Unified scaling for the optimal path length in disordered lattices

Area of Science:

• Complex Networks
• Statistical Physics
• Materials Science

Background:

• Optimal path calculations are crucial in networks with scientific and technological applications.
• The influence of lattice geometry on optimal paths, especially in disordered systems, remains under-explored.

Purpose of the Study:

• To propose a unified scaling ansatz for the mean optimal path length in D-dimensional disordered lattices.
• To investigate the role of lattice geometry in determining optimal path scaling, particularly in the strong disorder limit.

Main Methods:

• Development of a unified scaling ansatz incorporating two new exponents, φ and χ.
• Comprehensive numerical simulations on 2D lattices and supplementary results in 3D.

Main Results:

• The proposed ansatz successfully unifies known results across strong and weak disorder regimes, including crossover behaviors.
• Novel scaling scenarios in disordered lattices are identified.
• The ansatz provides insights into universal exponents, such as the fractal dimension of the optimal path (d_opt = φ + χ) in the strong disorder limit.

Conclusions:

• The geometry of the embedding lattice significantly impacts optimal path characteristics in disordered systems.
• The unified scaling ansatz offers a robust framework for understanding optimal path behavior across different disorder strengths and dimensions.