A network is a countable, connected graph $X$ viewed as a
one-complex, where each edge $[x,y]=[y,x]$ ($x,y\in X^0$, the
vertex set) is a copy of the unit interval within the graph's
one-skeleton $X^1$ and is assigned a positive conductance
$\mathsf{c}(xy)$. A reference "Lebesgue" measure $X^1$ is
built up by using Lebesgue measure with total mass
$\mathsf{c}(xy)$ on each edge $xy$. There are three natural
operators on $X$: the transition operator $P$ acting on
functions on $X^0$ (the reversible Markov chain associated
with $\mathsf{c}$), the averaging operator $A$ over spheres of
radius~1 on $X^1$, and the Laplace operator $\Delta$ on $X^1$
(with Kirchhoff conditions weighted by $\mathsf{c}$ at the
vertices). The relation between the $\ell^2$-spectrum of $P$
and the $H^2$-spectrum of~$\Delta$ was described by
{Cattaneo} \cite{Cat}. In this note we describe the relation
between the $\ell^2$-spectrum of $P$ and the $L^2$-spectrum
of $A$.