The linear framework.

A. A labelled, directed graph, G, gives rise to a system of linear differential equations by treating each edge as a first-order chemical reaction under mass-action kinetics, with the label as rate constant. The corresponding matrix is the Laplacian of G. B. In a strongly connected graph (note the difference to the one in A), there are spanning trees rooted at each vertex, the roots being circled. The MTT gives an element of according to the formula in the box, as explained in the text. C. In a general directed graph, G, two distinct vertices are in the same strongly connected component (SCC) if each can be reached from the other by a path of directed edges. The SCCs form a directed graph, , in which two SCCs are linked by a directed edge if some vertex of the first SCC has an edge to some vertex of the second SCC. has no directed cycles, allowing initial and terminal SCCs to be identified.