9. Hyperflow Model

The hyperflow model on derivation graphs accessible through mod::flow::Flow (C++) and mod.Flow (Python) consists of many different variables and constraints. This page is not meant to fully describe the model but to describe how the variables are available in the C++ and Python interface.

9.1. Variable Specifiers

The following information is common for both the C++ and Python interface. Though, in Python the specifiers are defined in module scope, while in C++ they are defined in the namespace mod::flow::vars. Thus, it may be useful to use the using-declaration using namespace mod::flow::vars; when defining linear expressions/constraints in C++.

Each defined specifier represents a sum of variables, and indexing the specifier retrieves a specifier for a single variable (or a sum of a subset of the variables) as indicated by the argument. E.g., inFlow represents the sum \(\sum_{v\in V}x^-_v\) while inFlow[x] retrieves the specifier for \(x^-_v\) where x is either the vertex \(v\) or a graph associated with that vertex in the underlying derivation graph \(\mathcal{H} = (V, E)\). In the table below, the “Type” column refers to the types flow::VarSumVertex/FlowVarSumVertex or flow::VarSumEdge/FlowVarSumEdge.

Specifier	Type	Extension	Description
inFlow	Vertex	Base	The input flow variables, \(x^-_v\).
outFlow	Vertex	Base	The output flow variables, \(x^+_v\).
isInUsed	Vertex	Base	Indicator variables for \(x^-_v > 0\).
isOutUsed	Vertex	Base	Indicator variables for \(x^+_v > 0\).
isInLessOut	Vertex	Base	Indicator variables for \(x^-_v < x^+_v\).
isInGreaterOut	Vertex	Base	Indicator variables for \(x^-_v > x^+_v\).
isInOutZero	Vertex	Base	Indicator variables for \(x^-_v = x^+_v = 0\).
vertex	Vertex	Base	Variables for flow through each vertex, \(\sum_{v\in V}\sum_{e\in \delta^-(v)} x_e\). The indexed variable specifier retrieves the inner sum for some \(v\).
isVertexUsed	Vertex	Base	Indicator variables for \(\sum_{e\in \delta^-(v)} x_e\).
transitInternal	Vertex	Base	Variables for flow through each vertex, which enters and exits the vertex from/to the networks. That is, excluding the flow that is covered by inFlow and that covered by outFlow. Note, the indexed specifier can only be used if the separation of the transit flow is possible in the underlying expanded derivation graph. The non-indexed specifier can only be used if all indexed specifiers exist. The necessary network expansion can be triggered by disabling IO flow reversal (flow::Flow::setAllowIOReversal()/Flow.allowIOReversal), or by explicitly calling flow::Flow::separateIOInternalTransit()/Flow.separateIOInternalTransit() to make the variable available for the relevant vertices.
isOverallAutocata	Vertex	Overall autocatalysis	Indicator variables for overall autocatalysis.
isOverallCata	Vertex	Overall catalysis	Indicator variables for overall catalysis.
edge	Edge	Base	The edge flow variables, \(x_e\), without the input/output edges.
isEdgeUsed	Edge	Base	Indicator variables for \(x_e > 0\) for the non-input/output edges.
isBothReverseUsed	Edge	Base	For each unique unordered pair of edges (not input/output) \(e_a = (S_a, T_a), e_b = (T_a, S_a)\) there is an indicator variable indicating \(x_{e_a} > 0 \wedge x_{e_b} > 0\). The indexed specifier can only be retrieved if the given dg::DG::HyperEdge/DGHyperEdge actually has an inverse. Two edges being each others inverse maps to the same indexed indicator variable.

9.2. Objective Function

The objective function is always minimised and can be changed to any linear expression. The default objective function is determined by the enabled extensions. Each extension will add some linear expression to the objective function if it is enabled.

Extension	Added Linear Expression
Base	edge \(+\) inFlow
Transit	\(0\)
Overall Autocatalysis	isOverallAutocata
Overall Catalysis	isOverallCata