2004/05/10 by Sameet Sreenivasan, Tomer Kalisky, Lidia A. Braunstein +3 · 1 citation
Mathematics · Physics and Astronomy · #Ansatz #Artificial intelligence #Combinatorics #Complex Network Analysis Techniques #Computer science #Crossover #Geometry #Lambda #Mathematics #Measure (data warehouse) #Opinion Dynamics and Social Influence #Physics #Quantum mechanics #Scaling #Statistical physics #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.70.046133
6 pages, 6 figures. submitted to Phys. Rev. E
arxiv created 2004/05/10 · openalex publication_date 2004/10/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path lopt in a disordered Erd\ifmmode \mbox\Ho\else \Ho\fis-R'enyi (ER) random network and scale-free (SF) network. Each link i is associated with a weight \ensuremathτi\ensuremath≡exp(ari), where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder. We find that for any finite a, there is a crossover network size N*(a) at which the transition occurs. For N⪡N*(a) the scaling behavior of lopt is in the strong disorder regime, with lopt\ensuremath∼N1∕3 for ER networks and for SF networks with \ensuremathλ\ensuremath\geqslant4, and lopt\ensuremath∼N^(\ensuremathλ\ensuremath-3)∕(\ensuremathλ\ensuremath-1) for SF networks with 3<\ensuremathλ<4. For N⪢N*(a) the scaling behavior is in the weak disorder regime, with lopt\ensuremath∼ln\phantom\rule0.2em0exN for ER networks and SF networks with \ensuremathλ>3. In order to study the transition we propose a measure which indicates how close or far the disordered network is from the limit of strong disorder. We propose a scaling ansatz for this measure and demonstrate its validity. We proceed to derive the scaling relation between N*(a) and a. We find that N*(a)\ensuremath∼a3 for ER networks and for SF networks with \ensuremathλ\ensuremath\geqslant4, and N*(a)\ensuremath∼a^(\ensuremathλ\ensuremath-1)∕(\ensuremathλ\ensuremath-3) for SF networks with 3<\ensuremathλ<4.