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Controllability, Matching Ratio and Graph Convergence.

Dorottya Beringer1, Ádám Timár1

  • 1Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053 Hungary.

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Summary
This summary is machine-generated.

This study proves key statements on the directed matching ratio in control theory, demonstrating its concentration around the mean for random networks. The findings confirm convergence for scale-free networks, advancing network analysis.

Keywords:
Benjamini–Schramm convergenceControllabilityMatching ratioRandom configuration modelRandom graphsScale-free networks

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Area of Science:

  • Control Theory
  • Network Science
  • Statistical Physics

Background:

  • The directed matching ratio is a critical parameter in control theory, linked to network properties.
  • Previous work by Liu et al. (2011) relied on numerical and heuristic evidence for statements regarding this ratio.

Purpose of the Study:

  • To provide rigorous mathematical proofs for two main statements by Liu et al. (2011) concerning the directed matching ratio.
  • To extend existing results on the convergence of matching ratios in random graph sequences.
  • To analyze the behavior of matching ratios in scale-free networks.

Main Methods:

  • Mathematical proofs for statements on directed matching ratio concentration.
  • Analysis of convergence for random graph sequences in the local weak sense.
  • Generalization of results from Elek and Lippner (2010).
  • Investigation of almost sure convergence for scale-free networks.

Main Results:

  • The directed matching ratio of directed random networks with fixed degree sequences concentrates around its mean.
  • The mean of the directed matching ratio converges to the matching ratio parameter of the limiting graph for sequences converging in the local weak sense.
  • Almost sure convergence of matching ratios is demonstrated for common scale-free network families.

Conclusions:

  • The study provides rigorous proofs for critical aspects of the directed matching ratio, validating and extending prior work.
  • The convergence properties of the directed matching ratio are established for various network models, including scale-free networks.
  • These findings contribute to a deeper theoretical understanding of network structure and its implications in control theory.