  We present an emergent stochastic flocking dynamics of the Cucker-Smale (CS)
ensemble under randomly switching topologies. The evolution of the CS ensemble
with randomly switching topologies involves two random components (switching
times and choices of network topologies at switching instant). First, we allow
switching times for the network topology to be random so that the successive
increments are i.i.d. processes following the common probability distribution.
Second, at each switching instant, we choose a network topology randomly from a
finite set of admissible network topologies whose union contains a spanning
tree. Even for the fixed deterministic network topology, the CS ensemble may
not exhibit a mono-cluster flocking depending on the initial data and the decay
mode of the communication weight functions measuring the degree of interactions
between particles. For the flocking dynamics of the CS ensemble with these two
random components, we first use a priori conditions on the network topologies
and uniform boundedness of position diameter, and derive the flocking estimates
via matrix theory together with a priori conditions, and then replace the a
priori condition for the position diameter by some suitable condition on the
system parameters and communication weight. The a priori condition on the
network topology will be guaranteed by the suitable spanning tree time-blocks
with probability one.
