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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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Detectability thresholds of general modular graphs.

Tatsuro Kawamoto1, Yoshiyuki Kabashima1

  • 1Department of Mathematical and Computing Science, Tokyo Institute of Technology, 4259-G5-22, Nagatsuta-cho, Midori-ku, Yokohama, Kanagawa 226-8502, Japan.

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Summary
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We studied how well we can detect community structures in networks. Some network patterns are fundamentally undetectable, no matter how clear they seem.

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Area of Science:

  • Network science
  • Statistical inference
  • Graph theory

Background:

  • The stochastic block model (SBM) is a fundamental tool for analyzing network community structure.
  • Understanding the limits of community detection is crucial for interpreting real-world network data.

Purpose of the Study:

  • To determine the detectability thresholds for various modular structures within the stochastic block model.
  • To investigate the influence of cluster hierarchy on these detectability thresholds.

Main Methods:

  • Theoretical analysis of the stochastic block model.
  • Derivation of conditions for the detectability of planted graph structures.
  • Examination of the relationship between modularity and inferential limits.

Main Results:

  • Established precise detectability thresholds for diverse modular configurations in the SBM.
  • Demonstrated that the hierarchy of clusters significantly impacts the detectability threshold.
  • Identified specific planted structures that remain undetectable irrespective of their clarity or 'fuzziness.'

Conclusions:

  • The detectability of modular structures in networks is not solely dependent on their apparent strength but also on their specific configuration and hierarchy.
  • Certain network structures possess inherent properties that preclude their reliable inference, posing fundamental limits to community detection algorithms.