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Topologically protected synchronization in networks.

Massimo Ostilli1

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Topologically equivalent nodes in a graph can synchronize easily when forming a connected subgraph, acting as independent pacemakers. This synchronization is independent of other nodes and graph changes.

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Area of Science:

  • Graph Theory
  • Network Science
  • Nonlinear Dynamics

Background:

  • Topological equivalence in graphs is defined by coinciding neighbor sets.
  • Synchronization of coupled oscillators is a key problem in network science.
  • Heterogeneous oscillators introduce complexity to synchronization phenomena.

Purpose of the Study:

  • To investigate the synchronization behavior of nonlinearly coupled heterogeneous oscillators on topologically equivalent nodes.
  • To determine conditions under which synchronization can be achieved within such groups.
  • To analyze the robustness of synchronization to changes in the rest of the network.

Main Methods:

  • Mathematical proof establishing conditions for synchronization.
  • Analysis of graph properties, specifically focusing on node neighborhoods.
  • Numerical simulations to validate theoretical findings.

Main Results:

  • Nonlinearly coupled heterogeneous oscillators on topologically equivalent nodes synchronize easily if they form a fully connected subgraph.
  • Synchronization within the group is independent of oscillators outside the group.
  • Synchronization is typically achieved when the internal degree is less than or equal to the outgoing degree (k_IN <= k_OUT).

Conclusions:

  • Groups of topologically equivalent nodes can act as independent synchronized units or pacemakers within a larger network.
  • The synchronization status of these groups is robust to alterations in the broader network structure.
  • The findings provide a theoretical framework and simulation-based evidence for predictable synchronization in complex networks.