  In this paper we consider the \emph{fully-dynamic} All-Pairs Effective
Resistance problem, where the goal is to maintain effective resistances on a
graph $G$ among any pair of query vertices under an intermixed sequence of edge
insertions and deletions in $G$. The effective resistance between a pair of
vertices is a physics-motivated quantity that encapsulates both the congestion
and the dilation of a flow. It is directly related to random walks, and it has
been instrumental in the recent works for designing fast algorithms for
combinatorial optimization problems, graph sparsification, and network science.
  We give a data-structure that maintains $(1+\epsilon)$-approximations to
all-pair effective resistances of a fully-dynamic unweighted, undirected
multi-graph $G$ with $\tilde{O}(m^{4/5}\epsilon^{-4})$ expected amortized
update and query time, against an oblivious adversary. Key to our result is the
maintenance of a dynamic \emph{Schur complement}~(also known as vertex
resistance sparsifier) onto a set of terminal vertices of our choice.
  This maintenance is obtained (1) by interpreting the Schur complement as a
sum of random walks and (2) by randomly picking the vertex subset into which
the sparsifier is constructed. We can then show that each update in the graph
affects a small number of such walks, which in turn leads to our sub-linear
update time. We believe that this local representation of vertex sparsifiers
may be of independent interest.
