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Approximating nonbacktracking centrality and localization phenomena in large networks.

G Timár1, R A da Costa1, S N Dorogovtsev1

  • 1Departamento de Física da Universidade de Aveiro & I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.

Physical Review. E
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Summary
This summary is machine-generated.

We introduce a novel degree-class method to approximate network dynamics, simplifying calculations for large networks. This approach accurately estimates key network properties, especially when short cycles are minimal.

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Area of Science:

  • Network Science
  • Statistical Physics
  • Dynamical Systems

Background:

  • Message-passing theories are crucial for analyzing dynamical processes on networks, including percolation and epidemics.
  • The nonbacktracking matrix and its properties (eigenvalue, eigenvector, centrality) are central to understanding these dynamics.
  • Analyzing large networks with the full nonbacktracking matrix presents significant computational challenges.

Purpose of the Study:

  • To develop an efficient approximation method for key quantities derived from the nonbacktracking matrix.
  • To reduce the computational complexity associated with analyzing large-scale networks.
  • To provide an alternative to the full nonbacktracking matrix for memory-constrained environments.

Main Methods:

  • Proposed a degree-class-based method to approximate network quantities.
  • Utilized a smaller matrix related to the joint degree-degree distribution of neighboring nodes.
  • Focused on approximating the largest eigenvalue, eigenvector, and nonbacktracking centrality.

Main Results:

  • The proposed method provides accurate estimates, particularly in networks with low degrees of short cycles.
  • Degree-degree correlations beyond nearest neighbors were found to be generally weak in most networks.
  • The scheme effectively captures the localization of nonbacktracking centrality, especially in large networks.

Conclusions:

  • The degree-class-based method offers a computationally efficient alternative for analyzing dynamical processes on large networks.
  • This approach is particularly effective when message-passing is a valid approximation of the underlying model.
  • The findings highlight the importance of local network structure (degree correlations) in approximating global network properties.