Online version of All pair shortest path when path weights are updated Given 


an undirected graph of $n$ nodes with weighted edges and 
a sequence $S=((e,w),...)$ of updates, always decreasing, of the weight $w$ of edge $e$.


What is the online complexity of computing the shortest pair between each pair of nodes in the graph after each updates? Is there any solution better than solving the problem offline in time within $O(n^3)$ (using Floyd-Warshall's algorithm) after each update?
I think adding the keywords decremental/incremental might help you find the relevant papers. 

See e.g. 
Camil Demetrescu, Giuseppe F. Italiano: Decremental All-Pairs Shortest Paths. Encyclopedia of Algorithms 2008 for an overview, Camil Demetrescu, Giuseppe F. Italiano: Fully dynamic all pairs shortest paths with real edge weights. J. Comput. Syst. Sci. 72(5): 813-837 (2006), and Camil Demetrescu, Giuseppe F. Italiano: A new approach to dynamic all pairs shortest paths. J. ACM 51(6): 968-992 (2004)

Approximate versions have been considered as well; the most recent result I know of is Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai: Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization, FOCS 2013.
Table 1 in their paper has an overview of results.