What does it mean that graph density indicates knowledge depth?

A densely connected area of your graph represents deep understanding.

Example: You have 40 notes about distributed systems. When you map the links between them, you find that almost every note connects to multiple others — consensus algorithms link to failure modes, which link to CAP theorem, which links to partition tolerance, which links back to consensus. The region is thick with edges. Then you look at your 40 notes about nutrition. Most sit in isolation — a few link to 'meal planning,' but the connections stop there. You don't need a quiz to know which subject you understand deeply and which you've merely collected facts about. The graph already told you.

Try this: Pick two subjects you know well and one you're just beginning to learn. For each, list 10 concepts from memory. Then draw the connections between them — every relationship you can articulate (causes, enables, contradicts, exemplifies, depends on). Count the edges. Calculate the density: edges divided by (nodes times (nodes minus 1) divided by 2). Compare the three numbers. The subject you understand most deeply will have the highest density, not because you tried to connect things, but because deep understanding is dense connection.