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Canonical fitness model for simple scale-free graphs
1Institut für Physik, Humboldt-Universität zu Berlin, D-12489 Berlin, Germany.
Summary
This study examines a fitness model for generating power-law graphs. Merging multiple edges in multigraphs results in a simple graph with a degree distribution P(k) proportional to k(-2) for large k.
Area of Science:
- Network science
- Statistical physics
- Graph theory
Background:
- The study considers a fitness model for generating random graphs with power-law degree distributions.
- These distributions, P(k) proportional to k(-1-α) with 0<α<1, lack a defined mean, posing challenges in analysis.
- The model is extended to scenarios involving multigraphs, where multiple edges can exist between nodes.
Purpose of the Study:
- To analyze the behavior of a fitness model generating simple graphs with power-law degree sequences.
- To investigate the impact of merging multiple edges in multigraphs into single edges within a simple graph context.
- To derive the relationship between the normalized fitness parameter (r) and the expected node degree (ν), and to determine the asymptotic behavior of the degree distribution.
Main Methods:
- Analytical derivation of the relationship between the fitness parameter and expected node degree.
- Mathematical analysis of the degree distribution resulting from merging multiple edges in a multigraph.
- Numerical simulations to validate the analytical findings regarding the asymptotic behavior of the degree distribution.
Main Results:
- The study establishes a relationship between the normalized fitness parameter (r) and the expected node degree (ν).
- Analytical results demonstrate nontrivial intermediate and final asymptotic behaviors for the degree distribution.
- It is shown that the model produces a degree distribution P(k) proportional to k(-2) for large k, irrespective of the parameter α.
Conclusions:
- The fitness model, when adapted for simple graphs by merging edges, exhibits a universal power-law tail P(k) proportional to k(-2) for large degrees.
- The analytical findings on the asymptotic behavior of the degree distribution are robust and confirmed by simulations.
- This research provides insights into the structural properties of complex networks generated by fitness models, particularly concerning degree distributions without a mean.
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