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Structural transition in interdependent networks with regular interconnections.

Xiangrong Wang1, Robert E Kooij1,2, Yamir Moreno3,4,5

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This study generalizes the critical coupling threshold for interdependent networks with regular interconnection matrices. It provides bounds for this threshold, offering insights into network behavior and potential failures.

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Area of Science:

  • Network Science
  • Complex Systems
  • Statistical Physics

Background:

  • Interdependent networks comprise multiple interconnected layers with varying dependencies.
  • Dynamical processes in these networks are often characterized by the system's Laplacian matrix.
  • Multiplex networks, a special case, exhibit a structural transition at a critical coupling strength.

Purpose of the Study:

  • To extend and generalize the structural transition threshold to interdependent networks with regular interconnection matrices.
  • To provide upper and lower bounds for this critical coupling threshold.
  • To explore the physical meaning of the transition and its existence.

Main Methods:

  • Analysis of interdependent networks with a regular interconnection matrix B.
  • Derivation of transition threshold bounds using mathematical formalism.
  • Application of quotient graphs for specific scenarios.
  • Investigation of the transition's relation to minimum cut.

Main Results:

  • Upper and lower bounds for the transition threshold p* in regular interdependent networks are provided.
  • Exact transition thresholds are derived for specific network structures.
  • The physical interpretation of p* in terms of minimum cut is discussed.
  • A counterexample demonstrating the non-existence of the structural transition is presented.

Conclusions:

  • The study generalizes the understanding of structural transitions in interdependent networks beyond multiplex structures.
  • The findings contribute to characterizing more realistic multilayer network behaviors.
  • Results are relevant for systems with deviations from multiplex network constraints, impacting areas like power grids and diffusion processes.