  We investigate the influence of the range of interactions in the
two-dimensional bond percolation model, by means of Monte Carlo simulations. We
locate the phase transitions for several interaction ranges, as expressed by
the number $z$ of equivalent neighbors. We also consider the $z \to \infty$
limit, i.e., the complete graph case, where percolation bonds are allowed
between each pair of sites, and the model becomes mean-field-like. All
investigated models with finite $z$ are found to belong to the short-range
universality class. There is no evidence of a tricritical point separating the
short-range and long-range behavior, such as is known to occur for $q=3$ and
$q=4$ Potts models. We determine the renormalization exponent describing a
finite-range perturbation at the mean-field limit as $y_r \approx 2/3$. Its
relevance confirms the continuous crossover from mean-field percolation
universality to short-range percolation universality. For finite interaction
ranges, we find approximate relations between the coordination numbers and the
amplitudes of the leading correction terms as found in the finite-size scaling
analysis.
