Strictly Monotone Weight Functions on Directed Graphs

OUTLINE

Introduction
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. 

Introduction
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''.
General...