Regression and Singular Value Decomposition in Dynamic Graphs

Most of real-world graphs are dynamic, i.e., they change over time. However, while problems such as regression and Singular Value Decomposition (SVD) have been studied for static graphs, they have not been investigated for dynamic graphs, yet. In this paper, we study regression and SVD over dynamic graphs. First, we present the notion of update-efficient matrix embedding that defines the conditions sufficient for a matrix embedding to be used for the dynamic graph regression problem (under $l_2$ norm). We prove that given an $n \times m$ 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. We show that for example if the matrix embedding of the graph is defined as its adjacency matrix, after an edge insertion or an edge deletion or an edge weight change in the graph, SVD of the graph can be updated in $O(n^2 log^2 n)$ time. Moreover, after a node insertion in the graph, rank-r approximation of the graph can be updated in $O(rn^2)$ time.