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Flows on graphs with random capacities.

T Antal1, P L Krapivsky

  • 1Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts 02138, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 7, 2007
PubMed
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We studied random capacities on graph links to understand maximal flow. For infinite binary trees, maximal flow vanishes beyond a capacity threshold, a finding applicable to other graph types.

Area of Science:

  • Graph theory
  • Probability theory
  • Network flow analysis

Background:

  • Understanding network flow is crucial in various scientific disciplines.
  • Randomness in link capacities introduces complexity in flow analysis.

Purpose of the Study:

  • To derive the probability distribution for maximal flow in binary trees with random link capacities.
  • To investigate the behavior of maximal total flux in such networks.
  • To generalize findings to other graph structures.

Main Methods:

  • Derivation of probability distributions for maximal flow.
  • Analysis of flow behavior in infinite binary trees.
  • Generalization of methods to simple graphs, hierarchical lattices, and complete graphs.

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Main Results:

  • The probability distribution for maximal flow from root to leaf in binary trees was derived.
  • For infinite trees, maximal flow vanishes beyond a specific capacity threshold.
  • Maximal total flux from root to leaves was examined.

Conclusions:

  • The study provides insights into the probabilistic behavior of network flows with random capacities.
  • The derived threshold offers a critical parameter for understanding flow limitations in infinite binary trees.
  • The methodology is adaptable to a broader range of graph structures, including lattices and complete graphs.