  A signed graph offers richer information than an unsigned graph, since it
describes both collaborative and competitive relationships in social networks.
In this paper, we study opinion dynamics on a signed graph, based on the
Friedkin-Johnsen model. We first interpret the equilibrium opinion in terms of
a defined random walk on an augmented signed graph, by representing the
equilibrium opinion of every node as a combination of all nodes' internal
opinions, with the coefficient of the internal opinion for each node being the
difference of two absorbing probabilities. We then quantify some relevant
social phenomena and express them in terms of the $\ell_2$ norms of vectors. We
also design a nearly-linear time signed Laplacian solver for assessing these
quantities, by establishing a connection between the absorbing probability of
random walks on a signed graph and that on an associated unsigned graph. We
further study the opinion optimization problem by changing the initial opinions
of a fixed number of nodes, which can be optimally solved in cubic time. We
provide a nearly-linear time algorithm with error guarantee to approximately
solve the problem. Finally, we execute extensive experiments on sixteen
real-life signed networks, which show that both of our algorithms are effective
and efficient, and are scalable to massive graphs with over 20 million nodes.
