  We investigate the validity of the "Einstein relations" in the general
setting of unimodular random networks. These are equalities relating scaling
exponents: $d_w = d_f + \tilde{\zeta}$ and $d_s = 2 d_f/d_w$, where $d_w$ is
the walk dimension, $d_f$ is the fractal dimension, $d_s$ is the spectral
dimension, and $\tilde{\zeta}$ is the resistance exponent. Roughly speaking,
this relates the mean displacement and return probability of a random walker to
the density and conductivity of the underlying medium. We show that if $d_f$
and $\tilde{\zeta} \geq 0$ exist, then $d_w$ and $d_s$ exist, and the
aforementioned equalities hold. Moreover, our primary new estimate is the
relation $d_w \geq d_f + \tilde{\zeta}$, which is established for all
$\tilde{\zeta} \in \mathbb{R}$.
  For the uniform infinite planar triangulation (UIPT), this yields the
consequence $d_w=4$ using $d_f=4$ (Angel 2003) and $\tilde{\zeta}=0$
(established here as a consequence of the Liouville Quantum Gravity theory,
following Gwynne-Miller 2017 and Ding-Gwynne 2020). The conclusion $d_w=4$ had
been previously established by Gwynne and Hutchcroft (2018) using more
elaborate methods. A new consequence is that $d_w = d_f$ for the uniform
infinite Schnyder-wood decorated triangulation, implying that the simple random
walk is subdiffusive, since $d_f > 2$ (Ding and Gwynne 2020). For the random
walk on $\mathbb{Z}^2$ driven by conductances from an exponentiated Gaussian
free field with exponent $\gamma > 0$, one has $d_f = d_f(\gamma)$ and
$\tilde{\zeta}=0$ (Biskup, Ding, and Goswami 2020). This yields $d_s=2$ and
$d_w = d_f$, confirming two predictions of those authors.
