Extension to structured or correlated networks

Determine whether the almost-sure law of large numbers for the principal Dirichlet eigenvalue and the associated sharp survival threshold, established for Erdős–Rényi binomial random graphs with sink vertices in the discrete diffusion model where survival is governed by the smallest eigenvalue of the principal submatrix of the graph Laplacian on non-sink vertices, extend to habitats modeled by structured or correlated networks with dependent edges or nontrivial topology.

Background

The paper studies population persistence on graphs with sink vertices via the principal Dirichlet eigenvalue, defined as the smallest eigenvalue of the principal submatrix of the Laplacian obtained by removing sink rows and columns. For Erdős–Rényi random graphs with many sinks, the authors prove a law of large numbers for the Dirichlet eigenvalue and identify a sharp survival threshold proportional to the expected boundary degree.

In the conclusion, the authors explicitly note that extending these asymptotic spectral survival results beyond binomial random graphs to structured or correlated networks remains open.