  In this study, we explore the fundamental principles behind the architecture
of the human brain's structural connectome, from the perspective of spectral
analysis of Laplacian and adjacency matrices. Building on the idea that the
brain strikes a balance between efficient information processing and minimizing
wiring costs, we aim to understand the impact of the metric properties of the
connectome and how they relate to the existence of an inherent scale. We
demonstrate that a simple generative model, combining nonlinear preferential
attachment with an exponential penalty for spatial distance between nodes, can
effectively reproduce several key characteristics of the human connectome,
including spectral density, edge length distribution, eigenmode localization
and local clustering properties. We also delve into the finer spectral
properties of the human structural connectomes by evaluating the inverse
participation ratios ($\text{IPR}_q$) across various parts of the spectrum. Our
analysis reveals that the level statistics in the soft cluster region of the
Laplacian spectrum deviate from a purely Poisson distribution due to
interactions between clusters. Additionally, we identified scar-like localized
modes with large IPR values in the continuum spectrum. We identify multiple
fractal eigenmodes distributed across different parts of the spectrum, evaluate
their fractal dimensions and find a power-law relationship in the return
probability, which is a hallmark of critical behavior. We discuss the
conjectures that a brain operates in the Griffiths or multifractal phases.
