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 in-teraction matrix and we highlight how they are related 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 communi-cation 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 char-acterise 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) proportional to (1+r2)-beta /2 and beta >= 1. In a particular case of the Motsch-Tadmor model, we also prove with our method that the flocking phenomenon occurs independently of the initial conditions and the communication function.