I have come across this phrase often, and a quick Google search indicates that it is common:

$ y $ depends weakly on $ x $.

I understand what a dependency is. If $ y $ depends on $ x $, that means it is possible to deduce $ y $ from $ x $, given all other factors:

$$ y = f(x) $$

But what does " $ y $ depends weakly on $ x $ " mean?

For context, I came across this phrase in my study of parallel algorithms:

An algorithm is said to be highly scalable if the efficiency depends only weakly on the number of processors when the problem size and the number of processors increase by the same factor.

(I'm including this as an example only, I'm not asking for a discussion of scalability.)

  • $\begingroup$ I'm guessing that the answer is $ y = c \times log(x) $, but I'm not sure. $\endgroup$ – Flimm Jan 15 '13 at 19:45
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    $\begingroup$ There is no specific formula as this is a soft term. In your example, some problems are easy to share between processors while others require much communication between the processors and the efficiency will fall rapidly as the problem size and number of processors increase. $\endgroup$ – Ross Millikan Jan 15 '13 at 19:55
  • $\begingroup$ @RossMillikan: I had a chat with my professor and you're right, it's a soft term without a strict definition. I've added that answer since it's the one that satisfies me. $\endgroup$ – Flimm Jan 16 '13 at 12:11

It means that the absolute value of the derivative of $y$ with respect to $x$, i.e., $|dy/dx|$, is small compared to some scale value.


A more rigorous way of stating this is that $y$ depends weakly on $x$ over an interval $[a,b]$ when

$$\max_{x \in [a,b]} \left | \frac{dy}{dx} \right | < s $$

where $s$ is some scale value.

  • $\begingroup$ What if the derivative of $ y $ depends on $ x $, would that mean that $ y $ does not weakly depend on $ x $? $\endgroup$ – Flimm Jan 15 '13 at 20:00
  • $\begingroup$ No, my comment applies regardless; it was also worded loosely. I might say that the maximum value of the absolute value of the derivative over an interval is smaller than a scale value, than y weakly depends on x over that interval. $\endgroup$ – Ron Gordon Jan 15 '13 at 20:07

As Ross Millikan said, there is no specific formula as this is a soft term. The idea is that $ y $ will only change to a small degree as $ x $ changes.

For example, $ y $ could be said to depend weakly on $ x $ if $ y $ depends on $ \log x $ or $ \frac {1}{\log x} $ or $ \sqrt x $ or $ \frac {1}{\sqrt x} $.


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