Littlewood-Offord theory and controllability of random structures

Motivated by recent developments in random matrix theory through the study of inverse Littlewood-Offord problems, we investigate the controllability of random binary symmetric matrices. We show that, as the dimension of the state space goes to infinity, the probability of (A,b) being controllable approaches one for many choices of the vector b including elements of the standard basis, the all-one vector, and random binary vectors. In particular, we verify a conjecture of Godsil [1] and show that most systems are controllable from single inputs.