  The mixing time of a graph is an important metric, which is not only useful
in analyzing connectivity and expansion properties of the network, but also
serves as a key parameter in designing efficient algorithms. We introduce a new
notion of mixing of a random walk on a (undirected) graph, called local mixing.
Informally, the local mixing with respect to a given node $s$, is the mixing of
a random walk probability distribution restricted to a large enough subset of
nodes --- say, a subset of size at least $n/\beta$ for a given parameter
$\beta$ --- containing $s$. The time to mix over such a subset by a random walk
starting from a source node $s$ is called the local mixing time with respect to
$s$. The local mixing time captures the local connectivity and expansion
properties around a given source node and is a useful parameter that determines
the running time of algorithms for partial information spreading, gossip etc.
  Our first contribution is formally defining the notion of local mixing time
in an undirected graph. We then present an efficient distributed algorithm
which computes a constant factor approximation to the local mixing time with
respect to a source node $s$ in $\tilde{O}(\tau_s)$ rounds, where $\tau_s$ is
the local mixing time w.r.t $s$ in an $n$-node regular graph. This bound holds
when $\tau_s$ is significantly smaller than the conductance of the local mixing
set (i.e., the set where the walk mixes locally); this is typically the
interesting case where the local mixing time is significantly smaller than the
mixing time (with respect to $s$). We also present a distributed algorithm that
computes the exact local mixing time in $\tilde{O}(\tau_s \mathcal{D})$ rounds,
where $\mathcal{D} =\min\{\tau_s, D\}$ and $D$ is the diameter of the graph. We
further show that local mixing time tightly characterizes the complexity of
partial information spreading.
