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.

翻译：本文提出了一个统一的框架，用于测量嵌入在网络交互中的加权系统的集中度。传统指标（如赫芬达尔-赫希曼指数）虽能捕捉权重分布中的离散程度，但忽略了承载权重的元素之间的关系拓扑。为解决这一局限，我们引入了一类拓扑感知的集中度指标，该指标同时考虑权重分布与网络结构。该框架的核心是一个基准网络集中度指数（NCI），定义为标准化二次型，用于测量沿观测网络链路实现的潜在加权互连比例。在此基础之上，我们构建了一个灵活的扩展指标类别，通过修改交互结构或归一化基准，衍生出加权、密度调整、零模型、度约束、数据变换及多层网络等变体。该指标家族保留了归一化、不变性和可解释性等关键性质，同时允许从不同维度（包括强度、高阶交互和极端事件）评估依赖关系中的集中度。理论结果刻画了这些指标的特征，并建立了其与经典集中度和网络测度之间的联系。实证与模拟证据表明，具有相同权重分布的系统可能因网络拓扑的不同而呈现出显著不同的结构性集中度水平，这凸显了所提框架捕获的额外信息。该方法广泛适用于加权元素通过网络进行交互的经济、金融及复杂系统。