Structural Concentration in Weighted Networks: A Class of Topology-Aware Indices

TL;DR

This paper introduces a new class of topology-aware concentration indices for weighted networks that integrate weight distribution and network structure, providing a more comprehensive measure of concentration in complex systems.

Contribution

It develops a unified framework and a baseline index for measuring structural concentration, extending classical indices to account for network topology and interactions.

Findings

Network topology significantly affects concentration levels.

The proposed indices capture dependence on network structure and weight distribution.

Empirical evidence shows different systems with similar weights can have different concentrations.

Abstract

This paper develops a unified framework for measuring concentration in weighted systems embedded in networks of interactions. While traditional indices such as the Herfindahl-Hirschman Index capture dispersion in weights, they neglect the topology of relationships among the elements receiving those weights. To address this limitation, we introduce a family of topology-aware concentration indices that jointly account for weight distributions and network structure. At the core of the framework lies a baseline Network Concentration Index (NCI), defined as a normalized quadratic form that measures the fraction of potential weighted interconnection realized along observed network links. Building on this foundation, we construct a flexible class of extensions that modify either the interaction structure or the normalization benchmark, including weighted, density-adjusted, null-model, degree-constrained, transformed-data, and multi-layer variants. This family of indices preserves key properties such as normalization, invariance, and interpretability, while allowing concentration to be evaluated across different dimensions of dependence, including intensity, higher-order interactions, and extreme events. Theoretical results characterize the indices and establish their relationship with classical concentration and network measures. Empirical and simulation evidence demonstrate that systems with identical weight distributions may exhibit markedly different levels of structural concentration depending on network topology, highlighting the additional information captured by the proposed framework. The approach is broadly applicable to economic, financial, and complex systems in which weighted elements interact through networks.