  We consider the problem of joint learning of multiple linear dynamical
systems. This has received significant attention recently under different types
of assumptions on the model parameters. The setting we consider involves a
collection of $m$ linear systems each of which resides on a node of a given
undirected graph $G = ([m], \mathcal{E})$. We assume that the system matrices
are marginally stable, and satisfy a smoothness constraint w.r.t $G$ -- akin to
the quadratic variation of a signal on a graph. Given access to the states of
the nodes over $T$ time points, we then propose two estimators for joint
estimation of the system matrices, along with non-asymptotic error bounds on
the mean-squared error (MSE). In particular, we show conditions under which the
MSE converges to zero as $m$ increases, typically polynomially fast w.r.t $m$.
The results hold under mild (i.e., $T \sim \log m$), or sometimes, even no
assumption on $T$ (i.e. $T \geq 2$).
