Full characterization of the bulk spectrum in the Multi-Scale Model

Characterize the bulk spectrum of the realized adjacency matrix A of the annealed Multi-Scale Model with node weights x_i drawn independently from a Pareto(α) distribution (0<α<1) and edge probabilities p_{ij}=1−exp(−ε_n x_i x_j) with ε_n=n^{−1/α}. Specifically, determine the eigenvalue density and the precise location of the bulk edge as functions of α and n, refining the current crude upper bound on the spectral norm of the fluctuation matrix H=A−P to include its dependence on α.

Background

The paper decomposes the adjacency matrix A into an expected component P and a fluctuation component H=A−P. The leading (outlier) eigenvalues and eigenvectors of P are analyzed and shown to scale as √n and to exhibit log-periodicity, accurately describing outliers in A up to a growing index on the order of ln n.

Beyond these outliers, the spectrum of A contains a bulk primarily generated by H. The authors derive a crude upper bound for the spectral norm of H that scales as √n, indicating that the bulk edge lies on the same scale as the structural outliers. However, this bound does not yet capture the dependence on the stability index α and the full spectral density of the bulk remains analytically unsettled.

The authors state that a complete analytic characterization of this bulk regime is highly non-trivial and leave it for future work, motivating a precise determination of the bulk spectral density and edge as functions of model parameters to fully understand the competition between structure and randomness in this infinite-mean regime.