  Richter and Thomassen proved that every graph has an edge $e$ such that the
crossing number $\ucr(G-e)$ of $G-e$ is at least $(2/5)\ucr(G) - O(1)$. Fox and
Cs. T\'oth proved that dense graphs have large sets of edges (proportional in
the total number of edges) whose removal leaves a graph with crossing number
proportional to the crossing number of the original graph; this result was
later strenghtened by \v{C}ern\'{y}, Kyn\v{c}l and G. T\'oth. These results
make our understanding of the {decay} of crossing numbers in dense graphs
essentially complete. In this paper we prove a similar result for large sparse
graphs in which the number of edges is not artificially inflated by operations
such as edge subdivisions. We also discuss the connection between the decay of
crossing numbers and expected crossing numbers, a concept recently introduced
by Mohar and Tamon.
