  We study Plurality Consensus in the Gossip Model over a network of $n$
anonymous agents. Each agent supports an initial opinion or color. We assume
that at the onset, the number of agents supporting the plurality color exceeds
that of the agents supporting any other color by a sufficiently-large bias. The
goal is to provide a protocol that, with high probability, brings the system
into the configuration in which all agents support the (initial) plurality
color. We consider the Undecided-State Dynamics, a well-known protocol which
uses just one more state (the undecided one) than those necessary to store
colors. We show that the speed of convergence of this protocol depends on the
initial color configuration as a whole, not just on the gap between the
plurality and the second largest color community. This dependence is best
captured by a novel notion we introduce, namely, the monochromatic distance
${md}(\bar{\mathbf{c}})$ which measures the distance of the initial color
configuration $\bar{ \mathbf {c}}$ from the closest monochromatic one. In the
complete graph, we prove that, for a wide range of the input parameters, this
dynamics converges within $O({md}(\bar {\mathbf {c}}) \log {n})$ rounds. We
prove that this upper bound is almost tight in the strong sense: Starting from
any color configuration $\bar {\mathbf {c}}$, the convergence time is
$\Omega({md}(\bar {\mathbf {c}}))$. Finally, we adapt the Undecided-State
Dynamics to obtain a fast, random walk-based protocol for plurality consensus
on regular expanders. This protocol converges in $O({md}(\bar {\mathbf {c}})
\mathrm{polylog}(n))$ rounds using only $\mathrm{polylog}(n)$ local memory. A
key-ingredient to achieve the above bounds is a new analysis of the maximum
node congestion that results from performing $n$ parallel random walks on
regular expanders. All our bounds hold with high probability.
