Unbiased selection of the scaling constant φ for α = φ N^{-1/s} in graph filtering

Determine an unbiased method for choosing the constant phi in the scaling rule alpha = phi * N^(-1/s), where alpha controls the graph filtering functional J = rho^alpha (1 - delta)^beta (with beta set to 1 in the simplified analysis), so that the resulting optimal connection density rho for spatial networks of size N in spatial dimension s can be specified without reliance on ad hoc or subjective criteria.

Background

To make large spatial networks visually readable when node positions are fixed, the paper introduces a graph filtering optimization J = rho^alpha (1 − delta)^beta and shows that optimal connection densities depend on how link lengths correlate with weights. To counteract the increasing difficulty of visualizing larger systems, the authors propose scaling the filtering parameter as alpha = phi N^−1/s, where s is the spatial dimension and phi is a positive constant.

While this scaling preserves the qualitative filtering behavior, the authors explicitly note that choosing the constant phi in an unbiased manner is unresolved. Lacking a theoretical prescription, they turned to an online human-perception experiment to estimate phi, obtaining an empirical value near 1. The open question concerns establishing a principled, unbiased rule for selecting phi independent of such empirical calibration.