Given the capacity matrix and the known flows at certain links, we can use these link flows to calibrate the parameter of IFN model \( \kappa \) such that the resulting flows at the the known links would be as close as possible to the given link flow. Let \( \mathbf{C} = [c_{ij}] \) be the capacity matrix, \( \kappa \) be the parameter, and \( \mathbf{S} = [s_{ij}] \) be the stochastic matrix. Then, each element \( s_{ij} \) of \( \mathbf{S} \) is given by proportional formula: $$ \mathbf{S} = [s_{ij}] = \frac{c_{ij}}{\sum_{k=1}^{n} c_{ik}} $$ where \( n \) is the number of columns (or rows, since it's a square matrix) in \( \mathbf{C} \). The node vector is computed as $$ \mathbf{\pi} =\begin{bmatrix} \mathbf{S}^{T}-\mathbf{I} \\ \mathbf{j}^{T} \end{bmatrix} \setminus \begin{bmatrix} \mathbf{0} \\ \kappa \end{bmatrix} $$ From the node vector \( \mathbf{\pi} \) and stochastic matrix \( \mathbf{S} \) we can compute the flow matrix with parameter total flow \( \kappa \). $$ \mathbf{F} = \mathbf{\pi} \cdot \mathbf{j}^{T} \circ \mathbf{S} $$ The notation \( \mathbf{x} = \mathbf{K} \setminus \mathbf{b} \) means \( \mathbf{x} = \mathbf{K}^{+} \mathbf{b} \) where \( \mathbf{K}^{+} = (\mathbf{K}^{T}\mathbf{K})^{-1} \mathbf{K}^{T} \) is the Generalized Left Inverse of the Moore Penrose Inverse.
Read: Graph Theory and Linear Algebra
to generate random capacity matrix. The input capacity matrix must be a non-negative square matrix and irreducible.
End each row separated by a semicolon. Separate each data in one row by comma or a space.
to add this flow into buffer. To delete from buffer, click 
in the Lab Tool below to calibrate IFN (right) based on the Capacity matrix (left) and the known link flow in the buffer.
Capacity matrix
Pattern of Capacity Matrix
Calibrated IFN
Pattern of Ideal Flow Matrix
IFN Lab: Calibrating IFN