From power grids to social networks to neuroscience, networks are increasingly important in science today. They are, however, inherently hard to study. On one hand, phenomena beginning in one part of a network can have complex and global effects on the rest of the network, and so behavior is frequently difficult to predict without simulations. On the other hand, modern networks are often massive, containing hundreds of millions or even billions of nodes. Due to this, network computations often require specialized algorithms that exploit network structure to perform their tasks efficiently.

In this work, we study matrix-based network computations and the relationship between network structure and linear algorithms. Out algorithms use either low rank upates or coarse grid projections to transform the problem into a smaller one that is exactly or approximately equivalent to the original. We refer to these techniques as error flattening methods.

We present three examples: a method for fast detection and identification of power grid topology errors; a nonlinear multigrid method to solve the power flow equations; and a two-part iterative method to solve graph Laplacian systems.