What is the name for types of games like Phage Wars?
Solution 1:
Typically this distinct game design element (played on a graph of nodes and edges) is referred to as a node-based game.
It's usually not seen as a distinct genre by itself, but tacked on as a qualifier to more distinctly describe a game.
For example, if your game is a node-based RTS, that means that the game takes place on a grid of nodes, can be controlled and evolves in real-time, and has some element of strategy such as unit types or economy.