  In this paper, we discuss the flocking phenomenon for the Cucker-Smale and
Motsch-Tadmor models in continuous time on a general oriented and weighted
graph with a general communication function. We present a new approach for
studying this problem based on a probabilistic interpretation of the solutions.
We provide flocking results under four assumptions on the interaction matrix
and we highlight how they relate to the convergence in total variation of a
certain Markov jump process. Indeed, we refine previous results on the minimal
case where the graph admits a unique closed communication class. Considering
the two particular cases where the adjacency matrix is scrambling or where it
admits a positive reversible measure, we improve the flocking condition
obtained for the minimal case. In the last case, we characterise the asymptotic
speed. We also study the hierarchical leadership case where we give a new
general flocking condition which allows to deal with the case
$\psi(r)\propto(1+r^2)^{-\beta/2}$ and $\beta\geq1$. For the Motsch-Tadmor
model under the hierarchical leadership assumption, we exhibit a case where the
flocking phenomenon occurs regardless of the initial conditions and the
communication function, in particular even if $\beta\geq1$.
