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The Optimal Pathin an Erdős-Rényi Random Graph

  • Lidia A. Braunstein
  • Sergey V. Buldyrev
  • Sameet Sreenivasan
  • Reuven Cohen
  • Shlomo Havlin
  • H. Eugene Stanley
Part I Network Structure
Part of the Lecture Notes in Physics book series (LNP, volume 650)

Abstract

We study the optimal distance \(\ell_{\mbox{\scriptsize opt}}\) in random networks in the presence of disorder implemented by assigning random weights to the links. The optimal distance between two nodes is the length of the path for which the sum of weights along the path (“cost”) is a minimum. We study the case of strong disorder for which the distribution of weights is so broad that its sum along any path is dominated by the largest link weight in the path. We find that in Erdős-Rényi (ER) random graphs, \(\ell_{\mbox{\scriptsize opt}}\) scales as N1/3, where N is the number of nodes in the graph. Thus, \(\ell_{\mbox{\scriptsize opt}}\) increases dramatically compared to the known small world result for the minimum distance \(\ell_{\mbox{\scriptsize min}}\), which scales as logN. We also find the functional form for the probability distribution \(P(\ell_{\mbox{\scriptsize opt}})\) of optimal paths. In addition we show how the problem of strong disorder on a random graph can be mapped onto a percolation problem on a Cayley tree and using this mapping, obtain the probability distribution of the maximal weight on the optimal path.

Keywords

Random Graph Optimal Path Minimal Path Optimal Distance Giant Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Authors and Affiliations

  • Lidia A. Braunstein
    • 1
    • 2
  • Sergey V. Buldyrev
    • 1
  • Sameet Sreenivasan
    • 1
  • Reuven Cohen
    • 3
  • Shlomo Havlin
    • 1
    • 3
  • H. Eugene Stanley
    • 1
  1. 1.Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215USA
  2. 2.Departamento de Física, Facultad de Ciencias Exactas y Naturales, Univ. Nacional de Mar del Plata, Funes 3350, 7600 Mar del Plata, Argentina 
  3. 3.Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-GanIsrael

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