  A now classical construction due to Kato and Nakayama attaches a topological
space (the "Betti realization") to a log scheme over $\mathbf{C}$. We show that
in the case of a log smooth degeneration over the standard log disc, this
construction allows one to recover the topology of the germ of the family from
the log special fiber alone. We go on to give combinatorial formulas for the
monodromy and the $d^2$ differentials acting on the nearby cycle complex in
terms of the log structures. We also provide variants of these results for the
Kummer etale topology. In the case of curves, these data are essentially
equivalent to those encoded by the dual graph of a semistable degeneration,
including the monodromy pairing and the Picard-Lefschetz formula.
