🤖 AI Summary
Traditional concentration measures, such as the Herfindahl–Hirschman Index, neglect network topology and thus fail to capture the joint effect of weight distribution and interaction structure in weighted systems. This work proposes the Network Concentration Index (NCI)—a family of topology-aware concentration metrics based on normalized quadratic forms—that explicitly incorporates network structure into concentration measurement. The framework ensures normalization, invariance, and interpretability. Through graph-theoretic modeling, null-model comparisons, and multilayer network extensions, both theoretical analysis and simulations demonstrate that even with identical weight distributions, differing topologies can yield markedly distinct concentration levels. These findings confirm that the proposed framework effectively captures structural information overlooked by conventional indices.
📝 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.