings

Derived an explicit convergence criterion for self-consistent measures.

Identified problematic nodes affecting convergence.

Proposed regularization scheme to improve convergence.

Abstract

Many quantities that characterize network elements are defined in an explicit form and calculated directly from the network structure; examples of include several centrality measures like degree, closeness, or betweenness. However, there are also implicitly defined quantitative measures, which are usually calculated iteratively, in a self-consistent manner, like PageRank or countries' fitness / products' complexity relations. The iteration algorithms involve calculations over the entire network; therefore, their convergence properties depend on the structure of the network. Here, we focus on investigating self-consistently defined quantities in bipartite networks of two sets of nodes where the quantities in one set are determined by the quantities in the other set and vice versa. We derive an explicit convergence criterion for iterations of these quantities and describe two different approaches to improve the convergence properties. In the first one, we identify "problematic nodes" that can be removed or merged while in the second one, we introduce a regularization scheme and show how to estimate the regularization parameter.