A Note on Directional Differentiability of Flow Network Equilibria with Respect to a Parameter

Abstract

Let (N, A) be a weakly connected directed graph, N denoting its set of nodes and A its set of arcs (both sets being finite). The node-arc incidence matrix of (N, A) will be denoted by D, and column α of D by D_α. (D has exactly one +1 and one -1 in each column, the rest of its entries being zero.) With each arc α ∈ A, we associate a oneparameter family of arc characteristics f_α (α;π), the parameter π varying between −$\hat{\pi }$ and $\hat{\pi }$ > o. This provides us with a one-parameter family of networks (N, A, f(…;π)). (For details on arc characteristics see e.g. IRI [1], who by the way speaks of branch instead of arc characteristics.)