Characterizing general scale-free networks by vertex-degree sequences

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems Analysis

Background:

• Many complex networks exhibit scale-free vertex-degree distributions following a power-law form: ck(-γ).
• Understanding the vertex-degree sequence is crucial for analyzing the power-law formation mechanism in scale-free networks.
• Previous work established that for scale-free networks with γ>1, the number of unique vertex degrees (l) is proportional to log(N), where N is the network size.

Purpose of the Study:

• To generalize the understanding of vertex-degree sequence length in complex networks beyond strict power-law distributions.
• To investigate if the logarithmic relationship between network size and the number of unique vertex degrees holds for more general degree distributions.
• To validate the theoretical findings with empirical data from diverse real-world networks.

Main Methods:

• Theoretical analysis of complex network vertex-degree distributions.
• Mathematical proof demonstrating the relationship between network size and the length of the vertex-degree sequence.
• Empirical validation using datasets from numerous real-world complex networks.

Main Results:

• The study proves that for complex networks with general vertex-degree distributions, the length of the vertex-degree sequence (l) is of the order of log(N).
• This finding extends the previously established logarithmic relationship beyond networks strictly adhering to power-law degree distributions.
• The theoretical results are corroborated by analyses of real-world network data, confirming the robustness of the logarithmic scaling.

Conclusions:

• The number of unique vertex degrees in complex networks scales logarithmically with the network size, irrespective of whether the degree distribution is strictly power-law.
• This generalized finding provides a fundamental insight into the structural properties of complex systems.
• The results have potential applications in diverse scientific, technological, and societal domains, aiding in the analysis and modeling of large-scale networks.