NodePotentials

class paminco.algo.efa.NodePotentials(net_nodes)

Class that stores values that relate to the potential in some region.

The potential \(\mathbf{\pi}\) for some region \(R_{\mathbf{t}}\) is given by:

\[\mathbf{\pi} = \mathbf{\pi}_{\mathbf{t}} + \lambda_{\mathbf{t}}^{\text{max}} \Delta \mathbf{\pi}_{\mathbf{t}} = \mathbf{L}_{\mathbf{t}}^{\ast} (\mathbf{\Gamma}\mathbf{d}_{\mathbf{t}}) + \lambda_{\mathbf{t}}^{\text{max}} (\mathbf{L}_{\mathbf{t}}^{\ast} \mathbf{b}),\]

where \(\mathbf{L}_{\mathbf{t}}^{\ast}\) is the generalized inverse of the weighted Laplacian, \(\Gamma\) the incidence matrix and \(\mathbf{d} _{\mathbf{t}} = \frac{\mathbf{b} _{\mathbf{t}}}{\mathbf{a} _{\mathbf{t}}}\). Here, \(\mathbf{b} _{\mathbf{t}}\) is the vector of offsets and \(\mathbf{a} _{\mathbf{t}}\) the vector of slopes of the linear marginal cost functions \(\tilde{f}_e\) for the region \(R_{\mathbf{t}}\).

A node potential \(\mathbf{\pi}\) induces a flow \(\mathbf{f}^{-1}(\mathbf{\pi})\) by:

\[\mathbf{f}^{-1}(\mathbf{\pi}) = \mathbf{C}_{\mathbf{t}} \Gamma^T\mathbf{\pi} - \mathbf{d}_{\mathbf{t}}. \]

See also

paminco.net.cost.PiecewiseQuadraticCoefficients

Attributes

node_idxndarray (n, )

Node indices, ndarray of int.

node_lblndarray (n, )

Node labels, ndarray of str.

pindarray (n, )

Node potential \(\mathbf{\pi}\).

pi_tndarray (n, )

Potential offset of the current region \(\mathbf{\pi}_{\mathbf{t}}\).

dpi_tndarray (n, )

Potential direction of the current region \(\Delta \mathbf{\pi}_{\mathbf{t}}\).

d_tildendarray (n, )

\(\tilde{\mathbf{d}}_{\mathbf{t}} = \mathbf{\Gamma}\mathbf{d}_{\mathbf{t}}\),