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In bromoethane, the three methyl protons are coupled to the two methylene protons that are three bonds away. In accordance with the n+1 rule, the signal from the methyl protons is split into three peaks with 1:2:1 relative intensities. The methylene protons appear as a quartet, with the relative intensities of 1:3:3:1.
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Modular tipping points: How local network structure impacts critical transitions in networked spin systems.

Daniel Reisinger1, Raven Adam1, Fabian Tschofenig1

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Network structure significantly impacts critical transitions. High modularity leads to cascading shifts, while low modularity results in unified transitions, highlighting the importance of component roles in networked systems.

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Area of Science:

  • Complex systems science
  • Network theory
  • Dynamical systems

Background:

  • Critical transitions are abrupt shifts in system states, relevant across ecology, sociology, and physics.
  • System network structure critically influences transition dynamics.
  • Network modularity is an understudied factor in critical transitions.

Purpose of the Study:

  • Investigate the impact of network modularity on critical transition behavior.
  • Analyze how altering degree assortativity affects transition dynamics.
  • Understand the role of local network structure in system-wide shifts.

Main Methods:

  • Simulated critical transitions in networked systems with varying modularity and degree assortativity.
  • Controlled manipulation of local network structure via degree assortativity.
  • Analysis of transition behavior across diverse degree distributions (e.g., power-law, Poisson).

Main Results:

  • High modularity networks exhibit cascading transitions; low modularity networks show unified transitions.
  • Networks with heterogeneous node connectivity transition earlier than predicted by average degree.
  • Exceptions to earlier transitions occur in low-modularity, high disassortativity networks.

Conclusions:

  • Network modularity dictates transition patterns (cascading vs. unified).
  • Node connectivity heterogeneity influences transition timing.
  • Understanding critical transitions requires analyzing individual components and their network roles.