On the Theory of Dynamic Graph Regression Problem

Most of real-world graphs are {\em dynamic}, i.e., they change over time. However, while the regression problem has been studied for {\em static} graphs, it has not been investigated for {\em dynamic} graphs, yet. In the current paper, 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, efficiently. We also show that some of the standard matrix embeddings, e.g., the (weighted) adjacency matrix, satisfy these conditions. Then, we prove that given an $n \times m$ update-efficient matrix embedding, 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. In particular, using the (weighted) adjacency matrix as the matrix embedding of $G$, it takes $O(n^2)$ time to update the optimal solution, where $n$ is the number of nodes of the revised graph. To the best of our knowledge, this is the first result on updating the solution of the graph regression problem, in a time considerably less than the time of computing the solution from the scratch. Finally, we study a generalization of the dynamic graph regression problem and show that it can be solved in $O(nm + mm')$ space.