Complexity and Stability in Compartmental Models

Compartmental models, by which flows through various systems can be studied, have a dual aspect: one structural and the other dynamic. The structural leads to a directed graph and may be analyzed by means of graph theory. The dynamic leads to a system of differential equations which are usually linear. The coefficient matrix in this case has a special form, that of the negative transpose of an M-matrix in which the off diagonal elements are nonnegative and the columns add up to zero. Hence the eigenvalues have a nonpositive real part.If the digraph is weakly connected, the differential equation has a stable equilibrium solution; if it is unilaterally connected, the solution is unique; if it is strongly connected, the solution is feasible as well. It is possible to define various indices of stability which may then be shown to be related to indices of complexity of the structure. However, it is also possible, by redirecting the flows, to show that a given model can be reduced to a mammillary system with the same equilibrium solution. Hence any index of stability based on the equilibrium solution has no relation to a complexity index based on the number of arcs per vertex. Other stability indices, however, increase with increasing complexity.