  This article studies stochastic relative phase stability, i.e., stochastic
phase-cohesiveness, of discrete-time phase-coupled oscillators. Stochastic
phase-cohesiveness in two types of networks is studied. First, we consider
oscillators coupled with 2{\pi}-periodic odd functions over underlying
undirected graphs subject to both multiplicative and additive stochastic
uncertainties. We prove stochastic phase-cohesiveness of the network with
respect to two specific, namely in-phase and anti-phase, sets by deriving
sufficient coupling conditions. We show the dependency of these conditions on
the size of the mean values of additive and multiplicative uncertainties, as
well as the sign of the mean values of multiplicative uncertainties.
Furthermore, we discuss the results under a relaxation of the odd property of
the coupling function. Second, we study an uncertain network in which the
multiplicative uncertainties are governed by the Bernoulli process representing
the well-known Erdos-Renyi network. We assume constant exogenous frequencies
and derive sufficient conditions for achieving both stochastic phase-cohesive
and phase-locked solutions, i.e., stochastic phase-cohesiveness with respect to
the origin. For the latter case, where identical exogenous frequencies are
assumed, we prove that any positive probability of connectivity leads to
phase-locking. Thorough analyses are provided, and insights obtained from
stochastic analysis are discussed, along with numerical simulations to validate
the analytical results.
