Flow graph (mathematics)
A flow graph is a form of digraph associated with a set of linear algebraic or differential equations:
Although this definition uses the terms "signal flow graph" and "flow graph" interchangeably, the term "signal flow graph" is most often used to designate the Mason signal-flow graph, Mason being the originator of this terminology in his work on electrical networks. Likewise, some authors use the term "flow graph" to refer strictly to the Coates flow graph. According to Henley & Williams:
A designation "flow graph" that includes both the Mason graph and the Coates graph, and a variety of other forms of such graphs appears useful, and agrees with Abrahams and Coverley's and with Henley and Williams' approach.
A directed network is a particular type of flow graph. A network is a graph with real numbers associated with each of its edges, and if the graph is a digraph, the result is a directed network. A flow graph is more general in that the edges may be associated with gains, branch gains or transmittances, or even functions of the Laplace operator s, in which case they are called transfer functions.
There is a close relationship between graphs and matrices and between digraphs and matrices. "The algebraic theory of matrices can be brought to bear on graph theory to obtain results elegantly", and conversely, graph-theoretic approaches based upon flow graphs are used for the solution of linear algebraic equations.
An example of a flow graph connected to some starting equations is presented.
The set of equations should be consistent and linearly independent. An example of such a set is:
Consistency and independence of the equations in the set is established because the determinant of coefficients is non-zero, so a solution can be found using Cramer's rule.
Using the examples from the subsection Elements of signal flow graphs, we construct the graph In the figure, a signal-flow graph in this case. To check that the graph does represent the equations given, go to node x1. Look at the arrows incoming to this node (colored green for emphasis) and the weights attached to them. The equation for x1 is satisfied by equating it to the sum of the nodes attached to the incoming arrows multiplied by the weights attached to these arrows. Likewise, the red arrows and their weights provide the equation for x2, and the blue arrows for x3.
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