We consider an environment for an open quantum system described by a ‘quantum network
geometry with flavor’ (QNGF) in which the nodes are coupled quantum oscillators. The
geometrical nature of QNGF is reflected in the spectral properties of the Laplacian matrix of the
network which display a finite spectral dimension, determining also the frequencies of the normal
modes of QNGFs. We show that an a priori unknown spectral dimension can be indirectly
estimated by coupling an auxiliary open quantum system to the network and probing the normal
mode frequencies in the low frequency regime. We find that the network parameters do not affect
the estimate; in this sense it is a property of the network geometry, rather than the values of, e.g.,
oscillator bare frequencies or the constant coupling strength. Numerical evidence suggests that the
estimate is also robust both to small changes in the high frequency cutoff and noisy or missing
normal mode frequencies. We propose to couple the auxiliary system to a subset of network nodes
with random coupling strengths to reveal and resolve a sufficiently large subset of normal mode
frequencies