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

L. Riso; M. G. Zoia

stat.MLMar 23, 2026

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

Authors: L. Riso, M. G. Zoia

Organizations: Department of Economic Policy, Universit`a Cattolica del Sacro Cuore, Milan, Italy

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.

Figures & tables

Variant	Eq.	$M$	$B$	P 3.1	P 3.2	P 3.3	P 3.4
Baseline NCI	( 9 )	$A$	$\mathbf{1}\mathbf{1}^{\top}-I$	✓	✓	✓	✓
Density-adjusted	( 28 )	$A$	$\delta(A)(\mathbf{1}\mathbf{1}^{\top}-I)$	✓	$\circ$	✓	$\circ$
Null-model	( 31 )	$A$	$\mathbb{E}[A\mid\mathcal{N}]$	✓	$\circ$	$\circ$	$\circ$
Degree-constrained	( 34 )	$A$	$\Lambda(w;\mathcal{G}(d))$	✓	$\circ$	$\circ$	$\circ$
Weighted NCI	( 36 )	$W=[\gamma_{ij}]$	$\mathbf{1}\mathbf{1}^{\top}-I$	✓	✓	$\circ$	$\circ$
Transformed-data NCI	( 38 )	$A^{(\mathcal{T})}$	$\mathbf{1}\mathbf{1}^{\top}-I$	✓	✓	$\circ$	$\circ$

Table 1: Variants of the Network Concentration Index within the broader network concentration framework. Columns P 3.1–P 3.4 indicate whether the variant satisfies the weighted-average representation (Proposition 3.1 ), normalization characterization (Proposition 3.2 ), assortative-connectivity representation (Proposition 3.3 ), and random-network benchmark (Proposition 3.4 ).

Scenario	HHI	$\delta$	$\psi$	$\psi^{(\mathrm{dens})}$	$\psi^{(\mathrm{null})}$	$\psi^{(\mathrm{deg})}$	$\psi^{(W)}$	$\psi^{(\mathcal{T})}$	$\psi^{(\alpha)}$
Core-periphery	0.176	0.267	0.514	1.929	0.366	0.433	0.357	1.929	1.683
Peripheral	0.176	0.267	0.095	0.355	0.049	0.068	0.066	0.355	0.877
Random	0.176	0.267	0.287	1.076	0.157	0.211	0.188	1.076	1.460

Table 2: NCI variants across three deterministic scenarios (

N=10

, fixed

\bm{\omega}

\mathrm{HHI}=0.176

\delta=0.267

Panel of Figure 1 — Figure 1: Three deterministic network structures ( $N=10$ , $\mathrm{HHI}=0.1758$ , $\delta\approx 0.267$ , fixed $\bm{\omega}_{\mathrm{ref}}$ ). Node size is proportional to $w_{i}$ ; colours are consistent across all figures.

Figure 2: All seven NCI variants evaluated across the three deterministic scenarios ( $N=10$ , fixed $\bm{\omega}_{\mathrm{ref}}$ , $\mathrm{HHI}=0.176$ ). Bars are grouped by index, ordered as in Table 1 ; colours identify the scenario.

Panel of Figure 3 — Figure 3: Monte Carlo analysis of network concentration measures. Panel (a) reports the kernel density estimates of the seven NCI variants under the first experiment ( $N=10$ , fixed $\bm{\omega}_{\mathrm{ref}}$ , $R=5{,}000$ replications per network type). Panel (b) shows the simulated mean $\hat{\mathbb{E}}[\psi(\bm{\omega},A)]$ as a function of the link probability $p$ under the ( Erdős et al., 1960 ) model, compared with the theoretical benchmark $\mathbb{E}[\psi]=p$ .

Panel of Figure 4 — Figure 4: Dependence structure of network concentration measures. Panel (a) shows the relationship between the seven NCI variants and the Herfindahl–Hirschman Index (HHI) based on the second Monte Carlo experiment. Panel (b) reports the Pearson correlation matrix of the NCI variants, pooled across the three network-generating mechanisms.

Measure	Value
HHI	0.0321
Gini	0.4710
NCI	0.0777

Table 3: Concentration measures, production network (WIOD 2014,

\theta=0.01

Figure 5: Sector weights $\omega_{j}$ , ranked by decreasing value. Colours denote macro-sector.

Figure 6: Symmetric production network, WIOD 2014 ( $\theta=0.01$ ).

