  We consider the all pairs all shortest paths (APASP) problem, which maintains
the shortest path dag rooted at every vertex in a directed graph G=(V,E) with
positive edge weights. For this problem we present a decremental algorithm
(that supports the deletion of a vertex, or weight increases on edges incident
to a vertex). Our algorithm runs in amortized O(\vstar^2 \cdot \log n) time per
update, where n=|V|, and \vstar bounds the number of edges that lie on shortest
paths through any given vertex. Our APASP algorithm can be used for the
decremental computation of betweenness centrality (BC), a graph parameter that
is widely used in the analysis of large complex networks. No nontrivial
decremental algorithm for either problem was known prior to our work. Our
method is a generalization of the decremental algorithm of Demetrescu and
Italiano [DI04] for unique shortest paths, and for graphs with \vstar =O(n), we
match the bound in [DI04]. Thus for graphs with a constant number of shortest
paths between any pair of vertices, our algorithm maintains APASP and BC scores
in amortized time O(n^2 \log n) under decremental updates, regardless of the
number of edges in the graph.
