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Inhomogeneous substructures hidden in random networks.

Dong-Hee Kim1, Hawoong Jeong

  • 1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 12, 2006
PubMed
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The load-based spanning tree (LST) of Erdös-Rényi random networks exhibits inhomogeneous structures, unlike the networks themselves. Its topology changes with network density, indicating correlated shortest path structures.

Area of Science:

  • Network Science
  • Graph Theory
  • Complex Systems Analysis

Background:

  • Erdös-Rényi (ER) random networks are typically characterized by homogeneous structures.
  • Understanding the properties of specific subgraphs within these networks is crucial for network analysis.

Purpose of the Study:

  • To investigate the structural properties of the load-based spanning tree (LST) that maximizes edge weight in ER random networks.
  • To analyze how the LST structure evolves with increasing edge density in ER networks.

Main Methods:

  • Defining edge weight using edge-betweenness centrality, representing the count of shortest paths through an edge.
  • Analyzing the topology of the maximum weight LST across varying edge densities in ER random networks.

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Main Results:

  • LSTs display highly inhomogeneous structures, contrasting with the homogeneous nature of the parent ER networks.
  • The LST topology transitions significantly with increasing edge density, exhibiting scale-free with cutoff, scale-free, and starlike configurations.
  • These structural changes imply a correlation in shortest path topology, despite the ER network's overall homogeneity.

Conclusions:

  • The structure of maximum weight LSTs in ER networks is not random but reflects underlying correlations in shortest path topology.
  • Edge-betweenness centrality as a weight metric reveals non-trivial structural organization within random network subgraphs.
  • Network density is a critical parameter influencing the emergent topology of these weighted spanning trees.