Methods, systems, and apparatus include computer programs encoded on a computer-readable storage medium, including a method for updating graphs. Labels associated with nodes of a graph are identified, including designators describing an attribute associated with a given node. The graph is provided, wherein labels have been assigned to each node in the graph. An initial set of weights for the labels are assigned reflecting a magnitude of a contribution of an associated label to a characterization of a respective node. A portion of the labels are assigned based on a propagation from other nodes. A change is identified in the graph that, when propagated, will affect other nodes. Sparse matrices, generated to describe the change, contain nonzero entries only in rows wherein connection weights and/or labels have changed. A new graph is generated using the sparse matrices without having to recalculate weights for other nodes not affected by the change.