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Shortest paths and load scaling in scale-free trees.
Béla Bollobás1, Oliver Riordan
1Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152, USA.
Summary
This study rigorously analyzes scale-free random trees, providing precise answers for node-to-node distance and node load distributions. Findings confirm previous heuristic predictions for these network properties.
Area of Science:
- Network science
- Statistical physics
- Graph theory
Background:
- The Barabási-Albert (BA) model describes scale-free networks.
- Random trees are a fundamental network structure.
- Previous heuristic analyses provided approximate answers for tree properties.
Purpose of the Study:
- To rigorously analyze node-to-node distances in scale-free random trees.
- To precisely determine the distribution of node loads in these trees.
- To validate and refine previous heuristic findings.
Main Methods:
- Rigorous mathematical analysis of scale-free random trees.
- Leveraging prior research on scale-free random graphs.
- Asymptotic analysis and convergence proofs.
Main Results:
- The node load distribution converges to a specific integer distribution, confirming a power law with exponent -2.
- The distribution of node-to-node distances exhibits asymptotic normality.
- A precise large deviation law for distances was derived.
Conclusions:
- The study provides mathematically proven, precise results for scale-free random tree properties.
- Heuristic mean-field approximations offer valuable insights for this model.
- Findings enhance understanding of network structure and dynamics.