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Area of Science:

  • Complex Systems
  • Network Theory
  • Control Theory

Background:

  • Controllability of linear systems was previously linked to maximum matching on bipartite graphs.
  • Analytical methods like the cavity method were used for size calculations in specific limits.

Purpose of the Study:

  • To develop a simplified and interpretable theory for estimating maximum matching sizes.
  • To explore the connection between system controllability and complex network structures.

Main Methods:

  • Utilized core percolation theory.
  • Introduced the concept of perfect matching of cores.
  • Developed an alternative analytical framework for random graphs.

Main Results:

  • Presented a simplified theory for estimating maximum matching sizes.
  • The theory is applicable to random graphs with varying degree distributions.
  • Provided a new perspective on the controllability problem.

Conclusions:

  • The new theory offers a more accessible approach to understanding maximum matching in complex systems.
  • Highlights the fundamental relationship between controllability and network topology.
  • Facilitates analysis of systems with asymmetric degree distributions.