For IOT, let \(w=(I-z)^{-1}\) and \(g=(I-b)^{-1}\). The four linkage measures are:

\[\begin{split}\begin{aligned} \mathrm{DB}_j &= \sum_i \widetilde{z}_{ij}, & \mathrm{TB}_j &= \sum_i \widetilde{w}_{ij}, \\ \mathrm{DF}_i &= \sum_j \widetilde{b}_{ij}, & \mathrm{TF}_i &= \sum_j \widetilde{g}_{ij}. \end{aligned}\end{split}\]

DB / TB are direct / total backward linkages; DF / TF are direct / total forward linkages. The tilde means that diagonal entries are excluded when cut_diag=True. SUT uses the corresponding unified activity/commodity systems.

With multi_mode=False and normalized=True, each measure is divided by its mean:

\[\ell_i^{\mathrm{norm}} = \frac{\ell_i}{n^{-1}\sum_j\ell_j}.\]

Amplifications use the raw values: \(\mathrm{TB}/\mathrm{DB}\) and \(\mathrm{TF}/\mathrm{DF}\). With multi_mode=True, each sum is split into Local and Foreign contributors; each share is its component divided by their sum. Normalization is not applied in that mode. Zero denominators in amplifications and regional shares give NaN.

db.calc_linkages(...) returns the standard backward and forward linkages.

When multi_mode=False, the dataframe also includes Forward Amplification and Backward Amplification ratios, defined as Total / Direct on the raw linkage values.

When multi_mode=True, the dataframe keeps the Local and Foreign components and adds their Local Share and Foreign Share for each linkage measure.

Sparse flow blocks use sparse coefficient systems and actual production nodes, including unequal regional Activity/Commodity classifications. Regional totals are obtained by solving against regional indicator vectors, without requesting full Leontief or Ghosh inverse views. The result keeps the existing index, columns, normalization and amplification/share definitions.

cut_diag=True excludes self effects exactly. Sparse arrays are not modified in place: direct diagonal terms are subtracted from the sums, and inverse diagonal terms use selected inversion when suitable sparse factors are available, or streamed unit-vector solves otherwise. Acyclic single-node components have a direct diagonal formula. Exact diagonal extraction can still be expensive for large strongly connected systems. The selected-inversion recursion follows Jacquelin, Lin and Yang (2017).

Detected singular production components raise WrongInput before an inverse is requested. The error identifies the system and its component nodes. MARIO does not remove those nodes, change source values or replace the diagnosed singular system with a pseudoinverse. This also applies when cut_diag=False: including self effects does not restore invertibility.

Database.calc_linkages(

scenario='baseline',

normalized=True,

cut_diag=True,

multi_mode=True,

)

Calculate backward and forward linkages for one scenario.

Parameters:

• scenario – Scenario to analyse.

• normalized – When True, normalize linkage indicators using the standard linkages convention.

• cut_diag – When True, exclude diagonal terms exactly, without changing source matrices. Sparse inputs do not require a full inverse view.

• multi_mode – When True, preserve the multi-regional interpretation of the linkages. This is valid only for multi-regional databases.

Returns:

Linkages payload returned by linkages_calculation(...).

When multi_mode=False, the returned dataframe contains total and direct backward/forward linkages plus Forward Amplification and Backward Amplification ratios computed as Total / Direct on the raw, non-normalized values.

When multi_mode=True, the returned dataframe keeps the Local and Foreign components for each linkage measure and adds Local Share and Foreign Share columns for the same measures.

Return type:

object

Raises:

WrongInput – When a singular production component is detected, identifying its nodes, or a linear solve fails its residual accuracy check.