directed graph math

Strictly Monotone Weight Functions on Directed Graphs



General Definitions
When can any weight function be turned into a strictly monotone weight function?
Weighted Directed Graphs in Applications

Autonomous Driving
Quantum Walks
Artificial Neural Networks


SUMMARY: We define a directed graph as a space with any binary relation, and we define a strictly monotone weight function on any directed graph. We prove when no weight function on a directed graph can be turned into a strictly monotone weight function, and we give a characterization of any strictly monotone weight function. Finally, we mention the use of weighted directed graphs in autonomous driving research, quantum information, and deep learning. 


Here is an example of a directed graph: 

A directed graph

Other examples are the real numbers with the strict order $<$ and any collection of subsets of any set with the binary relation $\subseteq$ of ``being a subset of''.

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