Relation between the degree and betweenness centrality distribution in complex networks.

Understanding the relationship between degree and betweenness centrality in complex networks remains an open challenge. Prior research has shown that for scale-free networks, the maximal exponent relating these measures is η_{max}=2, as proposed by Barthelemy. This led to a conjecture that δ ≥ γ + 1/2, where γ and δ describe the scaling of degree and betweenness distributions, respectively. However, some models and systems have shown deviations from this rule. This gap motivated a closer examination of how network structure influences these scaling relations. Visibility graphs of time series offer a unique framework to test these relationships. The BTW sandpile model, FBM, and Lévy walks are examples of systems where correlation strength and scaling exponents can be controlled. These models allow researchers to explore how structural properties affect centrality measures. No prior work had resolved how anomalous scaling behaviors might emerge in these contexts. This paper aims to clarify the conditions under which the Barthelemy conjecture holds or fails. The study focuses on visibility graphs derived from correlated time series to test the validity of the conjecture across different systems.

The study aims to evaluate the validity of the Barthelemy conjecture in visibility graphs of correlated time series. The conjecture links the scaling exponents of degree and betweenness centralities in scale-free networks. The researchers investigate whether this relationship holds universally or breaks down under specific conditions. They use visibility graphs from three models: the BTW sandpile model, FBM, and Lévy walks. These models allow precise control over correlation strength and scaling parameters. The goal is to determine if the conjecture fails in certain correlation regimes. The study also explores the implications of deviations from the conjecture. By analyzing the scaling exponents η, γ, and δ, the researchers assess the structural properties of these networks. The motivation stems from the need to understand how network topology influences centrality measures in complex systems.

The researchers employ visibility graphs to transform time series data into network structures. These graphs map time series into nodes connected by edges based on visibility rules. The study includes three models: the BTW sandpile model, FBM, and Lévy walks. Each model is parameterized by a scaling exponent—H for FBM and α for Lévy walks. The visibility graphs are analyzed for degree and betweenness centralities. Scaling exponents γ and δ are computed from the distributions of these measures. The maximal exponent η is calculated to test the Barthelemy conjecture. The researchers compare the observed values of η and δ to the predicted δ ≥ γ + 1/2. Deviations from this relationship indicate violations of the conjecture. The study also examines the universality of the generalized degree distribution across models. This approach allows the researchers to assess how structural properties influence centrality scaling.

The study finds that the Barthelemy conjecture fails in certain correlation regimes. For the BTW model and FBM with H ≲ 0.5, the exponent η exceeds 2, indicating a breakdown of the conjecture. In these cases, δ is less than γ + 1/2, contradicting the expected relationship. The BTW model shows η > 2 and δ < γ + 1/2, while the Lévy process adheres to the conjecture. The failure is attributed to large fluctuations in the scaling b-k relation. These fluctuations violate the hyperscaling relation η = γ - 1/δ - 1. The study identifies anomalous behavior in the BTW and FBM models. Despite these deviations, the generalized degree distribution remains universal across models. The scaling behavior of the generalized degree matches that of the Barabási-Albert network. These findings suggest that the structural properties of the time series models influence centrality scaling.

The researchers conclude that the Barthelemy conjecture does not hold universally across all network types. The conjecture fails in specific correlation regimes of the BTW model and FBM when H ≲ 0.5. In these cases, the maximal exponent η exceeds 2, and δ is less than γ + 1/2. The failure is linked to large fluctuations in the scaling b-k relation. These fluctuations violate the hyperscaling relation η = γ - 1/δ - 1. The study attributes this to anomalous behavior in the BTW and FBM models. Despite these deviations, the generalized degree distribution remains universal. This universality aligns with the scaling behavior of the Barabási-Albert network. The findings highlight the importance of structural properties in determining centrality scaling. The researchers propose that network topology and correlation strength are key factors in determining the validity of the Barthelemy conjecture.