Weighted Directed Graph

A Weighted Directed Graph is a Directed Graph that has assigned weights to their edges.

• AKA: Directed Network, Edge-Weighted Graph.
• Example(s):
  • .
  • a Semantic Similarity Network (SNN),
  • ...
  • …
• Counter-Example(s):
  • Weighted Network,
  • Rooted Direct Graph,
  • Flow Network.
• See: Directed Acyclic Graph, Natural Transformation, Weighted Graph, Rooted Graph, Control-Flow Graph, Signal-Flow Graph, Flow Graph, State Diagram, Directed Multigraph, Finite State Machine, Commutative Diagram.

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References

2021a

• (Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Directed_graph#Digraphs_with_supplementary_properties Retrieved:2021-7-11.
  • Weighted directed graphs (also known as directed networks) are (simple) directed graphs with weights assigned to their arrows, similarly to weighted graphs (which are also known as undirected networks or weighted networks).
    • Flow networks are weighted directed graphs where two nodes are distinguished, a source and a sink.
  • Rooted directed graphs (also known as flow graphs) are digraphs in which a vertex has been distinguished as the root.
    • Control-flow graphs are rooted digraphs used in computer science as a representation of the paths that might be traversed through a program during its execution.
  • Signal-flow graphs are directed graphs in which nodes represent system variables and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes.
  • Flow graphs are digraphs associated with a set of linear algebraic or differential equations.
  • State diagrams are directed multigraphs that represent finite state machines.
  • Commutative diagrams are digraphs used in category theory, where the vertices represent (mathematical) objects and the arrows represent morphisms, with the property that all directed paths with the same start and endpoints lead to the same result by composition.
  • In the theory of Lie groups, a quiver Q is a directed graph serving as the domain of, and thus characterizing the shape of, a representation V defined as a functor, specifically an object of the functor category FinVct K F(Q) where F(Q) is the free category on Q consisting of paths in Q and FinVct K is the category of finite-dimensional vector spaces over a field K. Representations of a quiver label its vertices with vector spaces and its edges (and hence paths) compatibly with linear transformations between them, and transform via natural transformations.

2021b

• (Oxford Math, 2021) http://math.oxford.emory.edu/site/cs171/directedAndEdgeWeightedGraphs/ Retrieved:2021-7-11.
  • QUOTE: In other cases, it is more natural to associate with each connection some numerical “weight". Such a graph is called an edge-weighted graph. An example is shown below. Note, the weights involved may represent the lengths of the edges, but they need not always do so. As an example, when describing a neural network, some neurons are more strongly linked than others. If the vertices of the graph represent the individual neurons, and edges represent connections between pairs of neurons, than the weight of an edge might measure the strength of the connection between two associated neurons.

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