Testing Central Structure for Complex Networks

The gist: This work introduces a network-level hypothesis test using the adjacency spectral radius to determine whether a network contains a statistically identifiable central structure before ranking individual nodes.

Problem and Motivation

Complex networks often exhibit pronounced central structures, prompting questions about whether small numbers of nodes occupy an exceptional structural position Many centrality measures presuppose the existence of meaningful central nodes, which can lead to reporting random artifacts when a network is nearly structurally homogeneous. The investigation should precede the selection of a particular centrality measure.

Methodology for Testing Central Structure

The core of the framework involves formulating the centrality existence issue as a hypothesis-testing problem.

(H0: the central structure is not pronounced relative to the specified null model, H1: the central structure is pronounced relative to the specified null model)

The test statistic used for quantifying concentration is the spectral radius of the adjacency matrix, defined as T(G) = ρ(A) = λmax(A). This statistic characterizes connectivity strength and spreading potential, as a larger spectral radius corresponds to a lower epidemic threshold in SIS models.

Null Models for Comparison

The null model specifies which network properties are treated as ordinary background structure. Three families of null models are considered:

1. E-R model: This samples uniformly from all simple graphs with N nodes and exactly M edges, preserving the average degree but not the observed degree sequence.

2. Configuration model: This preserves the observed degree sequence in generated random networks, offering a more realistic null model than G(N, M).

3. Chung–Lu model: This preserves a specified degree sequence in expectation by connecting nodes independently with probability p ij = min (didj / PN k=1 dk, 1).

p-Value Calculation and Evaluation

The empirical p-value is calculated by comparing the observed statistic Tobs = T(G) with an empirical null distribution generated from B randomized networks under a selected null model. The p-value is the proportion of randomized networks whose spectral radius is at least as large as that of the observed network. Rejection of H0 occurs when the p-value is less than a significance level α, such as 0.05. The results demonstrate a clear negative association between central-structure strength and the empirical p-value.

Experimental Findings

The study evaluates the framework through two complementary experiments.

Experiment 1:

The test was applied to synthetic and real networks across different random-graph null models and randomized network numbers. Identification performance on the simulated benchmark showed an F1 score of approximately 0.98 under the G(N, M) null model.

Experiment 2:

Controlled synthetic networks were used to assess systematic response to structure strength. The results show that increasing the number of central nodes generally lowers the p-value. Furthermore, increasing the connection weight w also generally lowers the mean p-value. The detection depends substantially on which structural properties are treated as ordinary background variation.

Conclusion

This work introduces a network-level hypothesis test for determining whether a network contains a statistically identifiable central structure. The test is intended as a diagnostic before applying node-level centrality rankings. Future work could extend the framework to weighted, directed, and temporal networks. The current study is limited to unweighted, undirected networks and the real-network collection does not provide independent labels for the presence of central structure. The framework to weighted, directed, and temporal networks, investigate alternative network-level statistics and more tightly constrained null models, and evaluate the test on real networks with external structural labels.

Author Contributions

Qilong Wu: Writing–review & editing, Writing–original draft, Visualization, Validation, Resources, Methodology, Formal analysis. Yuxiao Li: Writing–review & editing, Formal analysis. Simeng Zhang: Writing–review & editing, Validation. Xinying Liu: Writing–review & editing, Methodology. Yan Liu: Writing–review & editing, Visualization, Formal analysis. Zengyou He: Writing–review & editing, Writing–original draft, Supervision, Conceptualization.

Acknowledgments

This work has been supported by the Natural Science Foundation of China under Grant No. 62472064, and Chronic Disease Management Research Project of National Health Commission Capacity Building and Continuing Education Center (No. GWJJMB202510022150).

Conflicts of Interest

The authors declare no conflicts of interest.

Data Availability Statement

The data that support the findings of this study are available at Pajek Datasets and Network Data Repository. The original Pajek Datasets website (http://vlado.fmf.uni-lj.si/pub/networks/data/) is unavailable; these networks were obtained from its GitHub mirror (https://github.com/bavla/Nets/tree/master/data/Pajek).

References

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1. Better node selection for intervention by using a hypothesis test instead of relying solely on centrality measures that generally presuppose the existence of meaningful central nodes. This allows AI systems to distinguish between statistically identifiable central structures and those arising from ordinary structural variation.

2. Improved network structure characterization by applying the spectral radius as a statistic, which characterizes the network’s overall connectivity strength and is closely related to spreading potential. This enables AI to prioritize nodes based on their potential for large-scale spreading or lower epidemic thresholds.

3. More robust identification of influential spreaders by utilizing a hypothesis testing framework where a small p-value indicates that the observed network has a more pronounced central structure than expected under that null model. This ensures that reported key nodes are not just high-degree nodes but represent a statistically significant structural role relative to random graphs.

4. More nuanced understanding of network organization by incorporating different null models, such as the configuration model and Chung–Lu model, which preserve a specified degree sequence in expectation or control for degree sequence, allowing AI to assess centrality under varying constraints on background structure.

5. Systematic assessment of central structure strength by tracking how the test responds to changes in planted structure: increasing either the number of central nodes or their connection weight generally lowers the p-value. This allows AI to quantify precisely how much stronger a suspected central hub must be to be considered statistically significant.