Figure 7: NCI, HHI, and Gini as functions of $\theta$ (log scale). Production network, WIOD 2014.

Measure	Value
HHI	0.1126
Gini	0.7256
NCI	0.2647

Table 4: Concentration measures, international trade network (WIOD 2014,

\theta=0.005

Figure 8: Country weights $\omega_{c}$ , top-20 economies plus aggregate of remaining countries (Other), ranked by decreasing value.

Figure 9: Symmetric international trade network, WIOD 2014 ( $\theta=0.005$ ).

Figure 10: NCI, HHI, and Gini as functions of $\theta$ (log scale). International trade network, WIOD 2014.

Measure	Value
HHI	0.0879
Gini	0.4320
NCI	0.1939

Table 5: Concentration measures, equity dependence network (S&P 500 top-20 by market cap, 2015–2024).

Figure 11: Market capitalisation weights $\omega_{i}$ , S&P 500 top-20 stocks ranked by decreasing value. Colours denote GICS sector. End-2024 market caps (approximate, billions USD).

Figure 12: MST-based equity dependence network, S&P 500 top-20, 2015–2024.

Figure 13: NCI, HHI, and Gini on rolling 252-day windows (step: 63 days), S&P 500 top-20, 2015–2025. The shaded band reports the pointwise 95% bootstrap confidence interval for the NCI ( $B=500$ resamples per window).

Mar 12, 2026

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Didier Sornette, Yishan Luo, Sandro Claudio Lera

Institute of Risk Analysis, Prediction and Management, Southern University of Science and Technology, Shenzhen, China · Warwick Business School, University of Warwick, Coventry, UK

Hawkes Processes Social Network Analysis Computational Social Science

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Oct 11, 2026

Testing Central Structure for Complex Networks

Centrality measures rank nodes even when the observed differences can be explained by ordinary structural variation. We therefore study a preceding question: whether a network contains a statistically identifiable central structure at all. We formulate this question as a network-level hypothesis testing issue using the adjacency spectral radius as the test statistic and estimate an empirical p-value from randomized networks generated under a specified null model. The framework is evaluated using G(N,M), Chung-Lu, and two implementations of the configuration model on simulated benchmarks and 55 real networks. Controlled experiments further vary the number and connection weights of designated central nodes while holding network size and density fixed. The results show a clear negative association between central-structure strength and the empirical p-value. They also demonstrate that rejection decisions depend substantially on the structural properties preserved by the null model: the detected network sets are not simply nested across models.

Qilong Wu, Yuxiao Li, Simeng Zhang +3

School of Software, Dalian University of Technology, China · College of Electrical Engineering, Sichuan University, China · School of Software Engineering, Dalian University, China

Statistical Hypothesis Testing

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Aug 21, 2026

Omega-N: Interpretable Structural Node Descriptors and Their Applicability Domain

A composite structural index summarises a network in one number, and for a triangle-based index it is spectrally redundant: Tr(A^3) is the third moment of the adjacency spectrum. The non-redundant content sits one level down, in diag(A^3), which depends on eigenvectors and is not spectrally determined. A corollary in the theory paper predicted that the global scalar should tie sharpened spectral baselines rather than beat them, while the node-wise attribution should do better where the number of structural epicentres is unknown. We construct Omega-N by localizing each of the four factors. The direct localization is badly conditioned; two corrections from published practice fix it, a configuration-null excess per factor and a personalized-PageRank neighbourhood at several scales, giving ten interpretable features per node, with no attributes, training or embeddings. Against a recursive feature engine at five levels of recursion, Omega-N wins on three and ties on two of the six in-domain evaluations, the sixth a declared null where every arm returns chance, with ten features against its 28 to 252 before pruning. Two statistics from the graph and labels, not from performance, partition the eight benchmarks without error, and the two they exclude are the two on which it loses. The strongest application is drug-target prioritisation on protein interaction networks: +0.032 to +0.103 AUPRC over a six-feature centrality battery and +0.084 to +0.208 over the four-feature one, across three constructions, replicated on an independent AP-MS network and label source (degree-matched: +0.0723 on STRING, +0.0560 on BioPlex, p=0.00195). Adding Omega-N to centralities plus Node2Vec changes nothing. The claim is narrow and it is the point: ten named features, computed without training, match or beat hand-crafted centralities and a recursive engine, and do not touch learned representations.

Alberto Acedo

Biome Makers Inc

Topic pending

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Abstract

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