We study the role of connectivity of communication networks in private computations under information theoretical settings in the honest-but-curious model. We show that some functions can be 1-privately computed even if the underlying network is 1-connected but not 2-connected. Then we give a complete characterisation of non-degenerate functions that can be 1-privately computed on non-2-connected networks. Furthermore, we present a technique for simulating 1-private protocols that work on arbitrary (complete) networks on k-connected networks. For this simulation, at most (1−k/(n−1))⋅L additional random bits are needed, where L is the number of bits exchanged in the original protocol and n is the number of players. Finally, we give matching lower and upper bounds for the number of random bits needed to compute the parity function on k-connected networks 1-privately, namely ⌈(n−2)/(k−1)⌉−1 random bits for networks consisting of n player
