1. A computer-implemented method for analyzing costs associated with a cost flow model, the method comprising: receiving data associated with the cost flow model, wherein the cost flow model is associated with one or more entities, wherein each entity is represented by a node, and wherein the cost flow model identifies costs associated with relationships among the entities; creating one or more matrices that are representative of the costs and the entity relationships, wherein the one or more matrices include one or more linear equations; reordering, using one or more data processors, the one or more matrices using a topological sort; constructing virtual nodes, wherein construction of the virtual nodes results in one or more modified linear equations; and using one or more sparse matrix operations to solve the one or more modified linear equations, wherein the solutions to the modified linear equations are used to determine real-time activity based cost flow and cost contribution amounts among arbitrary subsets of the entities associated with the cost flow model.