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Frequency response analysis in electrical circuits provides vital insights into a circuit's behavior as the frequency of the input signal changes. The transfer function, a mathematical tool, is instrumental in understanding this behavior. It defines the relationship between phasor output and input and comes in four types: voltage gain, current gain, transfer impedance, and transfer admittance. The critical components of the transfer function are the poles and zeros.
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Renormalization of complex networks with partition functions.

Sungwon Jung1, Sang Hoon Lee1,2, Jaeyoon Cho1

  • 1Gyeongsang National University, Department of Physics and Research Institute of Natural Science, Jinju 52828, Korea.

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We introduce a new method to renormalize complex networks by overlaying a physical model. This approach provides a clear physical meaning for network renormalization, revealing scale-invariance in scale-free networks.

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Area of Science:

  • Complex networks analysis
  • Statistical physics
  • Network science

Background:

  • Renormalization groups are crucial in physics but lack clear definition for complex networks.
  • Existing methods for network renormalization are often vague and lack rigorous methodology.

Purpose of the Study:

  • To propose a novel and rigorous strategy for renormalizing complex networks.
  • To establish a transparent, model-dependent physical meaning for network renormalization.
  • To analyze the impact of renormalization on node strength distributions in different network types.

Main Methods:

  • Overlaying a generalizable physical model onto the complex network structure.
  • Extracting a renormalization group transformation from the overlying physical model.
  • Defining node strength based on the physical model and tracking its distribution under renormalization.

Main Results:

  • Developed a rigorous renormalization group transformation applicable to arbitrary networks.
  • Demonstrated that node strength distributions in scale-free networks remain scale-invariant.
  • Showed that node strength distributions in homogeneous random networks do not maintain scale-invariance.

Conclusions:

  • The proposed physical model overlay provides a robust framework for complex network renormalization.
  • Network renormalization reveals distinct behaviors in scale-free versus homogeneous random networks regarding node strength distribution.
  • This method offers a transparent and physically meaningful approach to understanding network evolution and properties.