Most of real-world graphs are {\em dynamic}, i.e., they change over time.
However, while problems such as regression and Singular Value Decomposition
(SVD) have been studied for {\em static} graphs, they have not been
investigated for {\em dynamic} graphs, yet. In this paper, we introduce,
motivate and study regression and SVD over dynamic graphs. First, we present
the notion of {\em update-efficient matrix embedding} that defines the
conditions sufficient for a matrix embedding to be used for the dynamic graph
regression problem (under l2​ norm). We prove that given an n×m
update-efficient matrix embedding (e.g., adjacency matrix), after an update
operation in the graph, the optimal solution of the graph regression problem
for the revised graph can be computed in O(nm) time. We also study dynamic
graph regression under least absolute deviation. Then, we characterize a class
of matrix embeddings that can be used to efficiently update SVD of a dynamic
graph. For adjacency matrix and Laplacian matrix, we study those graph update
operations for which SVD (and low rank approximation) can be updated
efficiently