To make the underlying math more explicit (if still handwavy), I see the thickness as the derivative of the parent with respect to the child; this is why we can multiply them together along a path (the chain rule). This perspective helps us see a few important things:
The thicknesses change over time and are based on marginal importance rather than absolute importance. Sometimes they change due to random external factors, but often due to your own actions—if Danslist now has a truly excellent network thanks to your efforts, improving it further may not be the most important thing anymore, even if having it is still the most important thing to the company’s success.
The thickness has units of something like [effect]/[work], where the unit of work is something like a person-hour. This means the thicknesses are based not just on importance, but on tractability; if transitioning to MySQL suddenly got 1000 times easier (resp. harder), the corresponding line is now 1000 times thicker (resp. thinner). In this example, even 1000 times thicker may not be enough to make the corresponding leaf relevant, but the general idea is important.
It’s not really a tree. Mental health is important for happiness directly. But it’s also important via many subgoals, since poor mental health can make working on other things more difficult. You have to sum all paths from the root to some node when considering its importance—a lot of the time, the tree is a good approximation (the gym is a separate realm from the office), but there are some things that are so important to everything that they demand this amendment.
The thickness has units of something like [effect]/[work]
I.e. presumably benefit/cost (work being a cost, whether financial or not), = the benefit-cost ratio (BCR) used in cost-benefit analysis in economics.
To make the underlying math more explicit (if still handwavy), I see the thickness as the derivative of the parent with respect to the child; this is why we can multiply them together along a path (the chain rule). This perspective helps us see a few important things:
The thicknesses change over time and are based on marginal importance rather than absolute importance. Sometimes they change due to random external factors, but often due to your own actions—if Danslist now has a truly excellent network thanks to your efforts, improving it further may not be the most important thing anymore, even if having it is still the most important thing to the company’s success.
The thickness has units of something like [effect]/[work], where the unit of work is something like a person-hour. This means the thicknesses are based not just on importance, but on tractability; if transitioning to MySQL suddenly got 1000 times easier (resp. harder), the corresponding line is now 1000 times thicker (resp. thinner). In this example, even 1000 times thicker may not be enough to make the corresponding leaf relevant, but the general idea is important.
It’s not really a tree. Mental health is important for happiness directly. But it’s also important via many subgoals, since poor mental health can make working on other things more difficult. You have to sum all paths from the root to some node when considering its importance—a lot of the time, the tree is a good approximation (the gym is a separate realm from the office), but there are some things that are so important to everything that they demand this amendment.
Your comment greatly adds to the value of the post for me, thanks!
I think this understanding of line thickness maps onto taut/slack constraints from linear optimization (also discussed by John Wentworth here).
I.e. presumably benefit/cost (work being a cost, whether financial or not), = the benefit-cost ratio (BCR) used in cost-benefit analysis in economics.