  Randomly evolving systems composed by elements which interact among each
other have always been of great interest in several scientific fields. This
work deals with the synchronization phenomenon, that could be roughly defined
as the tendency of different components to adopt a common behavior. We continue
the study of a model of interacting stochastic processes with reinforcement,
that recently has been introduced in Crimaldi et al. (2016, arXiv:1602.06217).
Generally speaking, by reinforcement we mean any mechanism for which the
probability that a given event occurs has an increasing dependence on the
number of times that events of the same type occurred in the past. The
particularity of systems of such stochastic processes is that synchronization
is induced along time by the reinforcement mechanism itself and does not
require a large-scale limit. We focus on the relationship between the topology
of the network of the interactions and the long-time synchronization
phenomenon. After proving the almost sure synchronization, we provide some CLTs
in the sense of stable convergence that establish the convergence rates and the
asymptotic distributions for both convergence to the common limit and
synchronization. The obtained results lead to the construction of asymptotic
confidence intervals for the limit random variable and of statistical tests to
make inference on the topology of the network given the observation of the
reinforced stochastic processes positioned at the vertices.
