  This paper investigates the role persistent arcs play for a social network to
reach a global belief agreement under discrete-time or continuous-time
evolution. Each (directed) arc in the underlying communication graph is assumed
to be associated with a time-dependent weight function which describes the
strength of the information flow from one node to another. An arc is said to be
persistent if its weight function has infinite $\mathscr{L}_1$ or $\ell_1$ norm
for continuous-time or discrete-time belief evolutions, respectively. The graph
that consists of all persistent arcs is called the persistent graph of the
underlying network. Three necessary and sufficient conditions on agreement or
$\epsilon$-agreement are established, by which we prove that the persistent
graph fully determines the convergence to a common opinion in social networks.
It is shown how the convergence rates explicitly depend on the diameter of the
persistent graph. The results adds to the understanding of the fundamentals
behind global agreements, as it is only persistent arcs that contribute to the
convergence.
